An Introduction to Teichmuller Spaces

euolerrug Euo;1Euo11 slrsd uopusl IroA /(eN Eraqleprell ullreg o,(1oa Eepan-reEulrdS suorlBrlsnflI€t qlla seredsrellnl...

Author: Yoichi Imayoshi | Masahiko Taniguchi

230 downloads 1373 Views 12MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

euolerrug Euo;1Euo11 slrsd uopusl IroA /(eN Eraqleprell ullreg o,(1oa Eepan-reEulrdS

suorlBrlsnflI€t qlla

seredsrellnluqclel o] uopcnporluluv rqrnEluel 'ntr . ilIsoferul 'A

Yorcut lvayosHr Departmentof Mathematics,college of GeneralEducation,osaka University,Toyonaka, Osaka560.Japan Ivfrrs.lHrxoTesrcucnr Departmentof Mathematics,Faculty of Science,Kyoto University, Sakyo-ku,Kyoto 606, Japan

ISBN 4-431-70088-9 Springer-Verlag Tokyo Berlin HeidelbergNew York ISBN 3-540-70088-9 Springer-VerlagBerlin HeidelbergNew York Tokyo ISBN 0-387-70088-9 Springer-VerlagNew York Berlin HeidelbergTokyo Tokyo1992 @ Springer-Verlag Printed in Hong Kong This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printing and binding: Best-set Typesetter, Ltd., Hong Kong

1661'lsnEny tqcn|tuol oqxqDspW tr1sofr,ou.tJ ?tplo1 '{ooq rno Jo uorlrpe qsrpug srql Sur,rerqceur l.roddns 1ryd1eqpue pur{ rterl} roJ o,t1o; Eepen-.re3ur.rdgo1 pue ,, 1rsra,rru1 o1o.{y ;o oua11rfuey JosseJordof s{u€rlt . 1reeq .rno ssa.rdxeo} a{I plnolrr a^\ ,f11eurg 'ldr.rcsnueuraq1 slred snorr€A ;o Sutpea.r,tq dleq qcntu sn perapuer oq.u ,ena.e3ruelftunr€H pue ,errreSngr4n,{rqso; 'o1ournsey41 olo{€tri{ 'eurrfoy rqso,epeg ,tlprcedse ,san3eellocpus spuarq II" ol apnlrler3 rno prorar osl€ eM .e)rlp" elqenp^ qcnur arle3pu" uorlrpe qsrl3ug eq1 ;o ldr.rcsnueu aql pear f,lpulrl e^\€{lo ore}ox pue epoJnx lqsepeJ, srossaJord 'pa3ueqcun sr aq1 yo .{poq ureur eq1 ,asaq1 lxel ueql Jeqlo 's1led pappe ol pat€lal fgerqc secueJaJarAreu Iereles apnlcur eM '?c"duror ,t1en1ce$ 'g Kpueddy ur palcnrlsuoc ,ece3rns uueureru lceduroc e;o acedsrTnpou pegrlreduroc eql leqt ,!rre,r o1 ^{oq e?errpur aal '.req1rng'(2 raldeq3 crrla(u uossreladlraM aq1 o1 qlrrr,raceds ?'S$) Jo lcadsar ,(1 raldeq3 rellnurqcral eql Jo sernl€Arnr 3o ,(1rnr1e3au Jo uorle^rrep € pue 3o 7'6$) .recueds pue €rr€poy ol enp se)eJrns uuetuarg Jo sarnlcnrls xalduoc eq1 ;o .{roaq1 uor}euroJep pue f.roaql rellmuqrral uaa^rleq uorlceuuoc ,(g .re1deq3 Jo t$) uap,tog Jo sueroeql pelerqele) ;o goord eql Jo qrle{s e epnl?ur osle eA\ '9 leldeq3 g$ ul suorl€ruroJsuerl J?Inpotu Jellntuqtrel uorl€curss"l? aq? uo Jo Jo ,troaql sreg pue uolsrnql eql pu€ ,g .ra1deq3Jo ?$ ul aceds .rallnuqcral aq? Jo ,s3urql .raq1oBuoury uorlecgrlcedruo? s(uolsJnqJJo uortcnrlsuoc eql pp€ e,ra, 'suorltasqns Jo suorlras Ieuorlrppe se scrdol '1xa1 Pal€ler ..{,aulera^es epnl)ur o1 pue leurSrro eq} ur sroJre roullu etuos lrerro:) o? uorlelsuerl srq? fq paproge flrunlroddo eqt Jo e3etue^p€ ua{e} o^€q ed\ 'asauedel ur uellrrltl acedsrallntuq)ref uo rlooq frolcnpoJlur Jno Jo uorl€lsue.r1qsq8ug eql sr srqJ

uoll.rPfl rlsrltug aq+ ol arBJard

snuaS;o saf,eJJnsuueluaru pasolo palueseJdaleH 'sJelaue.red xalduroc I - fg ,tq pezrr?eu"r€d rl (Z ? 6) 6W l€rlt pelrass€ uueuarg 'Lg8l uI 'aue1dxalduroc eql rlll^{ peul}uapl sl W leq} s^roqsselJnc crldqa pue suoll?unJ ot1dt11e;ofroeql eql '.r(11ecrsse1c u^rou{ IIe^{ $ sy '1utod auo Jo slslsuo) 0W ,(lsnonqo 'araqds uuetualg aql o? tual€^rnbe flleorqdrotuoloqlq sr oJez snua3 yo a?tsJrnsuu€ruerg pesol) ,trerraacurg '6 snua8 Jo saceJrnsuueurelg pasolc Jo sasselr ecuale,rrnbeclqdrou -oloqlq IIeJo las eql'q t€rlt'f snua3;o ecedsqnpour s(uusrualg aq.6W p"I 'se)"Jrns uuetueru (uollnlo^a sll Surceqtr pasolr ro; flesolc erou uelqo.rd srql aurrrrexesn lel 'uralqord lnpour s(uueruerg pell€? sI sIqI 'ploJrusru alqerluereJrp l"uorsueulp-o^\? peluerJo ua^rS e uo peuSrsseeq plnoc seJn+rnrls xalduroc lrurlsrp fueur ilroq ol se saslJ€ d11ern1euuorlsanb eqa 'luele -arnba fllecrqd.rouroloqlq fltressacau lou ere feql'seceg.rnsuuetuelg o/r{}uae^t}eq usrqd.rouoagrp Surrrrasard-uorl"luerroue sr ererllJr ue g'ploJrueur alqetlua.ragrp peluerro e se pap.re3al osle eq ue) er€Jrns uueruarg y leuorsuerurp-o^rl I€er 'lual€^Inbe flecrqd.rouroloqq are faql JI arnl)nr?s xeldluo? eur€s eql a^eq zU pue rU teqt PIes sI ?I'zU otuo Ig uror; Surddeur ctqd.rouroloqrq e slsrxe ereql Jr lualerr,rnbe,tllecrqdrotuoloqlq erc zA pu€ IU sec€Jrnsuu€urelg o^rl 'ploJrueur xalduroc leuorsuaurp-euo pel)auuo) e $ af,sJrns uuetualg V 'sacedsrallnurlf,rel Jo aruelsqns eqt dseJ3 rapeer eq1 dlaq sueld eseql t"q1 edoq e11 'spoqleur ?rlsrrneq pue saldurexe 1ecrd,t1q8norql slcelqns 'suor?e^r?oruall1ufl.rd pue slcedse freluaurala lsour eql tuoq lueserd osle eM urSaq a7y1'alqrssod s€ paurcluoc-Jless€ {ooq eq} e{eu ol palduralle e eq elA 'f.roaq13ur.r1saq1 uI selor 'f11uacag 'droeql luelrodrur Surdeld un3aq seq sacedsrallnurqrre;,;o f.roeql aq1 crpo3re pue 'sctueuf,p xeldruoc 'suorlenbe l€IluaraJlp 'suorsuauttpeeJql pue o^rl ur d3o1odo1'frlauroaE Fllualeltp 'frlauroe3 cre.rqaSle'srsdleue xelduroc 'sunog: crqd.rourolne'sdnor3 et1 'sdnor3 uelulely 'sdno.rEuelsqt\{ 'splogrueurxelduroc uorlcesJelul eql uI saII s?al€ luelrodun ,tueur apnlcul asaql 's?rl€lueqletu Jo ;o 'eceJrns ua,rr3 e uo seJn]f,nJlsxalduoc eq] 1cafqns srqtr IIe ;o uorlezrrlaurered e a,rr8 qrrqrrr sareds Jallnruq)ra; ;o froaql aql luasard arrr '4ooq $qt uI 'f.rlatuoe3 cre.rqeSle'sace;rns urr€urerU ,,{roaq}aq} '.{rlaruoa3 Jo Jo l€rluareJrp pue seeJ€aql ur elq€reprsuocuaeq seq ssarSo.rdpue 'suotle3llselur elrsualxe perrnds seq uralqord srqtr'uuerueql pr€qureg 'J 'C qfl^r peleurSrro ecsJrnsas€q pexg e uo soJnlcnrls xalduoc Jo suorl€rre^ eql ezrrlaue.red o1 1lrotlJo uralqord aq;

arBJard

vlll

Preface

g as finite branched covering surfaces of the Riemann sphere, and determined the number of parameters of Mo by the number of degrees of freedom of the branch points. In this book, we treat moduli spaces through Teichmiiller spaces and Teichmiiller modular groups as follows. Let R be a closed Riemann surface of genus g, and let X be a marking on ft, i.e., a canonical system of generators of a fundamental group of .R. Two pairs (R,D) and (B', D') arc defined to be equivalent if there exists a biholomorphic mapping f : R--- -R'such that /.(X) is equivalent to Dt. Denote by [E,X] the equivalence class of (R,E). Such an equivalence class [R, I] is called a ma"rked closed Riemann surface of genus g. The Teichmiiller space ?o of genus g consists of all marked closed Riemann surfaces of genus g. It is verified that ?, has a canonical complex manifold structure, and it is a branched covering manifold of the moduli spaceMn.Its covering transformation group is called the Teichmiiller modular group Modo which corresponds to the change of markings. It turns out that Mn is identified with the quotient space TofModr, which has a normal complex analytic space structure. The Teichmiiller space4 h* appeared implicitly in the continuity arguments of Felix Klein and Henri Poincar6, who studied Fuchsian groups and automorphic functions from the 1880s.Robert Fricke, Werner Fenchel and Jakob Nielsen constructed Tc k 2 2) as a real (69 - 6)-dimensional manifold. Fricke also asserted that ?, is a cell. Their method was based on the uniformization theorem of Riemann surfaces due to Klein, Poincar6, and Paul Koebe: every closed Riemann surface of genus S (> 2) is identified with the quotient space H f I of the upper half-plane .I/ by a Fuchsian group f which is isomorphic to a fundamental group of .R. Then each point [R, I] in ?, corresponds to a canonical system of generators of l- . Hence we see that [.R,X] is representedby a point in R6g-0 which is called the Fricke coordinates of lR,t). Moreover, the Poincar6 metric on f1 induces the hyperbolic metric on .R, and the conformal structure defined by this hyperbolic metric corresponds to the complex structure of .R. One of Oswald Teichmiiller's great contributions to the moduli problem was to recognize that it becomes more accessibleif we consider not only conformal mappings but also quasiconformal mappings. A quasiconformal mapping means a homeomorphism which satisfies the Beltrami equatiotr ut7 = pu". A Beltrami coefficient p measures the magnitude of deformation of a complex structure or a conformal structure. Around 1940 Teichmiiller discoveredan intimate relation between extremal quasiconformal mappings and holomorphic quadratic differentials, and asserted thatTn is homeomorphic to R6g-0. He also introduced the Teichmiiller distance o\ Ts. In the end of the 1950s, Lars V. Ahlfors and Lipman Bers developed the fundamentals of the theory of Teichmiiller spaces,and they gave rigorous proofs for Teichmiiller's results. They also showed that To @ 2 Z) has a natural complex structure of dimension 39 - 3, and can be embedded in A2(R) as a bounded domain, where ,42(R) is the space of holomorphic quadratic differentials of a closed Riemann surface E of genus g. From the Riemann-Roch theorem, it is

ur Surddeu leruJoJuof,lssnbIeuerlxa eql t"ql sI €apl s(rellnruq?IeJ Jo e?uassa 'asues ulel er{J 'u uo l?r}uereJrP tlterpenb rrqdrouoloq e tuou Peul"}qo Pu? -Jec ur ((eug€ f11eco1,,sl q)IqA\ Eutddeur l"ruJoJuosls€nbe sueatu g'3o Surddeur rallnuqclatr y 'sgurddetu leruroJuof,IsenbSulsn fq palcnrlsuoc sr (6 Z) f snuaS aql 'g raldeq3 u1 Jo U ec€Jrnsuueruorll pesol?€ Jo (U)J eceds .rellntuqcletr, 'sluarclgeoc nuerllag uo suollnlos;o acuepuadaprtqdrouoloq eql pue 'uorlenba ''e'l (sreg rurerlleg ue.rlr8e 8ur.{;sr1esSurddeu leuroJuorlssnb e;o eruelslxe eql orrrl a,rord el!\ uaqtr 'poqleur ctrlauroeS pu€ sJoJIqv ol anP sueroaql lelueusPunJ '11e;o ,sSurddeur e pu€ seJnpaco.rdcr1f1eu" o^rl Eursn leurro;uoarsenbeugap eira 's3urddeu 1s.ug leulo;uocls€nb Jo sltsluaulsPunJol pelo^ep sr 7 .ra1deq3 'tgyo aldunsJo suoll?unJ pesol) scrsepoa3 eql euluralap slutod 's?Isapoa8asaql 3uo1e qlSual crsapoa31€qA\Jo uralqord eql q fpnls oqe eM srelaruersd lsrlrl pu€ s)Isepoe3pesol) eldurls I - rt Jo suolllunJ ql3uel llsapoa3 'scrsapoa3 ,tq peugap ere 6a uo saleurpJoor ueslelN{eq)ual eql uaq;, Pasolc aldurrs ,{q slued Jo srred G - 67,Jo }es e o}ur saceJrnsuu"ruaru esodurocapo1 sr f.rlauroa3 cqoq.red.'tq eql 'f,Irletu elstulod aq? ,tq poqleu Pacnpul leluauepunJ slsaP I raldeq3 ;o 1u1odr*ar,raql tuo.r; (Z 7 0) fu ;o uorlcnr}suot aql q}l^{ (suol?eurroJsuerl 'sdnorE uelsqcnJ snlqgl tr Pu€ '1ce; 'sace3:rnsuueurarg Jo uraroaql uotlszltuJoJ-Iuneql ,{gar.rqureldxa oqe ar* '.1'3o srole.reua3 srrll /r{oqso} rapro uI 'tJr qlyl\ t;;o uorlecgtluePl eq} sarrr3srq; d snuaSJo [3'U] q ;o surals.{sl€?nrou€?Surpuodsar.ro,eq} qll^{ pagl}uePl (A -) '.re1nct1redu1 '.7 dno.r3 uelsqlnd e dq H aceJJnsuu€uralg pesoll pa{r€u q)"a ,P aueld-;1eqraddn eql Io J lH aceds luatlonb e ,tq paluasa.rdaru sed,t1^ aJ roJ 6J sluaserdar ldecxe a?eJrnsuueluaql f.rarra1eq1^roqs aA\.g-rg?I Jo lesqns e se '6 reldeq3 u1 qclqa (U ?) 6 snua3 3o tg aceds e{rlrJ eI{} 1?nrtsuoa em 'sclrle(u ueluueruelu 'sSurddeur fq pacnpur selnlf,nlls leluJoJuot Jo Pue leuloJuoclsenb Jo slulod/rrell or{} uro{ tg ,(pn1s osle a71\'6 snueS Jo sec€JJnsuueurelg Pasol) Pe{relu Jo }as 'sac€ds eseqt eql s-e,' snuaSJo t; aceds rellnuqf,ral eql lcnrlsuoc at',r Jo lepour eql Surluesa.rdregy'1 snua3Jo sec€Jrnsuuetualg PesolcJo aceds rellmuq?Iel 's.raldeqo 8uu'ro1o; eql roJ '11e aql pue aceds rlnpou aql lpcar ar* Jo lsrlJ spunor3lceq pue suolls^llotu a^Illtulrd ar'r3 ol q I .ra1deq33o esod.lnd eqa 'peqlrtsep ,{garrq ^lou $ {ooq slql ;o raldeqr q)"a Jo lua}uol aqJ, 'sdrqsuorlelar rreql f;rre1c ol pue ,lrl,arrr slurod snorJ€Aaseql tuorJ seceds rellntuqole1, e?sSllselul ol sI selPnls 3.o aseql Jo ecuasseeqI 'reqlo qlee o1 Palelal(1eso1c er" suoll€ulloJaP aseqtr .sSurddeur leuroJuorrs€nb fq pacnpur salnlcnrls xalduroc Jo suorleuroJap (gl) pue 'sctllaru crloq.redfq fq pacnpur selnltnrls IstuJoJuo?Jo suorleuroJeP (I) sdno.r8uelsqcnJ Jo suolleuroJaP (l) :pasn era!\ sac€Jrns uu€uralu 'setpnls asaq} uI Jo suorl€rrrroJapJo suorldrr?saP aarql 3ul^rolloJ el{l }s€al }e 'r; ;o sursrqdrourolne crqd.roruoloqrq ;o dno.r3 alercslP e eq o1 lno surnl 6a uo 0poy1 dnorS relnpou rallntuq)Ietr, aql Jo uollc" aql 'alour.reqlrng 'eceds rollal ultou{ l€uolsueutP-(g 0S) xelduroc € sl (U)zy leql XI

af,"Jard

Preface

those which determine a given point of "(E) is a Teichmiiller mapping. Then it turns out that Q k > 2) is homeomorphic to the space Ar(F) _ofholomorphic quadratic differentials on .R. Hence, ?s is homeomorphic to R6c-0. We also show that "(.R) is complete with respect to the Teichmiiller distance. In Chapter 6, using the Schwarzian derivative, we construct the Bers embedding of "(R) into a bounded domain in ,42(.R.), the space of holomorphic quadratic differentials on ft*. Here, E* denotes the mirror image of .R. By the Riemann-Roch theorem, Az(R-) is also identified with the (3g - 3)-dimensional complex Euclidean space C3r-3. Using this embedding, we see that "(ft) has a natural complex manifold structure of dimension 3c - 3. It is also proved that the Teichmiiller modular group M odo is a discrete group of biholomorphic automorphisms of ?r, and acts properly discontinuously on "0. This shows that the moduli space Mo =Ts/Modc has a normal complex analytic space structure of dimension 3C - 3. Chapter 7 treats the Weil-Petersson metric on 4. The holomorphic tangent space of To at a point [.R,X] is identified with the dual space of ,42(R). Then the Petersson scalar product on.42(R) induces the Weil-Peterssonmetric on ?n' We give two proofs for the fundamental fact that the Weil-Petersson metric is Kihlerian. Both of them a.redue to Ahlfors. In Chapter 8, we establish a beautiful formula due to S. Wolpert, which states that the Weil-Petersson Kihler form on 4 h* a simple representation with respect to Fenchel-Nielsencoordinates. We also give two appendixes. Appendix A deals with Schiffer's interior variation from the viewpoint of quasiconformal mappings. We explain Ahlfors' construction of the complex structure for Ts, which was the first construction of its natural complex structure. We also discuss variations with respect to degenerations of Riemann surfaces. In Appendix B, we explain briefly the compactification of moduli spaces. At the end of each chapter, there are bibliographical notes of books and articles to which we referred in the text. The bibliography is not complete. There is a vast literature relating to the theory of Teichmiiller spaces.We hope that this list helps the reader to begin to explore these researchpapers. Any omissions of references,or failure to attribute theorems, reflects only our ignorance. The authors are extremely grateful to Professor Osamu Takenouchi who recommended that we write this book. They also gratefully acknowledge the generous contributions of our friends and colleagues Makoto Masumoto, Hiromi Ohtake, Hiroshige Shiga, a^ndToshiyuki Sugawa, who read the original manuscript, and made many helpful mathematical suggestionsand improvements. Yoichi Imayoshi Masohiko Taniguchi October, 1989

LII III q0I Z6 LL , t

9L

u

L9 69 I9 I9

seloN ureroeql pumuEf2-ugrePleC Jo Joord V'V sluel?SeoC turerlleg uo ecuapuadeq t'V s3urddel4l leuroJuocls?n$ uo $uaroeql atuelslxg Z'V sarlrado.r4 frelueurelg pIrB suolllugeg I'V sturddel4l

luturo;uocrsen$ 7 raldeq3 seloN

sa?BurprooC

uotlecgrlceduoC s(uolsrnr{I t'8 Sutppaqurg ulaly-e{?q{ 8'8 sel€ulProoc uaEalN{erlcued 7,'t frlauroag cqoq.raddg Pue clrlel{ eJe?ulod I'8 cqoqJad'{g pue f-rlauroag uaslarN-Iaqcual

g .ra1deq9

seloN acedgerlct.rg 9'Z Eapow uslsqrnJ v'z suolleIuroJslle{l snlqg'Itr t'Z sSutrer\o3Iesra^Iuo z'z ueroar{J uoll€zrurroJlun I'c acedg eryltr

09 LV 8t t8 LZ 9Z 96

6 .ra1deq3

vz OG 9I vl 8 I I

seloN sernlcnrls l"urroJuoC Pue sernlcn.llS xalduro3 9'I acedg .relpurqcleJ Pue s3urddel4l pruro;uocrsen$ V'l f snueS;o acedS re1nwqcleJ, 8'I 4'I I snueg;o acedg rellnuqttal seceJJnsuu€tualu I'I aaedg Jallnurqcral f snuag;o

1 raldeqS

slua+uoc

Contents

Chapter 5 Teichmffller

Spaces

5.1 Analytic Construction of Teichmiiller Spaces 5.2 Teichmiiller Mappings and Teichmiiller's Theorerms 5.3 Proof of Teichmiiller's UniquenessTheorem Notes Chapter

Chapter

Analytic Theory of Teichmiiller Spaces Bers'Embedding Invariance of Complex Structure of Teichmiiller Space Teichmiiller Modular Groups Royden's Theorems Classification of Teichmiiller Modular Transformations Notes

135 144

146 r47 r52 r62 r67 'l'71

179

7

Weil-Petersson

Metric

7.I Petersson Scalar Product and Bergman Projection 7.2 Infinitesimal Theory of Teichmiiller Spaces 7.3 Weil-Petersson I\{etric Notes Chapter

r27

6

Complex 6.1 6.2 6.3 6.4 6.5

119 119

t82 i83 189 loo

2t7

8

Fenchel-Nielsen

Deformations and Weil-Petersson Metric 8.1 Fenchel-NielsenDeformations 8.2 A Variational Formula for Geodesic Length Functions 8.3 Wolpert's Formula Notes

219 219 224 226 232

Appendices A B

Classical Variations on Riemann Surfaces Notes Compactification of the Moduli Space Notes

233 243 244 253

References

254

List of Symbols

271

Index

274

secBdsrellntuqrlel ol

uopcnporluluv

'eJnlcnJls xalduroc leuolsuetulp-auo qlrm ecedsJJopsneH pelceuuoc € srreeruplottuout xe\ilu.tocr)uorsueurp-euo to 'U uo emlxnJts xalduoc e saugeP }! }€tl} acn{.tnsuuoul?rg f)uoNsueurtP'?uo v s 91r3f{ (!z'!2) } te{t {es all pu€'U uospooqroqUfnu aToutp.tooc {o u.ta1sfi,e 'ursrqdrouroauroqcrqdrouroloq e ''a't 'Surddeur crqdrouroloqlq e sI (r2U !2)tz *

('VlU h)lz : rJzotz - lrz Burddeur uo-rtrsuertaql 'g * qnu rn U (IlI) 'aue1d

xalduroe aq1 u1 f6, ?asqnsuado ue oluo f4;o ursrqdrouroeuoq e s lz ,treag (rr) 'g;o 'fpt>ln lesqns uedo ue sr f4 fra.tg (r) U pue aq1 3ut,$st :(t'f 'El.l aas) suorlrpuoc eerql 3uu'ro11oy -les rrf{ (fz'fg) eq A P1 'slrolloJ i ,(tf.ttl e ql,ta ecedsgropsn€H Pelceuuoc e 'sSurddeu crqdrotuoloqlq fq s'p sr ef,"Jrns uueruelg Jo uoIlIuSeP PrsPuels eq; (fla^lqn?ul peqcled are qf,rqa{ aueld xelduro? aql uI suleruoP Jo uoll?allo3 3 sl }I 'acnttns uuounty e pell€c sI ploJlueur xaldruoc Palceuuo? I€uorsuaulP-auo Y sacBJrns uuBrrrarlr Jo uol+Ilrsac

'I'T'I

s a J e J J n su u E r r r a r u ' I ' T 'aceJJnspeluelJo u€ uo crJlaru usruusrueru e fq pecnpur ernlrnrls leuroJuo) eql Pue alnlcnrls 's3urddeur xelduroc eq? uaealeq drqsuorlela.raql sl€ert g uollces leuro;uoctsenb r; ,(pn1s am 'p uorlcas uI 't snue3 ;o seceJrns uuetuelg ;o lulodarall eql urorJ pesolc pe{r?tu IIe Jo tes eql se I uotlcas uI PaugaP st t5 eceds renntuq)Ial eqJ 'I 'd t; snuaS;o eql aceds reilnurqttal frerltqre snuaS Jo lePoul e sa,il3 qllq/tr ;o 'salduexa aceds rellnruqcleJ ar{} pue eeeds Inpour aq} q1^{ sl€eP A uol}?es lecr '1sr1g -df1 auros pue sac€Jrnsuueruarll uoltrugap aq1 errrSar* '1 uorlcag ut Jo 'fqc1e4s 'pattluo eq II€I{s etuos Pu€ Jeql€r are s;oord aurog 'sraldeqc 3urno11o;eq? roJ spunorSlceq puts suotls^Ilou 'raldeqc s-rql uI arlr3 pue '6 snua3 ;o t; aceds rellnurqclal aql lcnrlsuoc aal

f snuag Jo aJBdS JaIInurIr.raI I ra+dBtlc

1. TeichmfillerSpaceof Genusg

z a- Plane

ar-plane

Fig.1.1. A coordinate neighborhood(U, z) of a Riemann surface .R is a pair of an open set [/ in ,R and a homeomorphism z of U into the complex plane such that for any element (Ui, ri) of a system of coordinate neighborhoods with U nU1 I $, the mapping zoziL: zi(U nU)--- z(U nUi) is biholomorphic. This [/ is also called a coonlinale neighborhoodof r?. Such a homeomorphism z is said to be a local coordinale ot a local parameter on U of R. A coordinate neighborhood ([/, z) with p e U is called a coorilinate neighborhood around p, and z is called a local coorilinateor local parameter arounil p. Local analysis on a Riemann surface ,R is reduced to analysis on domains in the complex plane via local parameters. For example, a holomorphic funclion on ,R is a function / on l? such that f oz-L is holomorphic on z(U) for any coordinate neighborhood (U,z) of ft. A mapping f of R into a Riemann surface,9 is said to be a holomorphic mapping if wof oz-r is holomorphic for all coordinate neighborhoods (U, z) of R and (V, u) of S with /(U) C V. A biholomorphic mapping f : R --- S means a holomorphic mapping f of Ronto,S which has the -1 : S - ft. Two Riemann surfaces l? a"nd S are holomorphic inverse mapping f biholomorphically equiualenlif there exists a biholomorphic mapping between .R and S. In this case, we regard ,? and ^Sas the same Riemann surface and write R = S. We say also that R and S have the same compler slruclure. Complex structures, biholomorphic mappings, and biholomorphic equivalencemay be and are actually often said to be confortnal straclures, conformal mappings, and conformal equiaalence, respectively(see$1.5). Remark. A Riemann surface is a two.dimensional real-analytic manifold, and the Cauchy-Riemann equation implies that local coordinates determine its orienta-

crqdroruoloqlq e sl U r- C : Jr leql ferrr e qcns uI araqds r } = ? e r e q a ' I - C - O u l " u r o pe q } o t u o { 0 > r r r , - I = H aueld-}1eq I C > ^l = *H aurld-;1eqra/(ol eql pus {0 < rnurl I C > t} raddn aq1 qloq fllecrqdrotuoloqrq sderu .rn - z uollcunJ ctqdrouroloq aq;,

'z'T'ttd eueld-or

aueld-z

'euo JeuJoJ eqt al"ts er* 'ala11'uollsnulluoc ct1f,1euefq ro Pue ln)r' ,,e1sed - (n)l - z 'dleotsselC 'panle^-e13utssr "n Jo Poqlau aql fq Pelrnrlsuoc sr 1r uorlcunJ crqdrouroloq eqlJo uorl)unJ asrelur aql qllqa uo e?"Jrnsuueruelg aql sI (lxaN slql '4 - rn uorlrunJ cre.rqe3peql Jo eeeJrnsuuetuelll at{} aes sn lal 'Qlt'{o} p"n (z'3) spooqroqq3raueleurProo?o'n1fq Peusep ?) sr C uo eJnlcnJls xaldurbc V'eceJJns uueurelg e osl€ sI'3 aueld xelduroc aql '{-} go iorlecgrlcedtuoc lutod auo eql q qc$Iir{ n C = ? ataqds uuvu'ery aq; '(r'O) eleulproo, auo fluo fq uarrrSsr O uo ernlcnrls xalduroc e pooqroqq3rau 'flaurep 'e?€Jrnsuueruolg e st aueld-z xalduroa aqt ul O ul€ruoP ,traaa'1p;o 1sltg sa"BJrnS uueurarlr 3o salduruxg

'Z'T'I

'[86-Y] '[OO-V] re3ut.rdg 1e3ar5'[37-y] ueur.re3uts Pus '[ga-V] €rN pue se{re.{ '[ZZ-V] ,tqoC pue seuof '[Ol-V] Suruung '[ag-V] ratsrog '[gt-V] s.reg '[g-y] oIr€S pue sroJIqV 'ecuelsut roJ (llnsuol 'sace;tns uuerualg 'pa1e1n3u€Ir}aq u"c pue slas uado;o ;o ,{roeq1 1e.raue8eql pue s}teJ esaq} .rog 'uoll€luelro sr$q elqelunot € seq e?eJrns uueuralg ,t.ra,ta1eq1 u^{onl-llaa s-ItI '.rageara11'uor1 sr eceJrns uusruarg € letll etunsse airr slql qtla paddrnba s,tea,r1e saf,"Jrnsuu?uraru'I'I

1. Teichmiiller Space of Genus g

mapping. This R is the Riemann surface of w = t/7. (See Ahlfors [A-4]' Chap. 8; Jones and Singerman [A-48], Chap. 4; and Springer [A-99], Chap. 1.) 1/7 is also Note that the Riemann surface R of the algebraic function w = z. u,2 equation by the defined regarded as the algebraic curve Finally, we seeelliptic curves, i.e., tori from the viewpoint of algebraic curves. For any complex number ) (# 0, 1), Iet .R be the algebraic curve defined by the equation w2=z(z-1)(z-.\).

(1.1)

In other words, .R consists of all points (z,w) e C x C satisfying algebraic equation (1.1) and the point p- = (oo, oo). We can define the complex structure of ,? by the complex structure of the z-sphere so that the projection r: E e, r(z,w) = z, is holomorphic. This r? is a two-sheeted branched covering surface over the z-sphere with branch points 0, 1, I' and oo. The mapping written as u, = f : R - e, fQ,u) = w, is holomorphic. This function / is and R is a Riemann surface on which the algebraic function \rc=W]

u - {z(z _tG

-,

is single-valued.

The Riemann surface ,R defined by algebraic equation (1.1) is rega.rdedtopologically as a surface illustrated in Fig. 1.5. Take two copies of the Riemann ,ph".", St, Sz with cuts between 0 and 1, and between,\ and m (fig' 1'3)' place them face to face (Fig. 1.4), and join along their cuts (Fig. 1.5). The resulting surface is homeomorphic to the Riemann surface -R. Hence, R looks like the surface of a doughnut. We call such a Riemann surface a torus. A torus is also called an elliptic curue; Lhis name comes from the elliptic integral (see $1.4).

Fig.1.3.

t=f{ !g'!V} lo srolor?u?6lo an1sfrsf)?ruouoc n

ro t=r1VAl'llV)}

'(od'a)rv 1ec e16

r=f

'(rlunaqr) r = r-[rB]r-tfvltrsltrvlL[ 6

uoll€ler l€lueu"PunJ eql sogsll€s Ptre 6g t6, t"''rg Iy uro.r;pecnpul luql'luVJ ' "' ' llgr] ' lly] sess€lcfdolouroq eq1 fq pale.rauaEsr od lurod eseq q?lar U 1o (od'g1ttt dno.r3 leluaurePunJ eqtr '(f't ;St.f) seprs dy ql1,u uoE,tlod xaruoc e o1 crqdroruoeuoq ul€tuoP e 1aBar* ' " ' 'r€l(I}t selrnf, uerll 'g'I 'EU ul se od lutod as€q I{lI^r ug'6V Pasolcaldurts od e4e; e lurod 3uo1e g, lnt pue d snuat Jo U ac"Jrns uueuell{ Pasolc e uo 'acottns uuourery uado ve Pe11eo y 'snua! allug Jo ac"Jlns uuetuerll pesolc € sI st a?"Jrns uueruarg lceduroc-uou u^rou{-lle^{ sl U'I snuaS;o $ snro} e Pue /tJa^a a?€Jrnsuu€r.uaru l€qt lceduroc '6 snuaE;o sr araqds uuerualg aql 'f snua6 to ecottns uuDut?NAPesop e PelIe? sl g'I '3lJ uI sB solPu"q d qtl,u alaqds e o1 crqdrouroeuoq ec€Jrnsuueuary Y

sa?BJrns rruBtuaru pasolc '8'I'I

'e'r'tt.r

'r'r'ttJ

sef,"Jrns uu"uall|I'I

l. Teichmiiller Space of Genus g

Fig. 1.6. (g = 3)

Fig. 1.7. (g : 3)

L.L.4. Lattice

Group

Representations

of Tori

We shall represent a torus as the quotierrt space C lf of the complex plane C by is a single-valuedmeromorphic a lattice group l-. since ur(z) = fiG4Q=U we can consider the complex (1.1), by equation defined function on the torus R p = (z(p),u''(p)) on 'R, th9 point any For paths on ft' along integral of Ilw(z) of algebraic function u(z) branch a selecting by defined elliptic integral @(p) is setting by and and a path from oo to z(p),

o(d=

f2lP)

J*

dz ,/r(t-l)(z-))

(1.2)

saprs Eurr(;rluapr fq peul"lqo aceJJnse ss Pazllear sl J/C eceds luarlonb sq; '[z] sasselc ecuele,rrnba lle Jo st$suoc.7 ,{q C P JIC eceds luarlonb aq;

'e'r'ttJ

'z fq paluasardar sselc acuale,rrnbeaqf [z] fq alouaq '(z)L= / {lla J ) L lueluele ue slslxe eral{}JI J raPun Tualoar,nba fes e6'a1ozau*rvut s l u r o do. / $ 1 Iz =(z)L uollelslr"rl€ are I 1 " t 1 1 C , z ' z '9 dnorS ursrqd qlr^r pagtuapl sl J ) qlu+rvrn, =,L fra,ra'1ce; u1 3o (9)ry -Jouroln€ arlrtpue aqt JoJ dnorS e?I11"1Y dnodqns 3 se papreSar sl ,7 u Jo 'U 'U roJ ilno.r,6acqyoye PIag raqunu I€er J qcns II€c alA fpeaurl ere faql '0 < (z)Llr)L)u1 ,$sr1eszv pue rl spollad aql ra^o luepuadaput '; aq? ecqs 'E, 1o pouad e Pall€? sI J Jo luauale fraag Jo slueuela dq raqlo qf,ee tuo+ rasrp qlrqAr senlel fueur .{1a1tugulseq (d)P uotlcun; eql t€q} ees a/tr '{Z>u'*l'ou*rYut}

= J 3ur11a5'flarrtlcadsar

,k__4$_4 - f zp

or ,l

z=iv

(u- z)(r- z)zf pue -rp

of ,l

,=tt

fq paluasarder are g'I '3rg ur Ig 'd pue -d Eurutof q1ed e uo (Iy salrn? pesol? eldtuts aq1 Suop O Jo sanle eq;, spuedep 11 '{lanbrun peururelap lou s-IO 1e.r3a1uratldqle slq} Jo enp^ eqtr 'U uo ea.r3aP;o I

g__,la_at zp l€IlueraSlp crqdrouroloq aq1 auII € s€ (6'1) pleSal o1 alenbape arour sr II'tlroureg 3o'q1ed e 3uo1e'1er3a1ur

-ralep sr (z)np

'1er3a1ur ql"d aq1 3uo1e uoll"nulluot cltfleue dq paunu Jo enle aql pue 9.1'31g ur -d lurod aq? o? spuodsarroe oo eJal{^r se)"Jrnsur"urerll'I'I

l. Teichmriller SPaceof Genus I

8

,4 with A' and B with 8' in the lattice of Fig. 1.8 by the translations 7r1,tr2, respectively. Now, we define a complex structure of C / f . Let r : C + C / f be the projection, i.e., "(r) - lzl fot z € C. Introduce the quotient topology on C/f , which is open if the inverse image r-r(Lr) is is defined as follows: asubset U otC/f open in C. It is verified that C/f is a connected topological space. For any two points [o],[6] € Cf l,we can take neighborhoods7o,V6 of a,b with r(I/") n r(%) - {. Since z is an open mapping, this shows that C/iis a Hausdorffspace. Moreover, for any point [c] e C/f , taking a sufficiently small neighborhood vo of a, we see that n gives a homeomorphism of v" into C/f . Let Uo = r(Vo) and zo: (Jo - Vo be a homeomorphism with zo(lzl) = z. Then (t/",2o) gives a coordinate neighborhood around lalin C/f - Thus C/f becomes a torus, i.e., a closed Riemann surface of genus I such that the projecgives an example of tion zr: C --- Clf is holomorphic. The triple (C,r,C/f) universal coverings, considered in $2.1 of Chapter 2. As is known in the theory of elliptic functions, the mapping [@]: r? '--' C lf sending a point p e R to a point [O(p)] e C/l- is biholomorphic. Hence we see that a torus defined by equation (1.1) is representedby a Riemann surface c/lr for a lattice group l-. In Chapter 2, we shall show that every torus is represented by a lattice group l- in c (see the corollary to Theorem 2.13). conversely, it is known that such a Riemann surface C /f is always biholomorphic to an elliptic curve defined by algebraic equation (1.1). For details, we refer to Ahlfors [A-4], Chap.?; Clemens [A-21], Chap. 2; Jones and Singerman [A-48]' Chap' 3; Siegel [A-98], Chap. 1; or Springer [A-99], Chap.l.

1.2. Teichmiiller Space of Genus 1 Let us construct the Teichmiiller space of genus 1.

L.2.1. The Moduli

Space of Tori

we use the fact that every torus is represented by a Riemann surface c/f, where ]- is a lattice group on c as in $1.4 (see the corollary to Theorem 2.13). On performing the transformation z r* zf 4,if necessary'we may assume from the beginning that the generatols ?r1and 12 Lor I a,re the ca"nonical ones I and r with Imr ) 0, respectively. Now, consider a lattice group f"={j=m*nrlm,n€Z}, where r € H = {r € C I Imr > 0}. As wa.sseen in $1.4, the lattice group I} corresponds to a subgroup of ,Aul(C), and the Riemann surface R, = Cf l, is Notice thal cf f, has the a torus. Denote by r, the projection of c to c/f,. group. structure of an additive

'(z'dtsalH =tw '(Z'Z)lSd fq g;o eceds 'sl teqt luarlonb eql qtl^\ PeUIluePtsl I,f41}€tll sarTdurrI'I uraroeqJ'IrolJo sassBl?af,ual -earnba rrqdrotuoloqlq IIe Jo tas eqt ''a'r'r.uoqto aeods,Ppout eqt aq rW P"l 'g reddn aql aue1d11eq 'ilno.r,6Jolnpou eq1 ;l ,L fra.rrg

3o ursrqdrourolne crqdrouroloqlq e s\ (Z'7,)'IS1

( l P + t ' c- , ' ' l = ( z ' 7 , ) r s d Q ) L| I\ t = " 9 - p o p u e z ) p ' ? ' q ' "ll '?7+=! o

) dnorS aq1 1ec a,u 'aaog

tr sl ,U -

'l'(P - ([z])/ fq ua'rt3 + n)) ,'A I t Surddeu crqdroruoloqrq e uaql 'splotl (8'I) ;t 'f1asra,ruo3 'I = cq - pD s^eq a1ll -lD + tcl '0< (, ,nD;q ,rwr _d

f = (t[or-;f

',! - (,t){or-/ pu" esuls 'I+ = ?q - pe l3q1 aas e^a ' (eJoureqlrr\{ 'srafielur eJ€ suotlela.reqt urorJ /p Pue ,?' ,9' ,D alaqlvr ,tP*'rP -" rQ* ,'t,o 'r-! o, lueurnS.reetues eql 3ut{1ddy 1aBen 'Ptt? - " ' 9*te ur€?qo ein 'a.ro;ereqa 'sre8alur ar€ P Pue

'? 'q'D aleqlr

'P+tc=n=(1)l 'q+tP=1a-(,t')l aleq a^{'ecua11'? rapun 0 = (0U o1 luap,rrnba are (1)/ pue (,-r.)rfqloq snq1, '0= d ecueq pue'0 = (0)1 leql erunsse,teur eaa'.re,roaro141 '(9'6 eufrua1 ,lc ra = (,)! lc) 0 + lc pue sreqlunu xelduoc 5rc ! pue ereIIA\'d + '(7'6 rue.roaql 15) s€ ualllr^a sr / uaqa 'j t, o. 'crqdrouroloqlq q .f esn€ceg 'l ,!)Lo! qanf 3 - C,! Eurddeurcrqdrouroloqe'sr 1eq1 P ! Wle lo"o-wrll 'pelcauuoc t(1durs sr acurg 1i slsrxe eraql teql salldtq ureroeqf fuorPouoru aq1 '1srtg 'ig oluo ,"A /oo"l4' Io / Surddeu crqdrouroloqlq e q arerll l€t{} alunss? 'c 'q 'o e.taym 'I = cq - p,Dqwn sta,aTut arD p PUD ,Plrc _ t , g*tp uoxlnpt, tf, fi,1uopuD fi Tuapamba frllocttlilloutopqrq erD 'tg puo at17filsr7os p puv t lt .r,edilneql w / pao t squr,oilony fiuo rol '11- tuaroaql uol onl 'g auold-{1ot1

(e'r)

'Z'I I snue5;o aoedg rall+urqrral

1. TeichmiillerSPaceof Genusg

l0

It is known that the quotient spaceHf PSL(2,2) is a Riemann surface (cf. $2.4 of Chapter 2) and that a fundamental domain (cf. $a.2 of Chapter 2) for PSL(2,2) is the shaded area in Fig. 1.9. Intuitively, we get the Riemann surface bV identifying the sides of this fundamental domain under the H/PSL(2,2) transformations z > z +1 and z e -lfz as is illustrated in Fig. 1.9. Hence we see that the moduli space of tori is biholomorphic to the complex plane. For more details, see, fot example, Ahlfors [A-4], Chap.7; and Jones and Singerman [A-48], Chap. 6.

Fig.1.9. Remark. A torus given by equation (1.1) depends on a complex parameter ,$f 0, 1), which is denoted by ,Sr. It is well known that two such tori 51 and S1, are biholomorphically equivalent if and only if there exists a linear fractional transformation which takes the set of branch points { 0, 1, }, oo } of Sr to the set of branch points {0, 1,^',m} of ,91, (see,for example, Clemens [A-21], Chap. 2.7). Thus we see that 51 and 51, are biholomorphically equivalent if and only if )/ is equal to one of the following numbers:

.\.

1 + , ,r

1-),

't r-)'

l-1 )

'

l ,\-1

Now, let G be a finite group of order 6, generated by cr()) = 1/) and - {0, 1 }. This fact Sz($ = I -.\ which are analytic automorphisms of D = C quotient spaceof Dby G (cf. $2.a shows that ML= D/G,where Df G means the of Chapter 2). Moreover, we find a biholomorphic mapping F : D lG ---+C, which is defined uy r([.1]) = /(.\) with

/()) =

(,\2-^+l)3 t2() - 1)2

r ueaaleq a)uaraJrp aql l€rll reprsuo) uec e1t{snql'sdno.r3 arr11e1ueemlaq I * ,'J:/ rusrqdourosreqt o1 spuodsauoc q?rq^,r'[((,r)tg)/] = ([(,r)tg])V pun = ([(,r)ty])V reqr qcns (od''A)ru * (od','g1rv: { ursrqd.rourosr l((,4trtll u€ sarnpur "A - ,"A:/ Surddeu ctqdloruoloqlq e^oqe aql leql ees e.tretueg 'l(,t)tV) srole.reua3 '{ go tua1s,{sle)ruou€) € seq qclqm (od' ,tA')r)Lqtrw ] [(,r),g] p a g r t u a p rs 1 , 1 . ( f p e p u r r S ' . { 1 e , r r 1 c a d s a r ' [ ( r ) t g ]p u e [ ( " r ) t y ] o t , p u e I s P u e s qclqlr\acuapuodseiloraql repun (od'"A)to o1 crqdrouroq sl iI ueql'(odtrU)I, Jo srol€reua3 go urals,ts letluoue? e a;rt3 ['g] Ptn [Jy] sasselcddolouroq aqa 'od ?urod es€q qtl^{'U uo (-r,)tg pue (z)ty sa,rrn, pesoll aldurts eulure}ap ''U '.{1a.l.r1radsar'C pue'I Pu" 0 uea^l}aq sluaru3asaql ul r pue 0 uae^4,laq dno.r3 leluetu"PunJ aql Jo lurod aseq € sB [0] - od a4ea'dno.r3 eql uorJ luelua?els e^oqe eql rePlsuoc el!\ eql Jo 1utod,r,ret.rr. 1o (od''g)rl I"lueu"punJ

'01'T'tIJ

+

,,1 -

'(Ot't'3t.f pu€ .{1a.,rr1cedsa.r',r. aas) I o? (,r)/ pue (1)1 '{q spuas (1)//z
'Z'Z'I

'Z'I I snuaD1o aoedgrallnurq)ra;

12

l. Teichmriller SPaceof Genus g

and r'corresponds also to the different choicesof generators of ur1(R',po), i.e.,

{ F'(")1, [B'(")]] and{ /.(Fr("')l)' /-([s'("')]) ] (seeFig.1.10)' E, = { Fr]' [Br] ] systemof generators Now,for anytorusft, takea canonical of the fundamental group r1(.R,,p) of R, and consider the pair (r?,Xo). Such a Xo is called a marking on r?. Two markings Ep = {[1t]' [Bt]] and .Do' _= curve Co { [,4i], [Bi]] are said to be equiaalenl when there exists a continuous t1(R,pr) T1(R,p) Ts": p' the isomorphism p which induces to f.oln ot R = an element sends Here, ?c'" = Tc"(lBr)). and Tc"(lAi) with [Ai] [C] of [Bi] 'C,] . product of the definition For ol r{R,p'). C an element r{R,p) to [Co-1 of curves, see $2.2 of Chapter 2. Next, two pairs (R, E) and (S, -Do) as above arc equiaalent if and only if there exists a biholomorphic mapping h: S ' R i s e q u i v a l e n tt o s u c h t h a t h . ( D ) = t . ( { l A ' , , 1 , [ B ' t ] ] )= { h - ( [ A i ] ) , h . ( [ B i ] ) ] of (R,D). We call class the equivalence = by Denote Ep [fi,X0] 1[Ar],[Bt]]. genus of 1 consists of T1 space Teichmiiller lorus. The marked such a lR, Do] a all marked tori. Theorem L.2. For euery point r € H, let E(r) = { [,41(r)], [Br(t)]] be the marking on R, - c/1, for which lA1(r)l and [Br(r)] conv,sponilto I and,r in = r' . 1,, rvspecliuely. Then [-R",X(r)] lR,, , E(r')l in T1 if and only if r Prool. Assume that [R",](r;1 - fR,,,E(r')]. Then there is a biholomorphic mapping h: R,, - R, such that h.(X(z')) = {h.([.41(r')]),h.([,B1(r')])] is = { [41(r)], [Bt(t)]]. W" may assume that h([0]) = [0] equivalent to I(r) by replacing h with hr(ltl) - n(lrl) - h(tO]) if necessary.Then, the definition of equivalence of X(r) and h.(X(r')) implies that h.([.41(ti)]) = [,41(r)] and h.([,B1(r')]) = [,B1(r)].Take a lift h of h with h(0) = 0. Then h() = az for some - ar' = r. complex number a. Hence we concludethat h(1) = d = 1, and h(rt) tr Therefore, we have r = T' , The converse is obvious. Since every marked torus [ft,Xo] is represented by [.R',X(r)] for some r in f/, this theorem shows that 7r is identified with I/' Another method to mark tori is realized via orientation-preserving diffeo' morphisms between tori instead of systems of generators of fundamental groups. For that purpose, fix a marking E = {Ft],[.B1]] on,R. Then any pair (S,/) s defines a of a torus.g and an orientation-preserving diffeomorphism f : Rma.rking f.(21 = { /.([At]), / ([.B1])] on S. Theorem L.3. Let R,S, anil St be tori, and let f : R -- S,g: R - S' be = [S',9-(t)] in T1 if orientalion-preserving diffeomorphisms. Then [S,/-(t)] --* S' is homotopic to a biholomorphicmapping h: S - St. and only if Sof-L: .9 = [S',g.(t)].Take two points r,r' € I/ for Proof. Suppose that [S,f.(t)] = where .R", R,', E(r), and which [,R,E]: fR7,D(r')], [S,/.(tI lR,,x(r)] and ':(z') are defined a^sabove. Rega-rd / and g as diffeomorphisms of E" to .R,,. We may assume that their lifts I and f send 0, l, and r to 0, 1, and r', respectively. Thus we obtain a homotopy between f and' i by setting

'(Z'dlSa l,;igaeeds /(ff); qrp pagltuaPrsI uol;o s\ (Z'Z)IS1'acuepuodsalrocslql qnpoutrr eql pu€'(U)J.to lc€ ol PelaPrsuoc (U)Z;o ursrqdrourolneerqdrouroloqlqaql ^A'(Z'Z)lSa I t ol Sutpuodsarroc sI *[fr] uaql '[r1oo"t''A) = (l't ''U])-["] fq (U),, uo uoll?e EI eugePPu" ""A = 'r-(("')L1o"'rt) =.'! tnd '(U)J A loursqdrouroaglPe sr qctq^r !1as1roluo ui 1t"llr;"""rrti'!,Ul = l(')tS,(r)!'] ulelqo "^ r("t)Llo"tqo'! = (')r;'o'r1ecurg 'l(i)'ri = (Ir])'rl fq ua,rrE'A <- Q)tU :'t1 Surddeu crqdrouroloqlqe sacnPul z(p+-tc) = (t)'! fq paugap(C)t"V;o'r1 luauralaaql'I{ 3 -r.1utod,(uero; (prr€qraqlo aql uO 'lQ)L! .(t)fu] of eql,(q uarrr3 [{']g] Surpuaseeuapuodsauo? ,tpeturrd q (U),2 uo l, go uoltc€ eI{J '(S'I) urroJeql w uol}"turoJsuerl l"uollc€rJ '(u)z u" spe (2'7,)ISd r€aurl e sr.(z'?hsd ) ,L luaurale.{ra a 1eq} 11ecell 'ruo11 dnorE .relnpou aql pq1 ureldxa e,$ (U)J p I7 uorlecgrtuePlaql Sursn 'arnlcnrls xalduroce seq oEe (U).f ttqt ''g] Surpuasacuapuodseuoc 'arogaraq;'l't ''Al ol er1]raPun lno surnl 11 [(r)3' (U)Z qU^ t7 fgluapl u?f, a/n 'g'1 uraroaql Pu" $lrsruar Sutpacardeql urord '5] '(/'5) sled qcns t(ef); fq pelouepq qcg,ra'g 1o [rf 1o acodsren!,uqrre; eql sasselcacualelrnbaIIs Jo les eql ilet e1l 'rS * S :q tutdderu ctqdrouroloqtq 'JI e o1 rrdolouroq sl ,.g .- ,g : ,-lo6 't'l ruaroaq;, ul se fpo pue y. Tueyoatnba ere eloq" s (f ',S) pue (/'5') s.rredoil,1 ?eql fes ean'putut ut stql qfl1l .(t)g = ((,r),<)'(,;) 1eBa,r,t'.re1ncr1red = (lrl)"1 "l'l?)"!l "2 : { ursrqd.rouroeJlPSur,uasard-uollelualJoII? sacnPul Eurllas ,(q 'U -

'c)z'6ffia=Q)'! Surddeu reeutl eql teql '"2f = 'S] pue aas eal 'l(")S'oA) = k'U] rsqr qsns l? ) !'o! slurod [(".r.)3' log = [d3'g] ret{t q?ns S' +- A i;f urstqdrouroa; om1 3ur1e1 ',(1en1cy't(f)T'S] -;rp Suur.rasard-uor1e1ue1ro lre PuU "^ '[d3'5] snrol pe{reur f.rerltq.re rr€ roJ '1ue1e'unbaele '5i] '[(f)'t'rS] =l(S)'l teql saqdurr qcrq'u tr s3ull.reur t"qt ees et 'og.{q pacnpur ((a)0',g1rt, ,g uo (g3)*f pu€ (3').(/"q) "c; rusrqdroruo$ aqt fg 't j ? ; 0'('d)tg fq uarr3 q qtlq/'a * (("it)1or1'r,S)r:r : ruor; ,S' uo elrnc snonurluo? e eg. oC ?e'I'3 Euqreu aqt ('d)6 oq ('i\!rrt rog lurod aseq e aq od 1e1pue 'f pue /or1 uear$leq ,tdolouroq 3 3q (I i t i O) rf, p"I'ctdolouroq e.reS: ol U uror; f pue ;l'or1sEutddeur o/'al 'V Eurddeur ,S * U : 'd1asra,ruo3 * g : crqd.rouroloq.rq€ of crdolouroq sI 1eq1 esoddns rS ,-to6 'f1r1uapreql ol ctdolouroq sr ,? - 'tg : r-to6 '[(z)rd'] = (lr))',t 3q?tnd (ecuag 'f ptre ;f uaenlaq rg fdolouroq e a^eq ear

'r; I;0 'c) "

'(r)0r+(z)!(?-t)=Q)'4 I snueC 1o acedg rapurqslal

8I

'Z'I

1. Teichmfiller Space of Genus g

L4

1.3. Teichmiiller

Space of Genus g

In accordance with observations in the previous section, let us construct the Teichmiiller space of arbitrary genus g in two ways. The first construction is given by considering marked Riemann surfaces. A of a fundamental group system of canonical generators Dp = {lAil,lBil}f=t is called a marking on E. Two markings n{R,p) of a closed Riemann surface E = are equiaalentif there on it and Dp,- {lAil,pf}fu Dp {[Ai],lBil]f=1 = Tc"(AiDand on .R such that exists a continuous curve Co [,ai] [Bj] = z r 1 ( . R , p t ) o r 1 (R,p') o f w h e r e ? s " i s t h e i s o m o r p h i s m 1 , . . . , 9 , fc"(Bil) for I . . be markings on closed Let Do and Eo to C C"). sending any [C] lC;r Riemann surfaces i? and ,S of genus g, respectively. Two pairs (.R,Xo) and (S,E) are said to be equiaalenl if there exists a biholomorphic mapping h; S 8 such that the marking h.(E) = {h,(lA!i),h.(lBjl) }j=r it equiv= (R,Dr) of is denoted by equivalence class alent to Ep {lAil,lBil}j=r.Th" genus g. Teichrn'iiller of The surface a markeil closed Riemann lR, Ei and called surfaces of genus g. g closed genus of all marked Riemann is the set spaceTo of The second construction is given by considering orientation-preserving diffeomorphisms. Fix a closed Riemann surface R of genus g. Consider an arbitrary pair (S, /) of a closed Riemann surface .9 and an orientation-preserving diffeomorphism f : R ---.9. Two pairs (S,/) and (S',g) are said tobe equiualentif gof-L:,S - S'is homotopic to a biholomorphic mapping h: S - S'. Let [S,/] be the equivalence class of (S, /). The set of all these equivalence classes[S, /] is denoted bv "(r?) and is called the Teichmiiller space of R. As in the caseof tori, we assert that the Teichmiiller space ?o of genus s(>- 2) is identified with the Teichmiiller space ?(.R) of a closed Riemann surface -R of genus9. To see this, first fix a marking D = {[1i],lBil1]oi=t on .R with base point po. Corresponding to a point [^9,/] in T(R), a marking /- (X) on ,S determines a point [.9,f.(t)] in "r. It is noted that this point [.9,f.(D)) in ?, does not depend on a representative of [S,/] in ?(.R), which is seen from Lemma 5.1. Hence we define a mapping A2: T(R) --- To by setting

az(lS,/l) = [S,/.(t)] for any [S,/] € T(R). Theorem

L.4. The mapping A 2 : T(R) ---+To is bijectiue.

Proof (an outline). The injectivity of @s follows from the se'called Nielsen's theorem (Ilarvey [A-41], p.43). It can also be proved by Lemma 5.1. However, we shall give an intuitive explanation for its injectivity. Suppoeethat two points s a t i s f y@ . r ( [ S f, ] ) = O E ( [ S ' , e ] ) , i . e . , [ S ' . f . ( t ) ] = [ S ' , e . ( X ) ] [,S,/],[.9'g , ler@) in Tn. Then we can take a biholomorphic mapping h of S' onto S and an orientation-preserving homeomorphism go of S onto itself, which is homotopic to the identity, so that gy - looho! coincideswith / on each Ai and Bi $ f i S g).

ueql'[/3' 'S] = [(f)V'5r] 3ur,{;sr1es.g*- g:rf ursrqdrouroeluoqSurrrraserd -uorleluarJo ue slsrxe araql 1€rll /r\ou{ e^r 'a.roqe palels uaroeql s.uaslelN .{g'smo1o; s"eparrord eq osle uec 7'I {uaroeql ut dlt,rtlcafrns aqJ'tlrDuev '[f 'S] =l(5).{ '5'l acuaq pue'1ua1e,rtnbe are ,3 Pu€ (g)Y tnql E s,r,roqs/ Jo uollrnJlsuol aql 'S - U : / usrqdroruoagrp Sur,trasard-uot1e1uat.ro ue o1 .{llerrdolotuoq zf ur.ro;aptlualoaql Eurqloours aq1 ureSeSursn .'tqecuaq pu€ 'rng z, slq;,'r11 uo 06 = 7'6pue lo pooqroqq3reu e ur qloorus dl.ressaeaulou sI rn - A uo f - z6 WqI os S r- g : zf ursrqdrouoegp Surrrrasard-uoll€lualJo u€ augeq 'r1lg uo Id uorJ tp uo o6 usrqd.rouroagrpEut,rreserd-uolleluelro ue lf,nrls -uoc '3[ ro; se fe,r,r atu€s eq? uI '{qp pesol? e o1 crqdrouroagp q Pue 72 sul€}uoc qcrqaiodgo t4 pooqroqq3tau e ar1e1'aroruraqlrng '(tD| - S * If -y :16 tustqd -rouoesrp Surarasard-uolleluelro ue o1 flectdoloruoq f ruro;ap '(6'1 ureroeql 'g 'd*qC '[ZfV] qsqg) ueroeql Surqloorus e Sursn 'tnqtr 'lg p* fy qcea;o <- n-A td urstqd pooqroqq3rau€ uI qlootus fpessacau 1ou st f sHI'(r)t-S -JouroeuroqSutrrrese.rd-uoll€luelJo ue ulelqo e&' I) -A o1 g 3ut1celord 'alo11 '8 'deqC '[Al-V] qsrlg) tV - "? t,! ursrqdrouroagrp '(g'g uraroaqJ, uo t ue o? spualxa sJ'$ql teql u^{oqs q }I'g -uyg Surarasard-uorleluarJo q1!^{ sepl)ulqx eJ' qcrqa roJ syg *- ive : e;l usrqd.rouoeglp e lcnrlsuoc ue1 '4srp pesol? e A ueql 'flu 5 ol Surpuodselroc UVQ uo 5i 1as eql raplsuo? pue e o1 crqdrouroaglp sl qrlq^a U ut od;o n pooqroqq3tau 11eus{lluatcglns e a{"J 'l,S pue ' !,V ' lg ' lV .aV lo uyg ,{repunoq eq} o1 a! p uor}crrlser aql aq ! pl ' " ' 'I sdool pe Jo uorleluelro eql selresard 3/ 1"ql eurnss€eir e.rag 6' f 1e rol {'d}

-

- !g)21 lS = (oit}

' { o d } - t , V= ( o d }

- !y)s1

Surf;sr1es{odl - ,C '1xap oluo {0d} - C Jo 3;| usrqdrouoagrp Sur,traserd-uotleluelro ue a1e1 '(,2'S) uor; fear aruesaq? ur peurclqo {srp lrun pesolc st{} "y fq elouaq 'uy {slP }Iun Pasol, eql o1 crqdroruoaJlp sl d leqt asoddns feur arvr'4,;o xalra,r qcee punor€ ernlcnrls alqerlueragrp Surceldag '(f 't '4.f ;c) aueld eql uo saprsf6 qll/'^ d uo3flod Pesolc e pue oU ueearleq ursrqdrotuoagtp Sur,trase.rd-uorleluetro ue s1srxe eraql ueqtr, .,C-S=oS,C-A

r=! =oA,CAn!,V))=,C'(gnlil)= 6

r=f " 6

'.raq1rng ' 1as ,3 ro1 op 'salrnc qloous pasolc aldurs "t" t=;{ = ,K wj lurod aseq aqt eq llal'l!,vjI pu" I=r{ llAl'llV)l = Z ur fg pue '!,V'lg 'ff I1eteqt aurnssefetu am'1srrg 'J ue qcns go 'l(6-)-l',gj = uorlcnr?suor e earE 1leqs ea\ 1nq 'lceJ u^\oul-ila/( € sr sIqJ [,9',9] qcrq^\ roJ S: oluo A p { ustqdroruoeuroq Surnresard-uorlelueuo u" slsrxe ereql 'f1r,rr1ca[rns eql a,rord o;, 'g] fue roJ uJ > teql /$oqs ol lu?Icgns sr 1r lB ',S] eleq e.rlrsnqa 'ctdolouoq er€ rd '16' Pue / feqt flrsea a,rord ,Sl = [/ uec a \ 'aue1d aql ul {s!p lrun aq} o1 crqdrouroeuoq $ U urorJ fg pue fy 1e Eur -1e1apfq peurc?qo uretuop aql aculs 'uorlrusap aqt fq [td'S] = [6',9] teqt atoN 6 snuag;o aeedg rallgurqrral 't'I

9I

1. Teichmriller SPaceof Genus 9

16

we find a qua.siconformal mapping /o homotopic to / (Bers [26] or Lehto [A68], Chap.5, Theorem 1.5). This fo is not necessarily smooth; however, there exists a real-analytic quasiconformal mapping homotopic to f, (the Corollary to Theorem 6.9). Finally, we define a canonical group action on the Teichmiiller space ?(R). Let Mod(R) be the set of all homotopy classes [o] of orientation-preserving diffeomorphisms ar: .R * -R. We call Moil(R) the Teichmiiller modular group or the mapping class group of .R. Every element [ar] acts on ?(R) by [r].([S, /]) = [S, f or-'] for any [S, /] e "(n). We call every lw)* a Teichmil,ller moilulor transformation. Let Mo be the moduli space of closed Riemann sarfaces of genus g, i.e., the set of all biholomorphic equivalence classes [S] of closed Riemann surfaces ,9 of genus g. since for a,n arbitrary closed Riemann surface .s of genus g there exists an orientation-preserving diffeomorphism of R onto ,S, the moduli space M, is identified with the quotient space T(.R)/Mod(R) of "(i?) by the action of. Mod,(R). Therefore, we can study the moduli space Mo via the Teichmiiller space ?(.R) and the Teichmiiller modular group Mod(R). In Chapter 6, we shall see that "(E) has a (3c - 3)-dimensional complex manifold structure and that M od(R) acts properly discontinuously on "(8) as a group of biholomorphic automorphisms. In particular, the moduli spare Mo has a (3g - 3)-dimensional normal complex analytic space structure.

1.4. Quasiconformal

Mappings and Teichmiiller

Space

Let us reviewthe Teichmiillerspace?(,R) constructedin the previoussection from the view-point of the theory of quasiconformal mappings. L.4.1. Deformation

of complex

structures

and Beltrami

coefficients

For a point [S, /] g ?(,R), we want to compare the complex structures of ft and s. Take a coordinate neighborhood (u,z) on I and a coordinate neighborhood (lz,to) on ^5with f (U) C V, and set F = ?r,ofoz-l. Then

p-

Fz F'

is a smooth complex-valued function defind on iur open set z(Lr) in the complex plane. Note that it is independent of the choice of a local coordinate u.'. Since ] i, .n orientation-pr"r"ruing diffeomorphism, the Jacobian of F, i'e., lF,l' F' is l&12 i" positive-definite on z(U). Thus we have lpl < 1 on z(U). Further, biholomorphic on z(U) if and only if F = 0 on z(U)' We call y' the Belt'rcrni coefficient of / with respect to (U , z). It should be noted that a Beltrami coefficient of / depends on the choice of a local coordinate z on R. How it depends is shown as follows: take coordina*e

. l ( o ) ' r l --I

i(g;ffi=(o)>r

u asdrlla srr{}Jo srxe rounu aqt o} srxe roleur aq] Jo ol]er el{l

'l"l(l(o)'rl - r)l(o)"/l + r)l(o)"/l i l(o)zl 5 l,l(l(o)'tl sarlrlenbeutaqt ,tg '(tt't '8t"f) aueld-rn aqt ul asdrlla ue o1 aueld-z aql ur 0 raluec qt-ra elcrlc e spues 7 deur 't > r€aurteqt 'raaoero141 l$)"t /(O)ttl = l(g)r/l pue O * @)"1 1eq1saqdunqcrq,ra

'o< - .l(o)"/l = (o)/r .l(o)"/l sagsrles0 - z le (6)f uerqocel s1t 'urstqdrouroagrpSutrlraserd -uorleluerro ue sr / acurs '0 - z Ie / ;o uorsuedxa ro1,te; eql Jo tural rapro lsrg eqt aq z(iltt + z(g)'t = G)l 1a1 'aue1d-rnxalduroc eqt ul /O uIPruoP e oluo aueld-z xaldruoc aql ul 6 urSr.roeq1 Surureluor O ureluop € Jo tuuqd.rouroagrp Surarasard-uorl€luarroue sr / l€ql etunssearra'spooqroqq3reualeurpJoo?Surraprs -uoc 'srql easoI 'sluer)lgeoc tusrtleg;o Surueeurcr.rlatuoa3eq1 ureldxa a.tr'1srrg s8urddetr4l lBruroJrrocrsen$'6'7'1 ',t1t1eur.royuoc uorJ 3[ 3o uotletaap aql ernseauro1 pesn sl / Jo luerrlgaof, lruprllag aql pue 'g uo arnlrnrls xaldtuoc eql '(U)"f ul Jo uorleruroJepe sluasardar (y)"6 ul U'S] lurod e leql su€errrlr ef,uag {y :l 7eq1 Wl'lAl = [/'^g] ?eql s^roqssrq;'Surddeur crqdrotuoloqlq€ q,S * pue 'tusrqd.rouoeJrp Surl.rasard-uolleluelro ue sr /U * A :p? deu flrluepr eq1 'tl)D{ 's1as se (Io"*'("1)r-t) 1eq1 U = /U leql eloN } spooqroqq3raueleurp 'deilr slqt uI -rooc ;o uals,ts qlurr paddrnba IU ateJJns uu€uIeIU A\eu e aleq ea,t 'U uo arnlf,nrls xaldruoc " seugep v>a{(toDm'("1)vt) } spooqroqq3raueleu ',9 - g : ursrqdrouoeslp Sut,uasard-uolleluelro ue roJ -rprooc rt ;o ualsIs B puts S uo vl"{ ("*'"A) } spooqroqq3rau eleutp.roocJo ualsds e rog '.Lrop

'U uo / 1o Tuata$aocnaDr?Iegeql pallec$ q)nl^t

(e'r )

''P ,t - trl

zp

,tq ,{ldurrs (t't-) ed{1 ;o ruroJ l"rluereJlp slq} a}ouap e^r snq;, 'U el€ulprooc uo ad{1 (I'1-) Jo urroJ leltuareJlp € se)npul U Jo spool{ro,Q{31au - l{z areq^l uo /go sluerrlgeoc rur€rllagJo les eqt leql s.lroqsslql'rizors

(r r)

'(trut2)tz uo (#)

l@).(rzotrl)=

trl

e^tsqa,lr'Q * qnU ln '(qz'qn) pue rurerllag uaq,11',,(learlaadsar ol qll/'^ Jo sluarf,lgeoc (lz'fn) leadsar ./ eqt eq 'trl pup ld lr,1't1 1 (r2)l pue ln > (dI ?€qt qcns g p (tn'tn1 '(!m'11) spooq.roqq3reu eleurproocpue gr 1o (tz'qn) '(tr '.!2) spooqroqq3rau L1

a*dg

ralnurqf,ral pu? s8urddul4l purro;uorrsen$

'7'1

1. Teichmriller Space of Genus g

18

This shows that any infinitesimally small circle with center 0 is mapped by / to an ellipse whose ratio of the major axis to the minor axis is K(0).

L(z)

--t

cp- 0+aref,Q) a : ( 1 +l p ( o ) l ) r l l , ( o ) l b: (r-lp(0)l)rlf,(o)l

o:lareu$)

Fig. 1.11.

This statement holds at every point in D. Thus we also call the Beltrami coefficient , \ ft(r)

p t Q ) = f f i , z eD ,

the complen dilatationof / at z. As we saw before, Ft = 0 on D if and only if / is a biholomorphic mapping on D. We call f a quasiconfonnal mapping of D to Dt if f satisfies

Kr lrrl'J!. .". ' = supl* r-lpt?)l ,eb

Further, f is called a quasiconformal mapping with Beltrami coefficient Lrt.W" call K1 the maximal d,ilatation of f . In this chapter, we only consider smooth quasiconformal mappings. We shall study more general quasiconformal mappings in Chapter 4. tansformation formula (1.4) implies that the absolute value lprl(z)l of the Beltrami coefficient W = pJQ)dzldz of an orientation-preserving diffeomorphism I : R - .9 does not depend on local coordinates on l?. Thus lpy I is a continuous function and lpty| < 1 on ,t. Since r? is compact, we get

pl py( z)
(1.6)

ln particular, we have

"'=::B HP)t=il+ffll:'*

(1.7)

'@)"ttpO nP s|uo1aq o7 Quo puv l? A lo dout fi1t7uaptaq7o7 ctiloTouroqsr, 6rl '(A)+ttlO aq7 uotTo1et' sp1ot1 oeluos.r.ot (d)*ott, lorlor-6'atou.t.r,eql,.rng + f, i0 { r f i 1 u op u o t ? r S - S : r 1 A u t d d o u tc t y i l t o u , r o l o t poq s q r n e u a l l ' t S puo S * A i{ stusttliltoutoa$tp 0untasa.td-uotToTueuorof, '9'I ureroaqJ 'r(A)g

Jo luauele ue sr /r/ pus (U)+//?O

Iil:a-I

',v l _ n o/ \ - r i l : l r l

(o'r)

g) r.)-

Jo luaruale ue sI r't areq^l

"-\ = ({rt)*n = "r-.o[rt

fq ua,u3 q r(2,)g uo (A)+ttlO Jo uollc€ aqJ .@)+l lgO;o dnor3qns '{1.rea13'pg deru flrluapr eq} ol ctdolouroq (u)+l tlo Ieurou e q (A)'! !16r ur s?ueutrrela II€ Jo slflsuoc qclqar dnor3 e aq, (U)"t lpO leT JIeslI oluo Ur uo sursrqdrouoagrp Suu.raserd-uorlelualroIIe Jo dnor8 aq1 (A)+t tlO fq elouag '(g't) fq uarrrSurrou-oo? eQt Eursn {q t(g.)g uo f3o1odo1e eugeq'seceJ -rns uueureru pesol, oluo ar ec"Jrns uuerualu pesolc pexg e;o sursrqd.rouoeJlP 'U uo Surlrasard-uorlplua-IroII" Jo stualrgao? Itu€rlleg Jo les aqt aq r(U)g larl sluer?lgeoc lru€rlleg go aceds eq1 Eursn dq sacedsrellntuqclal replsuof,eJsn lerl sluarcsaoC

rtuertlag

'8't'I 3o sacedS

'seceds rellnuqtlal go froaql t.Ildleue xelduroc e dole,rap eirr eraqm '9 reldeq3 ur elor luelrodur ue sr(e1dslqJ'tr/ uo rtllertqdrouoloq spuedap /orrl luarclsao) rurcrllag aqt'./ paxg e roJ l€q? suoqs (g'1) elnruroJ'ta,roaro141 '((g'Z) "lnurroJ'9'6 eurural) y ut lutod e fl o pu" requnu Ieer " $ d aleq^r 'aue1d xaldruoc eql ul y cslp ?tun eql Jo Z D _ T

"a

ursqdrouolne

= (z)L

'tlrDu.Iay ctqdrouroloqlq e se ruroJ eruss aql seq (8'I) "1nlurod

.z{il --'trt t g : r } l o z l | u r i l d o u . te W ' ( 7 , ' l f i n p o p u o l t c t y i f u o r a o p q Nsqp z g + u1 |utatasatd-uotToTuauo.tot '.t'o7nct'7-tod = !) lS 1- U : lt sutstrld.tou.toa$tp 'spptt {ooyl.[il-I'l

(a'r )

-H:j;Frt

_ r^oe

T-t-"

uotlllar eqt 'J + g :6 'S ,- A : t su,tst'r1d.t'ou-toa[9p 'g secol.tnsuuour?tg Jotr 'g'T uollrsodor6 puD 'J'g |um.tasand-uorlDlueuo 'elnl uleql eql ,tq pe,rord ,,(lsee sr Surddeur elrsodtuoc e Jo sluelcseoc rtu€rlleg roJ elnuroJ 3ur,u,o11o;eq5 'g ;o Surdderu l€ruJoJuoctsenb g, : / usrqdrouoe$IP Sur,r.reserd-uoll€lueuo ue leql {es feru ai ecuaH

e s-I,g -

'l'f acedg relnurqf,reJ Pu? s8urddeyl TeurroJuof,rsenb

6I

1. TeichmiillerSpaceof Genusg

20

Prool. Suppose that there exists a biholomorphic mapping h: S * u - g-Lohof, we see that formula (1.8) gives

S'. Setting

F g = F h o J o r - r= P J o u - t= u * ( P t ) . Conversely,if there exists an elementu e Dif fa(R) with pc = u*(pt),then Proposition 1.5 shows that h - gowof-r: ,S* S'is a biholomorphic mapping. D The second assertion is clear from the definition. Corollary. The mapping of sending (S,f) to pt e B(R\ identif,cations:

iniluces lhe following

r(R) = B(R\lDif I"(R), Mo e B(R)r/Diff+(R).

1.5. Complex Structures and Conformal Structures In this section, let us reconstruct the Teichmiiller space ?(R) by means of conformal structures induced by Riemannian metrics on R. 1.5.1. Riemannian

Metrics

and Conformal

Structures

Suppose that a Riemannia.n metric ds2 is given on a real two.dimensional oriented smooth manifold M. This metric is represented as ds2 = Ednz -f2Fdxdy * Gdy2 on a coordinate neighborhood ([/, (r,V)) of M. SettinE z = t f iy, we see that it is written in the form

ds2= \ldz * prdZl2,

(1.10)

where ,\ is a positive smooth function on U and pl is a complex-valued smooth function with lpl ( 1 on U. Actually, l and p are given by

^=i ("+G+zJ-nc-r), E_G+2iF

t'- E +G+2 \/E E=7t' Local coordinates (u, u) on Lr are said to be isothermal coordinate.sfor ds2 if ds2 is represented as

ds2=p(duz+du2)

(1.11)

on U, where p is a positive smooth function on [/. Here we €Issumethat both of the orientations induced by the coordinates (c, y) and (u, u) on U coincide with the one on M. The complex coordinate u) = 1r* iu is also called an isothermal coordinatefor ds2.

fq uarr3 q ;4I uo (z'fr'r) - d lurod e'T,I'I'3ld rl pel"rlsnllr n (d'il seleulproot I€col aql 3urs1 'r11 uo )ulaur ueqJ crrlau ueruuetuelg € s€q u"ep{?ng eqt dq pef,npur J,tI Ietluouec zsp '(Zt't'ft.f) D > q > areq^4, 0 'srx€ z eql punor€ aueld-(z'f) "qt uo = zz * z(D - f) ala.rrcaql 3u-I^lo er zg fq paurelqo fl r{?rqr'r gll af,Bds ueaprTcng aql u! e?eJrns e aq W p1 'aldutoxg 'droeql pue drqsuorleler slql paztuEo uollcunJ ctrlatuoa3 aql papunoJ 'g 'ses?c -rer .raqSrq roJ enrl lou $ qtlq^\ 'sployrueur uu€uerg lsrs leuorsueurp ''e'l 'sp1o;rueur xalduroc I€uorsuaurp-1 .ro; ,tl,radord elqe4reur Iear l€uorsueurp-A -eJ e sr uorlrasse stq; 'Surdderu l"ruJoJuor € pells? sr Sutddeur cqdrouoloqtq e teql uos€er " $ srqJ '1ua1e,rrnbaare aln?cnrls IsruroJuo? Jo pu" aJnltnrls (teql s^\oqs uaroeql slqJ xaldtuoc ;o sldacuoc 'esea leuorsuaurp-oirrl aql q 'att1d.rouro1ot1lq tl fi1aopuo s! ([sp '.,1I)- Qsp'W):l s? *A lr lout^,t,otuoc S t! '([sp'N) splottuotu uvruuDutary uaql 'fi1aar7cedsar puo (zsp'W) louorsueurrp 'Z'I uraloaql -Z pa?ueuo frq pacnput,sacottns uuou?tg eq S puv A pI 'ureroeql 3ur,no11o; eql ol speal (61'1) uoll"lues -e.rda.raq1 go ssauenbrun eql esef, ar{l uI aas o1 fsee sl 'U uorsuaurlp ll 1eql Jo 'uaql uaa&rlaqSurddeur FluroJuol s slsrxa ereq? J! ernptury loru.totuoc euros eq1 eler1 ro Tuapatnba Qlou.t.r,oluoc a r e ( [ s p ' N ) p u e ( z s p ' W ) 1 e q 1{ e s a 1 l ' N u o ( z C ) t p u e ( t 5 l ; ; ' u e a a l l a q ' l s p fq pa.rnseaur 'a1Eue aql slenba W uo 7'C pue I, selrn? qloours fue uaeiu,leq 'f1arrr1rn1ul'W uo uorlcunJ rlloorus 'zsp fq, pernsperu 'e13ueaq1 leql srrcetu 1r penle^-lear v 4 6 araq,n '_;211 uo .sp(d)dxa o1 lenbe q ./ ,(q ltp to lceq 1nd e q 1y f u t t l d n u t p u t l o t u o c " s l N * W i t u s r q d r o u r o e g r p B u r , r r a s a r d - u o r l e l u a r r o ue '(|sp'N) poe (csp'A[) sploJrueurueruuetuel]r leuolsuetulp-U palualro rod 'zsp clrleru u€ruueruerg eql fq pecnpw ?rnpnr?s lout^to{uoceql palle) aq ,(eu U uo arnlcnrls xaldruoo eq; 'deu, slql q pautelqo aceJrns uueurelu eql g itq eloueq 'W uo ernlcnrls xalduroc s saugep r>!{(!n'h)} l€q} f;r.re,ro1 (,;ig;o qncJlp lou sr lI '!2 qcee uo lm elsurpJoo? I€r.uraqlofl ue slsrxe araql r)!{((n'!x)'12) eleurproocJo uralsdse ro; 'acua11'I > -llr/ll } spooq.roqqErau '7 .ra1deq3 ?eql paphord slsrxa s.ile,rale01 uorlnlos " qcns Jo A$ ul pe,rord st sy 'uo4onba ,urDrlleg aql pell€r sr uorlenba slql 'rn uorlnlos crqdrotuoaglp e seq

(elr) esp roJ /'1 al€urprooc l€uraqlo$

'

t

z

o

9a=9 m8

m8

uorlenba prluereJrp lerlred eql JI s?s1xe ue leql apnlcuoc erra'(0I'I) qlrrn Sur.reduroc t

, l z ni +

r

l

z P l z l z m=l"dl n n l d

sagsr??sesp roJ nl eleurProoc leruraqlosl u3 a?uIS sernlrnrls I"ruroluoC pu€ sarnlf,ulg xalduro3 'g'1

tz

22

l. Teichmriller Space of Genus g

\

a

Fig.1.12. , = ( o * 6 c o sg ) c o s 0 , y = (a * Dcos
,1,= rl,@)= [' ---!4o, A+OCOS/P Jo

then the metric d,sz= (a * bcosg)2de2 + b2dgz has the form dsz=^({)(d02+drl}). Thus to = 0 + fry' is an isothermal coordinate for ds2 on M, which defines a complex structure on M. Hence ft is a torus. A little more calculation shows that R is biholomorphic to C/f, where l- is a lattice group generated by 1 and

ib/\/i, -F.

L.5.2. Reconstruction

of Teichmiiller

Spaces by Riemannian

Metrics

Fix a closed Riemann surface r? of genus g ) L. Take any local coordin ate z on R. For an arbitrary Riemannian metric ds2 on rR, from the uniqueness of the expression in (1.10), we obtain a globally defined Beltrami coefficient p on R, being a differential form of type (-1,1) and llpll- < 1. Such a p is called the Bellrarni coefficientinduced by a Riemannian melric. Let us observe the relationship between the Beltrami coefficient of an orientation-preserving diffeomorphism and the one induced by a Riemannian metric.

'@)+llvo/(a)w = uw '@)'llto l(a)w = @)t :suotToc{zyuapt, 6urmo11ol ayy aatf acuapatnba6uo.t7spuD nuelD { o7 0utpuodser,ror)uleru o to'fr1aa4cadsa.r,'sso1c 'g'I uraroaqJ - a m b aa q 7o t ( A ) J u ! l t ' S l l u e u e p u o p u e sq n q m s f u r d d v u e q J 'uorlress€ 3ur,rao11o; eql of p€al g'I ruaroeql o1 ,t.re1oro3 eql pu€ uorle^Jasqo srql ',,t1e,rr1cedser'(A)W p sass'elc acuaprr,rnba 3uor1sIIe pue ecuele.,lrnbe IIeJo las eq? pue (a)+IlpO/(A)W rq eloue6l '(A)"lIlO o1s3uo1aqo srql @)"ttpo/(A)W y Tualoamba fi16uo.t7s eq o1 paugepa.re|sp pue usp 'reqlrnJ 'l€ruroJuocq (I*p'U) - (zsp'A),o ?tsqtqrns (A)+ttlO ur r,l luerueleue slsrxe areqlJr Tualoaznba aq 01 paugap erc (g)4r ur |sp pue 6s'pslueuale o rI 'g Jo (U)J eceds rallnurqcraa aql llnrlsuocer a^r '9, uo scrrlaur u€ruueurelg Jo (U)hf les eql Sursn 'no51 'w uo s)rrleru ueruu€ruaru ,{q parnput sluarrlg:aof,ttu€rllag Jo las aql ol lenbe sr Ur uo srusrqdrouroagrp Surarasard -uorleluarroJosluer)Ueo, rurerllagJo I(U)5. ?eseql leql eesa,r.'.{e.nsqt uI 't o7 0utpuodsatloc uo culeur uvruuoulery A e eq ol pr€s sr (|rp)-t crrlaru e qcns'S'Jo ernlcn.rlsxalduroc eq1 Surcnpur[sp crrleru € Jo ecroq) eql uo puadep lou saop uotlress€ slql l€r{l eloN '/ Jo }€r{t se '(g luel)cgeor rur€rlleg erueseql se,rr3/ .repun |sp l" (trtp) -l >1ceq11ndeq1 uaql 'crrleru er€rurod .ro 'ueaprlcng 'pcrraqds eql ,(q raldeq3 Jo t'I$ 'gc) ,t1a,rr1cedse.r pe)npul sr qcrq/( S' uo crrleu eql e{€t alA '(U .ra1deq3Jo 6'Z ueroeqa) eueld-;pq raddn aq1 ro 'aue1d xalduoc eql 'araqds uu€ruerg eqt ol crqdroruoloqlq sl S Jo ac"JrnsSurra.roclesrelrun aq1 '1ce; uI 'SJo auo leur8rro aql ol lual€^rnbe sr Ltp fq pa?npur arnl?nrls xelduroo eql leql os S' uo |sp or.rlaurueruu€ruerll " a{€l uer eM 'ue,rr3eq * :/ ursrqdrouoegrp Surrr.raserd-uorleluerro ue 1e1'd1asra,ruo3 .g U 'rl qtl* saprcurocluercgeoc rureJllag sll pue ursrqdrouroagrpSurrr,.rasard-uorleluarJo u€ sl ,U * g. : { Eurddeur ,t1r1uepr aq? ueqJ '(^'n) eleurproof, leurreqlosl aql ^q pernpur sr arnlcnrls xaldtuoc asoq^r /U a)eJrns uuer.uerge urclqo all.'(z 'p) pooqroqq3rau eleurpJoof,q)Ba uo

zo --:- fl m8

zo n8

= ::-

uorlenbe rurcrlleg aql Surr'1os'I'g$ ul uaes ueeq s€q sV 'A to zsp rrrleur ueruueruerll e ,tq pecnpur luerrlgeor rtuerllag eql s! r/ 1eq1 esoddns '1srrg 8,2

sarnlf,nrls l"urroJuoC pu" sarnl)nrlg xaldurop 'g'1

24

1. Teichmriller Spaceof Genus g

Notes The geometric function theory originated with Riemann's 1851 Gottingen dissertation [181] and his 1857 paper [182]. In connection with multi-valued analytic functions such as algebraic functions, he introduced the concept of the Riemann surface as a branched covering surface over the Riemann sphere. He also recognized clearly the intimate relationship between holomorphic functions and conformal mappings on a domain in the complex plane. In [181], he proved Riemann's mapping theorem which asserts that any simply connected domain in the complex pla.ne with mor'e than one boundary point is biholomorphic to the unit disk. In [182], he obtained the Riemann-Roch theorem. By using this theorem, he determined the degree of freedom of finite branched coverings over the Riemann sphere which represent closed Riemann surfaces of genus g, and he obtained the complex dimension mo of the moduli space of closed Riemann surfaces of genus g, that is, ms = 0, 1, and 39 - 3 for g = 0, l, and g > 2, respectively. For more complete exposition of Riemann's work, we refer to Ahlfors [a] and Klein [A-53]. The standard definition of a Riemann surface, that is, a one-dimensional complex manifold was introduced for the first time in Weyl's classic [A-111] "Die Idee der Riemannschen Fl6che" in 1913. The material of this chapter is classical. Some of the many celebrated books on Riema.nn surfaces are Ahlfors and Sario [A-6], Bers [A-13], Cohn [A-22], Farkas and Kra [A-28], Forster [A-32], Griffths and Harris [A-39], Gunning [A-40], Jones and Singerman [A-48], Schlichenmaier [A-95], Siegel [A-98], and Springer [A-99]. For details of topology on surfaces, there are further books by Birman [A-18], Harvey [A-4lj, Chapters 1 and 6, Moise [A-75], Stillwell [A-101], and Ziescha,ng,Vogt and Coldewey [A-114]. For algebraic curves, we refer to the books by Arbarello, Cornalba, Griffiths and Harris [A-9], Grffiths [A-38], Mumford [A-78], Namba [A-82], and Shafarevich [A-97]. The moduli space of tori considered in $2 isstudied in the context of elliptic curves, elliptic integrals, and theta functions. For an interesting exposition on this subject, see Clemens [A-21], and Jones and Singerman [A-48]. For textbooks on Teichmiiller spaces,there are Abikoff [A-1], Ahlfors [A-2], Gardiner [A-34], Harvey [A-41], Krushkal' [A-60], Lehto [A-68], and Nag [A-80]. For expository papers on this subject, consult articles by Ahlfors [8] and [11], and Bers [22], [29], and [40]. The approaches to Teichmller spaces as in $4 and $5 are found in Earle and Eells [62], [63], and Fischer and tomba [7L], respectively.

a?rtn e1t lo auo o7 TualoatnbaQparydloruoptfq fi1du.tzsfueag (aqeox pue ?recurod '.r!"IX)

' H J o ' J ' 3 s e c o t t n su u D u r e l y st acoltns-uuouaty peloauuo? 'tuaroaqtr, uoltBznuJoJrun

'(w6d '[y-y] sroylqy eas) {$p }run aq} ot crqdrouroloqrq sr lurod f.repunoq euo u"q? arotu qlra C ur ul€urop pelceuuot fldurts dra,ra 1€rll slresse qcrq/rr uraJoall ,utdrlou,t s.uuDur?ta ol Pa)nPer sI rueroaq? sHl'c aueld xelduroc eqt ur sur€urop ro; 'r(gercadsg 'sac€Jrns uu€tuerlf roJ sploq uaro -eql uollezlruloJlun eql Pallec sl q?lq,$ ?oeJalqe{r€IueJ e }sq} u^lou{ 11ams! 1I

rrreroaql uorlBzlruroJlun'I'z '6 snuaS;o tg aceds rellnurq)Iel eql ql!,u PagluePl q ll }"qt alord pue'g-ngll uI lasqns e se td aceds alcr.rg aql eu$ep aal '(6 l)f snueS3o sereJJnsuu€r.uarlr pasolc Surluasardar suelsfs Ierruorr€cEursn'g uorltas ur 'fgeurg sdno.rEuersrl?ndJos.ro1e.raua3;o 'ralel pasn ele qtlq^\ 'sdno.r3 uersqcr\{ yo satl.redord freluaurala euros a,rord airrr'7 pue t suolltes uI '; dno.r3 u€rsrlf,nde Aq H Jo J/H aceds luatlonb e fq paluasardar q (Z ?)n snua3 ;o ereJrns uuetuerg pasolc fre^e leql epnl?uot a,rl 'dem srql uI 'suolleuroJsueJl snlqgl 'dnor3 'relnarlred uersqrnd e uI tr J II€r ell'' 11 = U uerl^r 'Il ro'C '? ;o Surlsrsuocdnor3 e sB g uo rtlsrionurluo?srpfpadord slce J pu" o1 lualelrnba fllecrqd.rouioloqlq sr g, 'ruaroaql uorleznuJoJrun eql fg '6 uorlaai ur palrnrlsuoc s-rJ dnor3 uorleur.r5gsue.rlSurra,roc sll 'U Jo U eceJrns Eutreloc tg, Iesra^run eql JePrsuo?arn aae;rns uueuaru frelrqre ue ol ueroeql uollezfluroJ -run eql {1dde o1 repro uI 'g aueld-g1eq.reddn eq} ro '9 aueld xalduroc eql 'e a.raqdsuu€rueq aql :sae"Jrns uueruenr eerql eql Jo euo o1 crqdrouroloqlq $ eceJ -rns uusruarg palreuuoc fldurrs .,(Jea leqt slrass'e qcrqilr 'aqaox pue 'atecuto4 'ura1y o1 enp rueroeql uorlsznuroJlun aql urc1dxa er,r '1 uotlcag u1 'sdno.r3 u€rsqrnJ pue 'suorleurroJsueJl snlqgl [ 'sece3:tnsSurralroc lesrellun uo s]c€J f,Issq 'asodtnd 'sace3:rns sn{l roJ uu€ruerg Jo ruaroaql uort"z-turoJlun eql apuro.rda,u 'd snua8 (e-rsg ur }asqns e se pezrper sr (6 l)f snua3 ;o eceds e{llq aq} pag"c fl rlcrq^{ ;o aceds rellnurq)ral eql l"qt ^{oqs ol u .raldeqc luesard eql Jo esodrnd aqa

ar€ds a{rlqjt u raldBrlc

2. FrickeSpace

26

Remark. These Riemann surfaces 0, C, and /y' are not mutually biholomorphically equivalent. The Mobius transformation tr = (z -i)lQ * f) maps biholomorphically 11 onto the unit disk 4, and hence we often use the unit disk 4 instead of the upper half-plane If . Corollary. A closedRiemann surface of genus0 is biholomorphicequiualentto the Riemann spheree . Thus the moduli space Ms of closed Riemann surfaces of genus 0 consists of one poinl. Proof. Since a closed Riemanil surface R of genus 0 is simply connected, the uniformization theorem implies that .R is biholomorphic to one of the three Riemann surfaces e^, C, and 11. Since .R is compact, it should be biholomorphi! cally equivalent to C. For proofs of the uniformization theorem, we refer to books on Riemann surfaces listed in the notes of Chapter 1. See also Ahlfors [A-3]. For historical and expository accounts, see Abikoff [2], and Bers [29] and [36]. Among standard proofs for the uniformization theorem, there is a method in which a mapping function is constructed by using Green functions. Let us elucidate the concept of Green functions by using an intuitive example from electromagnetism. We rega.rd a Green function on a Riemann surface .R as the electric potential on R where a positive charge is given at a point p and whose boundary is earthed. Mathematically, when z is a local coordinate around p on R, we define the Green function g(.,p) on ,R with pole at p as the minimal function in the family of positive superharmonic functions which are harmonic in .R - {p} and have the singularity -loglz - z(p)l at p. The existenceof a "capacity" of the boundary of rt and is indepent of Green function depends on the choice of a point p. For example, the Green function on the unit disk 4 with - z")1. On the other hand, there are no pole at zo is given by logl(l-z;z)/(z Green functions on C or C. Now, assume that there exists a Green function g = g(,p) on R. Then we obtain a biholomorphic mapping f : R'-'+ A;

f k) = exp(-s(q)+ ic.k)),

(2.1)

where g* is the conjugate harmonic function of g on R - {p }. Note that g* is a multi-valued function whose periods are 2ntr (n e Z) becauseof the singularity of g and simple connectednessof R. Hence, / itself is a single-valued holomorphic function on .R. By the argument principle we see that / is univalent, i.e., a biholomorphic mapping of -R onto 4 (see Fig. 2.1). Next, we deal with the case where there exist no Green functions on R. We take a sequence{ n' }Lr of simply connected subdomains of E such that .R' is a relatively compact subset of E +r for each n, that Uf=rR" covers.R except for at most one point, and that every ftr has the Green function g,. with the common pole p. Then, in the same way as before, we can construct a biholomorphic mapping fn: Rn - Alor every n. By multiplying each /' by a suitable constant, we get a sequenceof biholomorphic mappings F' : R- - { tr e C I lr.rll< r' } so

'U I€sJelrun y Jo acottns |ut.taaoclDsrearun" U pue'g 1o 6uuaao) IDsrearune (A'o'A) II€) e^\ 'palcauuoe flduns sr U uaqm '.raillrnd 'U otuo g 1o uo4cato"tdeq| pa11ec6qe sr z deur Surra,rocaql 'Ujo acn!.rns|uuaaoc e g iue 'g 1o |uttaaoc e (A'v'U) IIec aaa'r1ooq srql uI 'crqdlouroloqlq q /? * ]1 :t deur palcr.rlsar aql '11 p (A)r.-" a3eun asra.rlura{l;o 1 luauodruoc pe}cauuor qcee roJ leql qcns pooq.roqqSreue s€q d fra,re y dout, Fuueaoc e eq ol pres sr lurod , A Jo 'sac€JJnsuueuaru aq pue Ur Ar ]erl A - U : ! Surddeur crqdroruo-Joqanrlcalrns y sdno.rg rrorlerrrroJsue4l, Eurre^oC puB saceJrns Eurraaog Jo suor+Iugo1, 'TZ'Z 'e?sJrns uueruarg f.rerlrqre ue Jo aceJrns Surrarroc pelceuuoc fldurs € lcnrlsuoc aaa'uraroeql uorlszrruJoJruneql ,f1ddeo1 rapro uJ

stur.ra,ro3 lesJa,rrun'e'Z

'Ur*n ;o Eurddeu erqdrotuoloqrq parrsape sef,npur

'c ro oluo c u uo uorl?unJ fr-uliaqf leql

'r'z'8tJ

l r > l m l l :v

fldurrs e

[€ruJoJuot --

lrnf/1 So1 (*)"6;o qder3

((4t)"0 - (r)6 uort?unJuaarg erlt Jo qder3

I I

+--I

sBurra,rog lesrel-ru1'Z'Z

LZ

2. Fricke SPace

28

"highest" covering surface of all coverings of covering of R mea,nsthat it is the r? (cf. Theorems2.2 and,2.4, and the remark in $2.2). Example I. We give a few simple examples of covering surfaces. (i) Let r: C - C - {0} be given by r(z) = e'. Then C is a universal covering surfaceofC-{0}. (ii) Let r: H + A-{0} be given by r(z) - e2'i".Then I/ is auniversal covering surface of A - { 0 }. * C - {0} be given by r(z)= zn, wheren is a positive (iii) Let r: C -{0} C integer. Then { 0 } is a covering surface of itself, but it is not a universal covering surface. (iv) For a given ^(> 1), set r = exp(-2r2/log,\) and A= {w € C l t < l.l < 1). Definer; H---+ Aby r(z) - exp(2trilogz/log.\), where logz denotes its principal branch. Then 11 becomes a universal covering surface of the annulus ,4. ( v ) t e t 4 b e a l a t t i c e g r o u p g e n e r a t e db y 1 a n d a p o i n t r € I / , a n d l e t r b e the projection of C onto the quotient space C f fr. Then C is a universal covering surface of the torus C/ lr. Any biholomorphic mapping '1, fr, - E wittr ro^l = z is-called a coaering transformalion of a covering (R,r,R). For a given covering (R,r, R), denote by l' the set of all its covering transformations. By the composition of mappings, l- forms a group, which is called the coaering transfonnation group of (,R.r,,R). In particular, we call I the uniaersal coaering lransformation group of (-R,r,.R) if ,R is a universal covering surface of r?.

(i)' (ii)' (iii)' (iu)' (")'

f = (rr) with 71(z)- z *2tri. f=(rr)with71(z)=z*1. 1= ('rt) with 71(z)- z exp(2riln),which is a finite groupof order n. r = (zr) with 71(z)= )2. 7 = (T,72) -- f,, where7{z) = z * 1 a,nd"fz(z)= z + r.

2.2.2. Construction

of tJniversal

Covering

Surfaces

First of all, we need several definitions. A path on a Riemann surface R means a continuous curve C: I - R, where.I is the interval [0, 1]. The points C(0) and C(1) are said to be the initial and lern'inal points of C, respectively. We also say that c is a path from c(0) to c(1). Throughout the book, if no confusion is possible, its image C(/) is also denoted by the same letter C.

fdolouroq € eleq am'f11eutg 'I ) n roy (n1)"g = (n)(c't)p fq g'uo (t't)p qled e ausep a,r'7 x / f (s'?) fue ro3'uaqJ,'1 3 s due .lo;'d = (I)"f, = (0)'J pue 'C = rI 'oI - og sagsrles tl'{ ("'.)uf = 'd } leql q?ns uaql uee^.rleq rldolouroq € eq U * 1 x I ii p1 'o7' o1 crdolouoq sI C leql su€etu q?Iqtl 'l"d'Cl,tq peluasarde.r 'q1ed pasoll " $ 'fod'Cl -C ecurs lod'ollleqt epnpuoa am sl C Jo lurod leurturel eql puie 'od lutod es€q IIII^a g uo qled Pasolee sI 5t uaql 'ei" '[od'olTol crdolouroq q C tnd [od''I] lutod eseq q]p\ g uo g qled Pesop ',r,r,o11 fierta 1eq1 aes ol luelrgns fl 1I 'Paleeuuos ,(ldturs q f€qt b,rord arr,r !f '[d'g] ol 'pelceuuoc q U leql satldtut qclqar lod'oll uro+ U uo I qled € a^eq al,r '7 ) s fle,ra iog [(s)g'"C] = (s)g 3ur11ag'I ) 1ro; (ls)j = (rt"C rq A uo'C Wed e eugep'1 I s q?ea ro,{'U uo qled e Aqlod'o1f ql.I^{ pal?auuoc sl lI f [d'g] lurod i(ra^a teql A{oqsol sjcgns }l 'U Jo ssauPa}ceuuoc aq1 a,rord ol1[t'0] - I ) ?,(ue ro; oa = (t)"1,(q paugap g-uo qled eq] aq oI p"I'too.t'4 'p?peuu@ fiylul'ts puD pep?uuoc st 'aaoqo pautoldxa sD peptulsuoc 'g acottns ?qJ 'T'Z Btutuarl 'ur uo ernlcnrls = 4z turl eql }eql ees err,r')Lodz xelduroc parrnba.raq1 splarf {(!z'!2)},tgurel -les 'gg ur ureurop pelcauuoc fldurrs e s\ dn l€qt qrns (or'on) pooqroqq3rau al€urprooc e e{sl 'U p [d'C] - gf lurod fue ro;'1ce; u1 'Surddeur ctqdrouroloq '1xe11 e setuof,aqA * U:1, l€qt os Ar uo alnltnrls xalduroc 3 euueP a,n 'deur e Eurra,roc Jo uorlrpuoc aql sagql"s pue U oluo Ur yo Eutddeur (uolllnrlsuof, eql ,cg 'a : (la'gl)r. rq eas o1 fsea q snonurluoc € sr l 1l 1eql ua,rr3 uorlceto.rd eql ee U * A:)L p1 'sceds 1ecr8o1odo1 JroPsneg € seuoteq g ueql 'A u\ 4 Jo spoor{roqq3rau pluaurepunJ Jo uralsfs e ouuep s^\ 'd2 aseql p;" dn uea/$leq aauapuodsauoc euo-ol-euo Isf,Iuouef, e a^€LI e,t 'uteutop ig,'h palcauuoc ,tldurrs e sr.d2 acurg 'D o1 d uroq d4 ur pauteluoc qled frerlrqre ue st d4 ';2 ul .{eirrr e qcns ul ur e sr b pw 1eq1 lutod 2I lb'bC.9] slurod II3 Jo les aqt dn pooqroqq3rau e 4p fq alouaq 'U ul ul€ruop pelceuuoc fldurrs-e $ qtul^r d lo '9, ol Peau an '1srrg e ecnPor?ul rog uo ,(3o1odol {ue e{el 'g lurod { Jo ld'Cl 's,lrolloJ se qqt leqt ees eA\ e satuoreq eceJrns Surra,roc U l"srelrun Jo B '[d'g] sasselcacuale,rrnba aseq?ileJo les eql eqU lef '(d'C) Jo ssplt acuele,unbaeqlld'CJ dq aloueq'g uo tC ol crdolouroq sr j pue d - d !\Tuapanba erc (d' ,g) pue (d'9) srted oarrl aseq; 'd o1 od uror; A uo C r{led fue pue U uo d lutod fue 3o rted e eg.(d'C) 'ecsJJns ulretuelg e 'U eosJrns uuetuell{ ualrE e uo od Jo lurod es€q 3 xld te1 'alotr1 ecsJrns Surre,rocl"srellun e i(lalarauoc lcnrlsuoc lleqs arrrsqled Sutsn fq 'd = (ilu tlA u\ il,"r;3:"9:r|"'; A uo CWed e sI Ar uo,, qled e o't11llv 61 pre6 sl q!. q d lurod y 'U ecsJrnsuueruelg e;o Sutrerroce eq (g')L'A) p"l '0),C lurod leururrel eql pue (0)C lutod prltut aq1 qf!^A U uo ,C . C rlled e 1aBaar 'rg;o lurod FIlluI eqt qq^a C;o lutod Ieunurel eq1 Eurlcauuoc ,(q '(g),9 = (t)C lsq? qcns A ao tC Pue C sqled oarrlrog s8urraaogl"sra^ru1'Z'Z

30

2. Fricke Space

F1r,";; I x I - E b"t*""n i andllo,polbysettingF1t,"; = [C1r,,;,r"(t)] for (r, s) € 1 x 1. Therefore, we conclude that E is simply connected.

B

On putting these observations together with the uniformization theorem, we obtain the following theorem. Theorem 2.2. Fo'reoerg Riemann surfaceR, there exisls a uniaersal coaering surfaceR of R, which is biholomorphic to one of the three Riemann surfacesA, C, or H. Throughout this section, 6 o universal covering surface E of a Riemann surface r? we always take the one constructed above. From the construction of such a universal covering, it is easy to get the following lemma by an argument similar to that used in the case of analytic continuation (Ahlfors [A-4], Chapter 8). Lemma 2.3. (Existence and uniqueness of a lift of a path) For any path C on R with initial point p, and for ang point F of R oaer p, there erists a unique lifl C of C wilh initial point fi. Ttreorem 2.4. (Litt of a mapping) For Riemann surfaces R and S, let (R,Tn,R) and (S,trs, S) be their uniaersal coaeringsconstrucled as etplained preaiously,respectiaely.Then giuen an arbitrary continuous mapping f : R- S, there etists a continuousmapping it fr,--- S with forp= osoi. Thit mapping - (1, where € fr. and I is uniquety tletermineduntler the condition that i@) fu 4r e S are such that rs(Q1) = f brn(Fr)) Morvooer, if f is differentiable or holomorphic, then f is also differentiable or holomorphic. This mappine i t Fl.- ^9 ir called a lift of f: .R *

S.

Proof of Theorem 2./. Setting fu = lCr,pr] and 4t = [Dt,/(pr)], we get a .f (C),/(p)l m a p p i n g d e f i n e db v f ( l c , p l ) _lDt.f (Ct)-t Jor all points [C,p] in R. Then it is obvious that /(f1) - {1 and fSrp - nsol. Since zrp a.nd n5 are locally biholomorphic and / is continuous, / must be continuous. It is also trivial that if / is differentiable or holomorphic, then so is /. the uniqueness assertion follows from Lemma 2.3. D Rernark. (Uniq:reness of universal covering) For any two universal coverings (R,r,R) and (r?1,11,R) of a Riemann surface r?, there exists a biholomorphic mapping g of R to r?q with Trog - n. See,for example, Ahlfors and Sario [A,-6], Theorem 18A of Chapter L

e Jo uorl-rugep ar{} ur uorlrpuo? aql sessrl€s qclq,$ u ur d 3o 2 pooq.roqqStau e esooqo 'ld'Cl = 4'@)" - d 1as pue 'g f gl tutod € a{€t '(rr) aes o5 " '(g),0 - i sagsp, 'l"C) -- L ', - o5l 3ur11n4 'A uo zg pue IC sqled euos rc1 t€rl? aas am IC . 7'C fd'z7l = p pue [d'rCj = 4 a,re{ er'ruaql 'd = (!)y = (4)a 1eq1esoddns '(r) a.,'o.rdoa '{oo.t4 'Q* X u ( y ) r p q l q c n s J ) L s l u a u a l a f i u o u t Q a T g u { I s o l l t?' o a . t ' oa " r ' a q y ' g sSaoi (rrr) lo y TasqnsTcoilutocfiuo.to!'st IDW:ry uo fipnonu4u@srp fr.1.r,ailo.ti1 'sTurodpat{ ou soy fiyquapt.ayy.tol Tilacxa - J ) L f r ^ r a a"ar o t Q = n U ( d L e ) D e ' " t o l n c q t o dq ' { p ! } 1 lo yueu,a1q ? D q l 1 l ? n sU u ! 4 ! o 2 p o o t l . r o q q \ n u a l q l p n s o f l e r e q l ' U ) g f . r a a a i o g ( n )

'@)L= D uo s?srseatayT'(p)tt.= (4)v q?!nU ) !'q fruo.tog o ltlln J a L \uau.ta1a

:sa4.tado.td6utno11otaq7 sa{stqos g acottns uuvur?tg eqJ 'g'Z BtutrroT 6ut.r,eaoc o Io (U':r-'A) to .1 dnotf uorTout.tolsuo"tT losJeaNutu 'a,rr1calrnssr acuepuodsa.rroc O srql ecueq pu€'.[C] - l, teql seqdrur7'Z ruaroeqJ'(d'A)ro Jo luetuele u€ = ["d'Cj = (l'd ''I])t reqr s^\oqs g'U €urureT snql sr [9] ecurg '(t'd'"tl).[C] 'flarrrlcadsar'g;o s1u1od pue', pue pue aql ar€ Ierlrur Ieurural ['d'oI] I'd'Cj C l€I{} Jo lJll € sl , acuaH'od lurod eseq qlu{ g. uo qled pesol? s sI Col, sarldurr y JLov uorleler aql ueql '(l"d'"tl)L ollod'oll uroq Ur rio qled e al Q 'a,rr1celrns slq] 1eq1 aaord o; sr ecuapuodsarJoc 3 ,L luaurela ,tue a1e1 'err,r1celur st eeuepuodseJJo?sltl] Ie.I'J

l€qt sA^olloJII'ed'a)rL Jo luatuale lrun eqt q [u ef,uaq Pu€'07 o1 ctdolouroq sr op 'snq;'[t'O] = 1 ) t fue to1 od - (l)'l reqr qrns U uo qled eq] sl o1 ereq!\ 'lod' o o " If = fod' Cf = (l"d' tl).|' Cl a^eq a^r ueql 'J Jo lueurele lrun eql q -[?] 1eq1asoddns 'arrrlcefutq ]l ]€q] e,rord oa '.7 o1 (od(g)tv;o rusrqdroruouor{e sr ecuapuodserrocsrq} }tsqt pI^Ir} sl rI'loord '(A'o'U) 0ut.taaoc losrearun n to 1 dnot'6 uotTotu.totsuo.r,T |uuaaoc losraarun?ql oluo A Io ("d'A)rv dnot6 pTuau,opunt aql uo spyaffi-['d - log)acuapuodsauocaaoqDeyJ'g'Z rraroaqJ to tustrlilrotuost. '(A'v'A) Jo uorleruJoJ -sue.r1Sur.rar'oce sr 1r 'sl leql '..,1o1 s3uolaq -[op] srql 'uo1]-ruuepeql fq 'fpea13

'ld'c.'c) = ([d'c1).1'c] ry> la'c) fq g uo -[op] uorlce eql eugep e^{'(od'U)rv>['C] ]ueuele fue rog 'f, 1o (od'g)rY eqt o1 crqd.rouroslsl J dnor3 Eurraaoclesrelrun slt l€q? aas dnorE leluaurepunJ 'p ace;rns uueuerg e II€qs alll Jo (g'.u'9,) ece;rns EutrarrocIesJeAIunuarrrEe rog sdno.rg uor+BrrrroJsuer;, EurreloC IT

lesrallun'e'Z'Z

s8urra,ro3l"sra^rufl'Z'Z

g2

2. FrickeSpace

covering map in $2.1, and denote by U the connected component of r-r(J) containing f. Actually, it is sufficient to take a simply connected domain U c o n t a i n i n ge . l t 1 ( 0 ) n 0 + { f o r s o m e1 € f , t h e n t h e r e a r e p o i n t s f u , f u e 0 with f1 = l(it). Since ro7 = n, we get T(fr1) = r(4), and hence Q1- fi1,for r is biholomorphic on U. Thus we have l(Ft) = id(Ft), where fd is the identity. By Theorem 2.4, we conclude lhat 1 - id. Finally, to verify (iii), assumethat there exists a sequence{ Z" }f,r consisting of mutually distinct elements of l- such that 7"(1() n I{ * / for all n. Then for each n, we can take two points [n,Fn € 1{ with fn = .ln([n). Since K is compact, taking a subsequenceif necessary,we may assume that { drl1T=r, {i" }Lr converge to Qo,io € /{, respectively, as n + oo. Since zro7, = ?r, we obtain r(4") = r(f") and o(4") = r(i,). Take a neighborhood [/ of r(,i'") in .R satisfyigg the condition of the definition of a covering map in $2.1, and denote by U and,I/ the connected components of zr- 1(U) containing f, and fo, respectively. Since { j"(q") }f-, convergesto fo, we hav,e.y"(y)n0 t' g for a sufficient-lylarge n. Since ro7"(0) = (J,it follows that 7"(U) = 7, namely, ("tny)-|,1n(0) = 0. By the assertion (ii), we conclude that 7,.11 - 7,. This is a contradiction. ! Exarnple 3. Here is a"nexample of a group which does not act properly discontinuously. Let a be a real number not equal to 2r multiplied by a rational number. Then the group generated by l(z) = edoz does not act properly discontinuously onC-{0}.

2.2.4. Representation

of Riemann

Surfaces as Quotient

Spaces

We shall explain a way to construct a Riemann surface Rlf fro a Riemann surface R and a subgroup l- of the biholomorphic automorphism group Aut(R), where f is assumed to satisfy the properties (ii) and (iii) in Lemma 2.6, that is, every element of f except for the unit element has no fixed points in E, and acts properly discontinuously on E. Two points F,C e Rare said tobe f -equiaalentor equiaalentuniler f if there exists an e-lement.f e f satisfying 4=t@). Denote by [f] the equivalence class of fi. Let R/f be the set of all these equivalence classesp], which is called the quolient spaceof r? by .i-. Define the projection r'. R * R/f by r(fi) = \fl. We introduce the quotient topology "n fr,/f . A subset U of R /f is said to be open if and only if the inverse image o-t (U) of [/ is an open subset of E. the p_rojectionr is readily seen to be a continuous mapping of E onto tr/f. Since ,R is connected, so is r?/f. Moreover, we see that Rl f is a Hausdorffspace, for l- acts properly discontinuously on .R. Now, we define a complex structure o" n/ f as follows: for any point f e E, take a neighborhood. Up of I satisfying the property (ii) in Lemma 2.6. We may assume that there exists a local coordinate zp on 0p. Then, putting p = r(fi), we see that r: 0O - tJ, is homeomorphic. Hence, setting zo = Uo =,tr(0), zpozr-r, we conclude that { (Up,zp)}rrrtl , definesa complex structure so ihut

G'z) ,

'29-r - (z)L ,'a , -; s, u?lprn oslo st slttJ 'T = "lql - "lrl ,ltp^ ) ) Q(o anqm

,p+z!_k)L

@'z)

9t zo fi.r'aag(nr) ru.ro!o soq (V)nV Tuaurap to '0*Dql?n C)q'oanym 'q+zo=(z)L

(s'z)

tuaag Q1) ut.rolo soq (g)wy to Tuaurala ' I = " q - p Dq w n ) c P ' c ' q ' oa n a y m ,P*zc _k\L

(z'z)

Q* zo

fr"raag(r) ur"roto soq (g)7ny lo Tuaua1a 8'Z stutuoT

erll e^sr{ Il pue 'V 'C'Q

:sturoJ 3utmo11o; sul"tuoP l€?Iuoue?;o surstqd.rouolne erqdrouroloqrg

srrrerrroq [BcruorrBC go sdno.rg rusrqdrourolny

'

crt1daoruoloqrg'1p'8'Z

1 g ) t n vp u e '( v ) t n v ' ( o ) t n v ' ( q ) n y

'puttu ul qql qllft sdnor3 ursrqdrourolne crqd.rouroloqlq aq1 fpnls sn 1a1 'g, uo flsnonulluo?$p fFadord eql roJ ldacxa g' ur slutod paxg E?" Pue'fllluepl '9'A €Luluel urord 'ur '19)tnv;o dnor8qns 3 sl Jo J 1noq1r,nslueruelaJo slsrsuo? 'i'Q sursrqdrourolne-clqdrouroloqtqlo dnorE aql (U)t"V,(q alouap eAyI/ to 'ureroeql uorlezrur.rd; seceJrns uuetuerg "".rq1 nq1 jo "uo o1 crqdrorioloqlq q U -run aql Jo anlnl fq 'era11 '.7 dno.r3 uolleuroJsuerl Sfrueaoc l€srollun 8ll ,(q Ur eceJrns Surra,roc lesralrun e p J/A eceJrns uusruarll luarlonb eqt fq Paluas lerdar sr U eceJlns uu"IuaIU ,(re,ra 16q1 ueas aAeq eal 'uollcas Surpeee.rdeql uI

suorleurroJsuBl,I,snlqgtrAtr'8'U

'@)".--,W ?Ul repun. A o7 Tuapanba filloatydtotuoto\?q q J nq V k I /U acuapuodsa.u,oc acottns uuotu?ty Tuat1onbeqt u?qJ 'tr dno.t6uotyont"totsuntT0uuaaoc losJearun 'Z'Z rrraroaq.1, qnn A acottns uuour?rg o to |uueaoc losrearun o eq (A')L'A) pI 'uorlresse Surrrrrollo;eq1 ,(lalerpeurul el"q a \ ueql 'J ,(q acopns uuouery tf 1o eW J /A eteJrns uueruelg sql ilec eM'J /A 3o Euirarroce st (tr f g';u'g) TuatTonb suorl"rurolsu?rl snlqol I 't'z

34

2. Fricke Space

where0eRandaeA. (iv) Eaery elemenlof Aut(H) has a form. t(z) =

az+b cd+d'

(2.6)

w h e r ea , b , c , de R w i t ha d -, b c= I . In (2.2), it is sufficient that complex numbers o,, b, c, and d satisfy the condition od - b" # 0. However, 7 does not change when a, b, c, and d are multiplied by a common constant. Hence,-we may normalize the expression of 7 by ad - bc - 1. Every element of Aut(e) is called a M1bius transformation or a linear fractional transformalion.In particula.r, an element of Aut(H) is called a real Miibius transformation or a real linear fractional transformalion. Proof^of Lemma 2.8. First of all, let us determine the form of an element 7 € Aut(C). If 7(oo) = oo, then in a neighborhood of oo, 7 has the Laurent expansion

tQ)=",+i

bnz-n,

where a I 0. Then tQ) - oz is holomorphic on e , and hence the maximum principle shows that lk) - oz must be a constant function, say 6. Thus we have l @ ) = a z l b w i t h o + 0 . I f z ( o o ) = z o # @ , t h e n s e t t i n gt { z ) - t l Q - z " ) , we see that both 11 and lpl are elements of Aut(e), and 71o7(m) = oo. Thus we havefolQ) = I/QQ)z o ) = a r z * b r , w h e r e0 r , 6 r € C w i t h o , * 0 . Therefore, 7 is expressedin the form (2.2). Next, every element t e Aut(C) is extended to an element of Aut(e) if we put 7(oo) - oo. By the above argument, it is obvious that 7 is represented in the form (2.3). Let 7 be an element in Aut(A). Set 7(0) - B. Then the Mcjbius transformation r(z) = (, - 0)/G - Bz) belongs to Aut(A). Hence .y2 - 1*.r also belongs to Aut(A) and 92(0) = 0. Schwarz' lemma implies that 72 is a rotation trQ) = eiqz,with real number d. Hence, T is expressedin the forrn (2.5). It is easy to see that 7 is written in the form (2.4). Finally, for any element 7 e Aut(H), taking a biholomorphic mappin gT(z) = (z-i)/(z+i) of fI onto .4, we have an elementlr = ToloT-r e AutlA). Thus 71 is a Mijbius transformation and is representedin the form (2.2). Since 7 sends ry' onto itself, we may assume that c, D, c, and d are real numbers, and ad- Dc ) 0. Therefore, this 7 is written in the form (2.5). tr For more on the fundamental properties of M6bius transformations, such as transformation of circles into circles, and the invariance of the crms ratio under them, we refer, for instance, to Ahlfors [A-4], $3 of Chapter 3; and Jones and Singerman [A-48], Chapter 2. Now, for every 7 e Aut(e) given by

'sl

ler{}1'L1o

'slueruel€?s3urno11o;eql e^€rl a,r,r'uor1e1nc1ec aldurs e ,(g 'oz - (oz)L Surr(;sr1es oz > C Jo tas aqt IIs 'r(1r1ujpreq} oz slurod pexgJo las aql eq (t)xrg 1a1 }ou sr qcrqn

, I = c q- p D , ) ) p , c , q , o

, ' . 1 1 , i= Q*zo

@)L

.{q ue,tr3 uor?euroJsuer} snrqory e aq I 1a1 suorlBurroJsuBrl

snlqgl trJo srrrroJ lBcruouBc

'z'e'z

'{1aar1aadsa.r '(1'1) atnTvufts to dno.r,6fil,opun lonads eq1 '1)29 pue (U'6)79 araqirr plle 7 aa.r6ap lo dno"r6nautl yonads lDa, eqt are (1

' { t+} lfi 'r)ns= G't)nsa = (v)wv

'{ r+

pue

}/(u'?,hs= (ll'z)tsa = (n)pv '{(e)t"v

a^eq a^{ 'flrelrurrg ) Ll 6oLor-.{ } = (ra)lny reqr pue

' = -, :; lz:ii';il li:1, | l

uorleuroJsuerl a.,rr1ce ford e o1 spuodser.roc(3)tnv Jo (p + zc)l&+ zD) - (z)1, lueurale ue 1eq1 ees e/tr uaq;,'rd Jo eleulProoc snoeueSouoqe sr [tz : 0z] araqrrr'rz/02 = (ftz : ozl)g * rd :J Surddeur crqd.louroloqrqe euuep 'paepul 'rd Jo uorleruroJsu€rl f,q ? a,rrlcefo.rde o1 spuodsauoc (3)lny Jo luetuela ue '1d aceds err,rlcalo.rd xalduroc Ieuorsuaurp-euo aql qlra pagrluapl q C a.raqdsuuetuerg eql ueq1\ llrDureq

'(c'dts ul yT sluauala o,rl!,1 ,tq peluasarder sI ,L 1eq1elop "L Jo uorlD?uesa.tdat rt.tTou e pelle)c$st(g)l"y 3 ,L luaruele ue rog (1,)r- W lo V lueuala wtr '4 aa.rfaplo dno.t6 = (C'Z)lSd toauq lotcads aar,Ttato.td eql tr lpc pue {1+}/6'dlS tas aIA

' { r + } / 6 ' z ) t s= ( q ) w v

ursrqdrourosrue secnpur (ruaroeq? tusrqdrouoruoqeql dq 'ecua11'xrr?Burlrun aql sI 1 ereq^r ltl '{ 'c'g 'o ereqrrl 'eloqe se y f + 1 sl W Jo loura{ aql ueqtr, Jo saulueeq?er€p pue '(C'Z)1S )>y .ro;(p+zc)/(q+zo) = (z)(y)W tq paugep(q)ny oluo (3'6)79 yo ;;4rusrqdroruoruoqe e^eqem'r(lasrarruo)'(C'dlS dnor3 reauqletcadsaq1;o lp c1 l'^ -l =V L q D) ,I =cq-pp

qll^{ j

> p'?'q'D

luauele u3 aAeq e^r D+z) (1__-i_ (z\L 9*zo suorl"urrolsu"rJ snrqol{'t'z

9t

2. Fricke Space

36

(i) The casewhere oo € Fix(7),i.e., c = 0. lf a/d = 1, that b, c - 6f = *1, then 7 has a sole fixed point oo and it is written in the form

1e)=z*b, where b is a non-zero complex number. On the other hand, if a/d f has another fixed point zo, and is represented as

1, then 7

u-zo=\(z-zo), where to =lQ),

and ) is a complex number equal neither to 0 nor to 1.

(ii) The case where oo f Fix(7), i.e., c f 0. If (a * d)2 = 4, then 7 has a sole fixed point zo and it is written as

w-%=;:1+o' where to = lQ), and a is a non-zerocomplexnumber.If (c* d)' # 4, then 7 has two fixed points z1 and 22, and it is representedin the form W - Z t

",- tr-

^, Z7 -- Z' l

where w = 1(z), and ,\ is a complex number equal neither to 0 nor to L. Now, two elements7r,jz e Aut(A) are said tobe Aut(X)'conjugate or conjagale in Aut(X) if there exists an element 6 e Aut(X) such that 1z = 6o"ho6-t , where X is one of C, C, H, or A. This leads us to the following lemma. Lemma 2,9. Eoery Mdbius tmnsformation 7(f id) has otueor two f,xed points on e , antl is Aut(e)-conjugate to the foltowing M|bius tmnsformation 7o: ( i ) I f t h a sa s o l ef i x e i l p o i n t t, h e n T . Q ) = z + d f o r s o m ea € C , a * 0 . ( i i ) I f 7 h a st w o f i t e d p o i n t s , t h e n T o ( z ) = ) , 2 f o r s o m e \ € C , ) + 0 , 1 . We call this 7o a canonical form of 7. Matrix representations of canonical forms in (i) and (ii) correspond to

[;?], tf ,f.^), respectively. A real Mobius transformation 7(l fd) whose fixed points a.rein fl = RU{ m } is Aut(H)-conjugate to a canonical form 7o such that the entry a or ) of a matrix representation.To is a real number.

'peuueplla,lr '1, sr sll a.renbs (1,)rr1 Tnq fq flanbrun peurrurelap olw peretlesr ec€r1slr ueql'V- f,q pacelda.r sr y 1ousr (l).r1 snql'(p+D)'.{. uorleurroJsusrlsnlqgl4le 'p ;1 Jo nD4 " pelpc q qcg/'r * o (l)r1 1nd a14

' T= ? q,C , l \ ' ^ l= , PD ) p,c,q,o L 9 DJ :,L;o uorleluasa,rdarxrrleru " Jo ecerl eql np+p tsql atoN 'Z+y/I+y= e , ( pa r ) u o r l e n b ea q l s e g s l l e sy r a q d r l p t u s 1 1 'ol,;o lurod pexg e^r1?€r11ea{} o1 Eurpuodsarroc,L;o lurod pexg sJo alloq? aql uo spuadep yy/I pue aqt roJ sarroqr o,lrl e eq a64'l;o 1''a'r'l,yo.rar1dr11ruu .taqd41nut, e pallsc sl y slql'0'O* y'C ) y) zy - Q)'L uroJ Iscruouer € ot apSnluoc-(g)tnv o. cqoqered lou sr rl?rqar ,L uorleurroysrrerl snrqotrtre 'alo1q 'g uo sTurodpae{ omy soq L fipo puo ctloqtedfitl s! ,t (H) ta tt, .zz = rz puV , *H 'H 'rz slutod par{ on\ sorl L ) zz ) rz pUl qtns zz tt fi1uopuo tt, ctTdqla st' L 'g uo sr L Tutod pat{ alos o soy L lr fiyuo puo lt cqoqn.r,od

(r) (r)

:p7or16utmo71olaq7 uayl 'fr,7t7uapt ?ql pu s, q)rqn uotgou.t.r,olsuorl sntqory loar o eq L pT 'Ot'Z eurtuarl 's11nsaresaql Surulqurop 'uor?ross€ aqt urelqo ea,r 3ura,ro11o; /tls"a 'cqoqradr(q ro 'cr1dt11a ro 'cqoqered reqlla sI ','[1t1uapt ro (11)7ny Jo luetuale fra^a 1€ql ^\oqs ol fsea st 11 'fla,rrlredsar '1,

eql lou sr q?rq^r'(V)lnV

go paxg Suqledar eq} !o pue r; ,(q alouaq pexg a^rlrer11e a{l pue lurod lurod 'flarrleadser 'L elql pve Turotl pax{ 0ur11ada,eq} pell€, 1o yut,odpar{ aatyco.tqqo elre ez pue rz uaqJ, 'I < lVl qll^ y euos to! zY - Q)'L ruroJ l"tluouec e Jo qurod pexg eql o1 puodsarroa 'fla,rrlcadser'zz pu€ Iz 1eq1 asoddng oo pu" 6 'l, uorleur.ro;suer? snrqoq cnuorpoxol e;o slurod pexg aql aQ zz pue rz lo.l 'seuo ?rloqredr(qapnlcur suorleurroJsusrl c[uoJpoxol eql airr '4ooq slql uI 'crldqe rou ttloqersd .raqltau sl ?l JI cruoJporol l€ql eurrrnss€ eq ol pres sr flrluepr eql lou s! qrlqa uorleurroJsuerl snrqory y'((oo'O] / V zV : ue,rrEsr aldurexauy 'cqoq.redfq rou 'cr1dq1a pu€ > I f) I lVl'C Q)L.{q 'cqoqered raq?reu are qf,rqra suorl€ruroJsu"rl snlqol are eraq+ teql aloN I 'I+y '0 = c t l o y , a d f i qq f ( U ) y e r u o s r o J z y u o r t e l l p e o l a l e 3 n f u o c s r l I ; r ( z ) o L < '(7 3 u) ttuT - (r)'L uorlelor e o1 ele3nluoc sr ycr,Ttlqla q t (I) ]I * 0'1g )f g auos tol zsp

'o+n'c>p

eurrros roJ a + z -

(z)ot uorlelsuerl e o1 ele3ntuoa s! 1l y cqoqotod q f (t)

'f1r1uapr eql tou $ qcrrlAr uorl€ruJoJ -suerl snrqontr e eq ,L 1a1 'sadf1 earql otq suorl"ruroJsusrl snlqotr tr fSsselc all

snlqg,trtrJo uorlBcgrssBlc '8,'8'z suorlBrrrroJsrrB.LT. suorl"urolsuerl snlqgl I 't'z

LT

38

2. Fricke Space

By a simple calculation, we see that Mobius transformations are classifiedby trace. Lernma 2.LL. Let 7 be a M6bius transfonnation which is not the idenlity. Then the following hold: (i) 7 is parabolic if and only if tf (7) = a. (ii) 7 is elliptic if and only if 0 f tf (1) < a. (iii) r is hyperbolic if and only if tf Q) > . (iu) r is lorodromic if and only if tf (1) e C - [0,4]. Finally, we define the axis of a hyperbolic real Mobius transformation 7. Suppose that 7 is conjugate to a canonical form U@) - )z with ,\ > 1, by a real Mcibius transformation 6. Namely, suppose that 7 = 6oloo6-1. The half-line L = {iy | 0 < y < oo} in the upper half-plane .Il is the geodesic,joining 0 and oo, with respect to the Poincar6 metric ldzl2I [m z)2 on I1 (see $3 of Chapter 3) . The image 6(L) of .L under 6 is called the oris of 7 and is denoted by Ar. Then ,4', is the geodesicjoining the fixed points r,, and a., of 1, which is characterized as a semi-circle which joins r., and o., and is orthogonal to the real axis. Similarly, we define the axis A, of a hyperbolic transformation 7 in Aut(A).

2.4. Fuchsian Models First, we show that a Riemann surface whose universal covering surface is not biholomorphic to the upper halfplane f/ is biholomorphic to one of e , C, C { 0 }, or tori. Next, we study some fundamental properties of discrete subgroups of Aut(H), i.e., Fuchsiangroups. 2.4.L. Riernann

Surfaces of Exceptional

Type

Let us determine Riemann surfaceswhose universal covering surfacesare biholomorphic to either 0 or C. Theorem 2.L2. A Riemann surface^R has a uniuersal coaering surface fr. biholomorphic to lhe Riemann sphereC if and only if R itself is biholomorphicto C. Prool. Assume that .E = e . Sitt"" every element 7 of its covering transformation group f is a M
'zry = tr @)rL ^q palereua3 sl J leql qcns Iy raqrunu a.Lrlrsode slsrxe a.req1'g uo flsnonurluorsrp dl.radord s+re J ecurs .y reqrunu aarllsod auos roJ zy - (z)L yr fpo pue yr o,Lqlyr e^rlelnuruoc fl (g)7ny ) L .19)WV ur uorle3nfuoc ,tq (g lueuala ue leql ^{oqs ol fsea sr < oy) zoy ,1 - (r)"L 'orToqred{q sr oL ,,no11 leql aunss€ {etu a,u uaql Wql asoddns 'rq z = (z)rL fq pale.reua3sl + J teqt q)ns rg requrnu a,rrlrsode s?srxaaraql 'g uo dlsnonurluoc$p dl.radord slce J a?urs .tI ) g auros .rogg * z = (z)L urroJ aqt ur uellrrA\ sl f ,,l1uopue gr o,L a^rl"lmuurot JI {tgl * (g)wV 3 l" luaurala u€ leqt ees ol fsea sr 14'(g)nV ur uorle3nfuoc fq > o g ) o q + z - ( z ) o L r y q 1 e r u n s s ef e u r e a ru a q l , c q o q e r e ds o , L ; 1 'cqoqrad{q @*'q'U

'H uo slurod pexg ou seq ol, acurg .ro crloqered sr o,L saqdurr etuure1 1eq1 0I'Z 'p! oL qtl,lr '{pgl1 o,L .too.t4 luaurale ue a{€tr * J 3 * J leqt arunsss,(eu e11 'ct1cfics, u?Vl 'uoqaqos? 'H uo snonut?uocsrp lN J II fr4.rado.rd s! J to uorl?o eW llUI q?ns puv g uo sTutoilpac{ ou sly {p?} - J .VTZ BururaT lo Tuaua1e fi.raaa Toqgqcns (g)nv {o dnolfiqns D eq J pI 'adfi7 TouorTdnr? lo eq ol pres sr rrol ro '{ O} - C 'C 'C Jo euo o1 erqd.rouroloqlqq qcrq^\ ac€Jrnsuueuer}I V ' .t / C ot cryd.toutoloqrq s! A pW q?ns J dno.tf ac47ol D slstr,? a.r,aq1'g sn"to7fr.tana"lo3r .z(.re11o.ro3 '3 o1 crqd.rouroloqlq sr snrol € Jo aceJJns D 'dno.r3erlc{c e aq pFoqs 3ur.ra.,loc I€srelrun e e)ueH J e qcns leq} slras$ qcrrl^r (p1'6 eurtual) eurural 3ur,rlo11og eqt s1?rp€rluoc srqJ .g. ;o dno.r3 FluauepunJ eq? o1 crqdrourosr sl J roJ (6 4uer;o dno.r3 uerlaqe aarJ € eq lsnur J uar{} ,.Fl o1 ctqdrouroloqlqq U JI'snrol e q g 1eq1asoddns'fleurg'C = U leqt ^rou{ a , $ ' I ' Z $ y o 1 e l d u r e f g u r u a e ss e A rs V . { O } - C = A 1 a 1 , } x a N. C = A 1 a Ba m 'palcauuoc 'C fldurrs =A fl ?eql aurnsselsrg,asra,ruoc aq1 lroils o1 C ecws 'flalrlcadsa.r 'snro1 e pue '{ '9 o1 crqdrouroloq 0} - C '(II) pue '(ll) ,(r) sasecur ,e.ro;araqa U ac€Jrnsuueuerg eql -lq sl J/C

-

'U re^o luapuedepur fpeauq er€ qcrq^r C f r g ' o g a u t o sr o y t g * z = ( z ) r L p r r s o g* z = ( z ) o l a r a q a r ' ( r t ' . t ) = J ( l l l )

'{o}-c)oq

oruos roJ oq*z

(sr - (z)ol uorlelsuerl e {q pele.reueErl ,('tl leq} J

--,t

(tr)

{ p t l = . t (r)

:(1'deq3 Jo Z$ '[t-V] sroJlqy ';c) sesecaerql 3ur,rao11og aql rncco ereql l"ql elo.rd uec e^{ ueql '(9'6 eruuel ;c) g uo flsnonurluocsrp '.ra,roaro1,1i 'g fpadord slc€ em uo slurod pexg ou seq {pl } -,f > f teql J [eeer fue asnereq'l = D ' r a q l r n g ' ( O " ' C g ' o ) q + z o - ( z ) L t u r o Je q l u r f f uellrr^,l\sl J ;l,L fre,re g'A"uure.I fq,(g)lnV;o dnorEqnse sl J acurg.dno.rE uol+€ruroJsue.rl Sur.ra,rocl€sralrun st! eq J lel 'C = U leqt aunssy '{oot4 sIePoI{ u"rsrlf,nJ 't'z

6t

2. Fricke Space

2.4.2. Fuchsian

Models

and f\rndamental

Dornains

The following is an immediate consequenceof Theorems 2.I2 and 2.13. Theorem 2.L5. A Riemann surface R has a unioersal couering surface fr, biholornorphic to H if and only if R is not ^of exceptional lype; that is, if and only if R is not biholomorphicloany one of C, C, C - {0}, ortori. If a universal covering surface E of a Riemann surface ,R is the upper halfplane fI, we call its universal covering transformation group I a Fuchsian rnodel of .R. In this case, f is asubgroup of Aut(H). However, identifying I/ with 4, we sometimes consider a F\chsian model f as a subgroup of Aut(A). Remark -1. By an argument similar to that in the proofs of Theorem 2.13 and Lemma 2.I4, we see that the fundamental group of a Riemann surface R is commutative if and only if .R is biholomorphic to one of C, C, C - { 0 }, tori, the unit disk .4, 4- {0}, or annuli {z e C | 1 < lzl < r}. In order to obtain a geometric image of correspondencebetween a Riemann surface R and its Fuchsian model f, we use a fundamental domain for f. An open set F of the upper half-plane 11 is a fundamenlal domain for f if F satisfies the following three conditions: (i) z({) oF = / for every 7 e f with 1 { id. (ii) If .F is the closure of .F in 11, then

a = [J ,r(F). 7el

(iii) The relative boundary 0F of F in H has measure zero with respect to the twodimensional Lebesgue measure. These conditions tell us that the Riemann surface R = H /f F with points on dF identified under the covering group l..

is considered as

Emmple y'. For each covering group l- in Example 2 in $2.1,we define similarly its fundamental domain. The following (i)", ..., (t)" give examples of fundamental domains for covering groups of (i)', . . . , (t)' in Example 2, respectively'

( i ) " r - { , e C | 0 < I m z< 2 r } . ( i i ) " r - { , e H | 0 < R e z< 1 } . ( i i i ) "F - { , e c - { 0 } | 0 < a r s z< 2 r / n } . ( i v ) " . F - { z e H l I < l r l< f } . ( u ) "F - { , e C l r - a } b r , 0 < a < 1 , 0 < D < 1 } . There is a simple way to construct canonically a fundamental domain for a Fuchsian model of a Riemann surface r?. First, cut -Ralong suitable smooth paths on ,R to get a simply connected domain Ro. Let .F be a connected component of

aa.rq13uo1e U ln? pue U 3 od lurod e e{e; 'tO pue 'zO 're self,rrc ea.rq1fq pepunoq 'aue1dxalduroe eql ut Ar ureuop e 'g'Z '3ld ur pal€rlsnll se 'raprsuo3

'8'Z'8tJ

'V I

'g alilu,oag

'z'z'Btr

€- )t

'6, u1 $ qcrqrr ureluop Fluarrr€punJ e ser{ J l"ql leeduroc f1a,rr1e1a.r smolloJ 'rslncur€d ul'?,'7,'3U ul patertsnll sB J roJ ureuop le]uetuepunJ e l.I sl ('U)r-l;o g luauoduoc pelf,euuoc V'oU ur€ruop pelrauuoe rtldurrse 1e3 aa,r 'lg pue fy 1yeEuop g,3ur11ng'od lurod aseq qlr&t salrn? pesolc alduns qloorus il€ arp {g pue fy trqt qans (6 l) d 6nua3 Jo Ur eceJrns uueruarg pasop " roJ (od,A)to Jo srol"reuaS;o ualsfs l"cruouec " nq,=f{ lg'lV } tet 'g a\duorg 'saldurexa 'fe,u, slql uI aa,rq1 eql u-r"lqo aal Eur.nolloJ 'J IoJ uleuroP IelueuepunJ e sI d slql 1eq1flrsea aeseAr'Z'Z$ul (A'y'A) Eurraloc lesJe^IuneJo uorl?nrlsuoc eql ,tg'.u detu Sur.ranoceql rapun tA p (;A)t-! a3eurr esra^ur eql sIePoII u"rsqf,nJ't'z

TV

2. Fricke Space

42

smooth curves Ct, Cz, and C3. By the same argument as that in example 6, we have a fundamental domain tr'for .R as is shown in Fig. 2.3. The elements Ir,''lz € l- corresponding to the elements [At],[Ar] E q(R,po), respectively,give a ca.nonicalsystem of generators of f. In $1.5 of Chapter 3, we shall describe another way of cutting .R to get a fundamental domain for this group.

Example 7. As a limiting case of Example 6, let each circle D; degenerate to a single point pi to obtain a Riemann surface,R biholomorphic to C-{n,pz,ps}. A Fuchsian model of .R is conjugate to the principal congraencesubgroup f(2) of leoel 2, which consists of all elements 7(z) - (o, + b)/(cz * d) such that a , b , c , d € Z , a d - b c = l , a n d a = d = 1 , 6 : c = 0 m o d 2 . A s a s y s t e mo f generatorsof f(2), we have fQ) = z *2 and fQ) = z/(22 * 1). The picture on the left hand side of Fig.2.4 shows an example of a fundamental domain for f(2). The picture on the right hand side of this figure illustrates a fundamental domain for a subgroup of Aut(A) conjugate to l-(2). For details, see Ahlfols [A-4], $2 of Chapter 7; and Jones and Singerman [A-48], Chpater 6.

----)

I

z-t

R

a fundamental domain in .F/ for

rQ)

a fundamental domain in 4 for f(2)

Fig.2.4.

Remark 2. As canonical fundamental domains for a subgroup of Aut(H) acting properly discontinuously on ff , we have Dirichlet regions and Ford regions. For details, we refer to standard text books such as Beardon [A-11], Ford [A-31], Jones and Singerman [A-48], Lehner [A-66], [A-67], and Maskit [A-71].

on <- (oz)uL I;:{ 'l q?ns sluatuela ecuenbase pu€ leql H ) lcurlsrp J Jo Jo } 'I1 uo dlsnonurluocsrp dpedord oz lutod e eAeq e^l ueql H ) lce lou saop J 'f1asra,r,uo3'uor?rugap eqt ,tq snor^qo sr (r) seqdu4 (ll) teql /oota leql aurnsse 'g uo filsnonur?u@stp filtadaul sTco J (II) 'uDtsq)nf, st (l)

.r

:Tuapatnbaa.ro6utmo11otayq (g)lny

.lI.U to tr ilno"t\qnso ro4

uraroaql

'a1e1duocfqaraq sl gI'Z eurruerl '(I) saqdrur D ;o;oord eql (r) ecuag 'elercsrp tou sr J '; go lurod palelofl ue lou sl J Jo p? luetuela lrun eql ecws pue 'u fue .ro;pg iL r+uLor_("f) p* J J I+ul,or_(u,[) acurg .Q1)lnV ul r-f o1 saS.rerluoctfi{ ,_("t) } W{t aes e,ra'suorleluase.rda.rxrrleur neql 3ur -raplsuoC '(U)t"V I ,L ue o1 saS.raauosr{rrq^\ J Jo sluauela ?cultslp luaurela '/'roN '(l) serldurr (rr) ecuaq I=":{ ul,} acuanbas ? s}srxa ararl} leq} arunss? Jo pue 'p1oq lou seop (rr) uorl.rasseeql ueql 'uersqcng lou sr J y'f1tea1a 'too.t4

'(11)wYuP

(s) (r)

efitaouocWyln J to sTuau,ep purlstp fi11onynu.t to sacuanbasou ?srseereqJ 'dno.t6 uvrsqcnf, o s! J .gT.Z BrrruraT lo ,7 dnol0qns D ro4 :Tuaparnbaato 6utmo71otaq7 (g)ny

'dno.t6 uvrsq?ng e se ol perraJer osle fl (V)l"V l" dnor3qns elercsrp '(C)lnV rc'(3)wV '(V)l"V;o dnol3qns elercsrp€ augep V jo Jaqunu elqelunoc e lsoru leJo slsrsuoc dnorE u€rsr1rr\f eira'f1.re1turg'sluaurala e '{1tlqe1unoe tuorxe puoees eql segsr}es (U'A)ZS ecurg 'dno.r,6uvrsycnl e Jo palp? q (H)InV;o dnorEqns eterf,srp O '19)tnV Jo leql uorJ pernpu! J uo f8o1odo1 e^rl"ler aql ol lcadsar qlrar slurod pelsloflJo slsrsuoc J ''a'l'Gt)lnv Jo lasqns alansrp e s.t J ! ?pr?erp eq ol pres sr.(11)7ny ;o .7 dnol3qns y 'oo ol sPuel u se 'dle,rrlcedsar'p pr. '" 'q'o o1 aE.rarruortp pue 'uc 'ug' ro yr fluo pue (U 'Z)I rrl JI S

fl il=,

ol saEre,ruoc

"rl

l"p t t " o l =o qll,!{ (U,Z)IS lo t;3{ "y } ecuenbasaq1 ,ara11.(U,Z)ZS;o f3olodol eqt,tq paenpursl (U'Z)ZSa;o fEolodol aql'I't$ ur ux\oqs $ sp uorl€rgrluepr eql repun (1g'Z)lSa dnor3 ar1 eqtJo euo eql 01 luale,rrnbasr f3o1odo1 slql'oo ol spual u * H Jo slesqns lceduroc uo I o1 .{pr.ro;run set.raauoc "L I (n)pV I I ol saEre,ruoc 1g)lnV Jo I=J{ u,L} acuenbas e teql sueeru srql '{to1odo1 uado-lcedruoc eql ''e'r.'(11)WV uo ,(Eo1odo1Iernleu e Surugap q1ral ur3aq e11

(n)pV

go sdnorEqns alarcsrq 'g'V'Z sIaPoII u"rsqsnd 't'z

t

2. Fricke Space

44

as n --+ oo. Since { r" }Lr is a normal family, taking a subsequence,if necessary, we may assume that { 7, }p, converges uniformly on compact subsets of f/ to a holomorphic function 7 defined in 11 . Flom the following lemma (Lemma 2.18), this 7 must be an element of Aut(H). Hence by Lemma 2.16, f is not F\rchsian, D and hence (i) implies (ii). Remark -/. For a subgroup f of Aut(d), the discretenessof l- does not always imply that it acts properly discontinuously on e . A typical example is given by (

az*b

.=ttQ)=*+d

a,b,c,d.eZ+iZ\.

Lemma 2.L8. Let { f" }Pr be a sequenceof Aut(H) which conaergesuniformly on compact subsels of H to a holomorphic function f defined in H . Here, f admits a constantfunction with aaluea. Then either one of the following holds: (i) / is an element of Aut(H). (ii) / is a constant function c with c € R. Proof. We consider the unit disk 4 instead of I/. Clearly, we have l/l S 1 on A.If lf(2")l = l for some point zo € A, the maximum principle implies that / is a constant function. Thus either l/l < I on 4, or / is a constant function c w i t h l c l = 1 . I f l / l < 1 o n ^ 4 , t h e n / b e l o n g st o A u t ( A ) . I n f a c t , { ( r " ) - t h a r being a normal family, taking a subsequence,if necessary,we may assume that it converges uniformly on compact subsets of 4 to a holomorphic function g defined in 4. In pa.rticular, we have gof = fd. By the same argument, we see D t h a t l 9 l ( 1 o n 4 a n d f o g = i d . H e n c e ,/ b e l o n g st o A u t ( H ) .

Proposition 2.19. Let f be a Fuchsian model of a closedRiemann surface of therc edsts a sequence g e n u sg 2 2 . F o r a n a r b i t r a r yp o i n t ( € R - R U { m } , conaerges to ( for any point zo € H. of f such that {6Q,)}f=t { f" }Lppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp Proof. Take a fundamental domain F for f such that the closure F of F is compact in I/ (Example 5 in $a.2). By the definition of a fundamental domain, of l- and a sequence{rr}f;=, of points in F we can pick a sequence{Zr}Lr is a normal family, we convergesto (. Since {'l'}Lr such that {l.Q^)}[, may suppose that {f" }Lt convergesuniformly on compacts subsets of ff to a holomorphic function f defined in 11. Thus Lemma 2.18 shows that / must be a constant (. e Remark2.Let L(f) be the set of all accumulationpoints of the set IlQ")lt l-], where zo is a point on f/. Lemma 2.18 implies that I(f) is independent of the choice of zo.It is well known that I(f) is a closed subset of fr.. We call I(f) the limit set of a Fuchsian group ,l-. iroposition 2.19 tells us ,(l-) = R provided that ft = H/f is compact.

,_(V)oV"V

uJoJ oql ur uellrr^\ sl "y lsql aas elt 'u raEalur errrlrsod ,(ue rog - rluV pus y - Iy Eurllnd 'ol,;o uorleluasardar xrrlpru e q qclqa

' f ] 9 l= ' n

[I II 'I tes e1l > lal > 0 qll,lr (1g'Z)IS 3 y uorleluasardar K:
t ? l"l sa{nqos # c to1t paprao.ul

' l' ^ ' ^l = ,

| - ?g- p D 1 g > p 'c'q 'o

Lq D)

'l uotToTuasa.til?J ruloru o 6unoq ) L frtaaau?qJ a z - (z)oL uotyo1 ,! -suDrln 0ututoyuoc dnafi uotsycri"l o ?q pI (lt1?,] nzturrqg) .IZ.Z BrrruraT J 'uorlrrpsrluoc € ol speel qcrqal'alarcsrp tr lou sr J '91'6 eurural ,(q 'aro;a.raqa'sluetuale --?{ l?qlqp Jo slsrsuor ? } leql ease,rl 'l + y p u e ' 0 < y ' 0 f g o a c u r g ' o o -t s u w C t s ? u a q l , I < y J I . o o + t s u s e

I r o l =' I'qt'; j "t ueqt'I > y > g y'acua11 o1 sa3ra.iluoc

o'l = ' I 'I | . ( , r ,- I ) e D; i arrrlrsod{ue.ro; u-VrBuVB 1eBea,r'u.ra3a1ur

, o* q e , u ) 9 , o

- ug 3ur11as'arrolq'flalrlcadsar

,

g] -

[,;,

, r + y, o < y , l ' ; .

i]

g

_V

dq uerrr3an g'L lo g'V suorle?uas '{ -e.rda.r xr.rleur ueql oo } = (g)*1.{ U (l)xrg pue { m'O } = (l)*tg t€rll eunsse feur aar 'uorle3nfuoc-(Hhnv fg 'sp1oq (rr) rou (r) raqlrau l€ql eunssv 'loo.r,4

.0 = (g)xr.{ u (r)4.{ (r) :sp1oy furmoilo! eyrlo euot"rl, 'plTg"r;r'::r:;"::, 'OZ'Z

s? L lI 'tr dno.r0uorsq?nf,o to sTuaualeonl ?q g puo L pI

BruuraT

'ralel pesn ar€ rl?rq,a,r, 'sdnor3 u"rsqcr\{ sarlradord auros luesard all ;o sdno.rg uBrsqcr\{ ;o sarl.radord Jaq}JqlI'V'?'Z sIePoI{ u"rsqf,nJ't'z

9t

2. Fricke Space

46

6".l "o" -- l o n d nJ ' l"n where o, = I - an-rcn-r, Thus

it follows

that

cr

bn = (an-!)2 , cn = -(cn-1)2, and d,. = I * a n - ( n - t . * = -"'n-t oo. Next, setting M = 0 as n *

max{ lol,l/(L - lcl) }, we obtain inductivelylo"l S M for any n. Thus each ant bn, and d, converges to 1 as n + contradicts the discreteness of l-.

oo. Hence, .Ar converges to Ao, which

tr

Theorem 2.22. Eaery element of a Fuchsian model of a closedRiemann surface of genusS (|=2) consislsonly of the identity and hyperbolicelements. Proof. Since every element 7 e f - { id } has no fixed points on I/, it is parabolic or hyperbolic. Assume that f contains a pa"rabolic element lo.By Aut(H)conjugation, we may suppose lhatT"Q) - z*1. From Lemma2.20, any element .y (+ id) of .i- with f(m) = oo is parabolic, which is written in the form = oo} is a ilz) = z *b for some real number 6. Hence, ]-- = { j e f I r(*) cyclic group. Replacing 7, with another element, if necessary'we may assume ad-bc= L, t h a t 7 o i s a g e n e r a t o rf o r f t . S i n c ee v e r y 7 ( z ) - ( a z * b ) l @ z * d ) , belonging to l- - ,i-- satisfies c + 0, Lemma 2.21 shows that lcl 2 1. Thus we obtain

rmTQ)S

1

1r-;pp

51

for all z with Imz ) 1. Set Uo - {z e H llmz > 2}. Then any two distinct points on [/o are not equivalent under any element of f - l--. Thus the quotient space Do = Uo/f* is biholomorphic to a domain Ro in r?. since 7o corresponds to a non-trivial element of the fundamental group of .r?,the closure E of R, in R is not simply connected. Since Do is biholomorphic to the punctured disk { z e C | 0 < l " l ( 1 } , w e i n f e r t h a t 4 m u s t b e h o m e o m o r p h i ct o { z € C | 0 < tr ltl S | ). This contradicts that R is compact. Remark. This theorem is also obtained by using the hyperbolic geometry discussedin $1 of Chapter 3. We present its outline. Let dsz = ldzl2l(Imz)2 be - H/f . the Poincar6 metric on 11, which induces the hyperbolic metric on R z * 1. For any positive number o, Assume that f has a translation 7o(z) denote by C" u closed path on ,R which is the image of the segment tro joining fo and 1o(ia) by the projection r: H -.R. Let l(C")be the hyperbolic length of Co,i.e., the length of .Lo with respect to the Poincar6metric. Then we see that t(C") - 0 as n + oo. On the other hand, .R being compact, we have a sequence + oo and r(l'o") + po as r, + oor { o" }L[r of positive numbers such that dn where po is a point on rt. Hence, if we take a simply connected domain u which contains po, then the closed path C,. is included in [/ for sufficiently large n. This implies lo - id, a contradiction.

,Hor{ro ,,u",u^ -,;,j"ii 1.j.,,i,",,; i";:;"Jn

'/G ol tuale^rnba sr (,<)Y r"qr q?ns /U * A : ! Surdderu crqdlouoloqrq / Utt V € slsrxe ereqt ueqtr '6Jul[g',A) = l€ql qcns uo,3 Bur4.retu€ pue lr [3',Ur] 6 snua3 Jo /Ar ec€Jrns uueruarg pesolc raqloue e{€I .6A.A uoxllsoilo.t4 lo !oo.t,4 'rldorloolgoln

= lld 'fo] eraqm

r=!

'P?= 't"]L[ lld

u uorrelaJ ,s.ro1o"raua6 uta7sfr,s Ieluau"punJ alos eq? sagsrl€s qcrq.lr to IoJ.uouDJ slr se ol parreJarrl I=;{ ld ' lo } s.roleraua3yo ualsfs eql .[3' ,g] acey.rnsuueruerll pesoll pe{r€ru e Io pporu uorsqrnl peztlDturou eq1 .7 dno.r3 u€rsqrnd srq} ilec e11\

fi.anbrun s?K otTcailsa.r-q7rm suortzpuor ,"n frriJE":Hti:'{:ri:::r"'r:;'Y;::

,(27)6 snuaf ppolu uvzstlcngo lo t=ou[!d'ln] s.tolo.tauaf lo ua7sfrslD?tuouD)p uant6 D rol .gZ.Z uorlrsodo.r4 lo g acottns uuDuery pesop D uo g 6utt1.tou, '(tt) p u n s u o r l r p u oc ( r ) uorlezrl€rurou aq1 f;sr1es ud pun to lr.I? etunsse.r[eura,ll ,r(ressaceu y,(g)Wy ul Surle3nfuor'.raq1.rng'Q = (6d)xllU(to)xrg seqdrur0U.Z€unueT ,arrrlelmuuroc ..,ir ud pun to acurg 'cr1oq.red,tq arc 6j pue to q1oq,Z7-(,tuaroeqtr dq ,1cegu1 lou ere 'suotllpuoc uor?ezrlsruJouaql segsrlss rIJlqA\u Jo lepo{u u"rsq)nJ 3 s}srxesferrrle araql 'f snuaS;o a)eJrns uu€urerg pesop e uo 3' 3ur>1reuuarrr3e Jod .tlJDurey '1 1e lurod paxg a^rlcerl?e sll ser{ t^a (I) ',{1anr1cadsa.r 'oo pue 6 1e slurod pexg a^r}ceJ}le pue 3ur11eda:s1 seq td (l) :suorlrpuoc uorlezrleurou aq1 esodurl e,!\ ,U uo 3 3ur4.reurue,rr3e o1 J Ieporu uersqrnd e ,,{lanbrunu8rsse o} rapro uI .ile,lr se Ur aures eqt Jo lapour uersqrnd e sl dnor3 eq1 '(g)wv fue ro; ,s1 ryql :(11)7ny go sursrqd ) ,.7 r_9J9 9 -rouoln€ .reuur{q pasnec ,{lrn8rqure egl seq ,lop A Io J Iapour uersqcnd e '6'"''Z'I = ! q c e e r o J ' f 1 a , r r 1 c e d s,e( .ord , A ) t o q [fg] pue [fy] o1 SurpuodsarrocJ Jo stueuela oq1 ld pue fo fq alouap ,9.6 utrreroeqlur pelels A lo J Iepou u€tsqcnd e pue (od,U)Iz uae.nleqursrqd.rourosr aql repun 'f snuaS;o U e)eJJnsuuetuerg pesol) e go (oa,g') t.u dnor3 plueur€punJ t=f{ aql srolerauaS;o uralsds q ''e'l 'U uo 3urr1.reu n rl Jo Ierruouec [lS],[!V]] 'f snuaE sec"Jrns uu€urarg pesol) pa{retu s}srsuo) 3' eraq^r Jo [3''U] II€ }o tg aceds rellnuqrrel eqt ,I raldeq3 ur peugap se^.rsv kZ) 0 snuaS;o Jo t$ 'QZ) 0 snue3go se)eJrnsuueuarg pesolc r; ecedsrefintuq)rel eql uo sel€urprooc e{]r.U pall€c Jo 'slapour u€rsqcr\{ sroleraueS;o ue1s.{s -os eugap fteqs ellr Jo l€)ruorr€c e Bursn fg

oreds a{rl{,{ '9'U acedg arpug '9'7

L?

2. Fricke Space

48

where {ot;,01 }f=, it the canonical system of generators of a Fuchsian model of .R' which iatisfiLs the normalization conditions with respect to ^D'. From condition (i), we have iQ) = )z for some l ) 0. Further, by condition (ii), a, and a! have the common fixed point at L, and hence ) = 1, i.e, i = id. Thus we get tr ai = ati and Bi - Fi. Lemma 2.24. Let {oi,gi}o;=, b" the canonical system of generators of the normalized Fachsian model i for a point lR, El in To. A an element t(z) = (az + b)/(cz + d) of {o;, |i}t=, do"t not coincide with Bo, then bc } 0. Proof. ln the case where 6 = c = 0, we have Fix(7) = Fix(Be) {0,*}, and hence 7 and Bn are commutative, a contradiction. Next, in the case where 6 = 0 and c * Q,we get Fix(7) = Fix(Be) = {0}. Thus, 7 and Bo being noncommutative, Lemma 2.20 implies that f is not F\rchsian. Hence we have a contradiction. By the same argument, in the case where b I 0 and c = 0, we tr obtain a contradiction. By this lemma, the canonical system loi,gi )f=t of generators of the normalized Fuchsian model l- for a point [.R,E] in To is written uniquely in the form

*, =

9i =

T#, atrz * b,,

ffa 6i,

ai di, ci € R,

ci ) 0,

aidi- bici= I,

o'i,our,C, eF., "'i ) 0' dili - fiCi= |

f o r e a c jh= 1 , i , . . . , 0 - , Now, we define the Fricke coorilinatesFo:Tn -* R6e-6 by f o ( [ R , l ] ) = ( 4 1 ,c 1 ,d 1 ,a l , c \ , d 1 ,. . . , as- | t ce - t , dg - r , a ' o- 1 ,c ' n -r , d ' s- r ) . The image Fn = fo(To) is called the Fricke space of closed Riemann surfaces of genus 9. The topology of Fo is introduced by the relative topology of Fo in R6c-0. In $2 of Chapter 5, we shall verify that.F, is asimply connected domain in R6g-0. By the following theorem (Theorem 2.25), Fs is a bijective mapping of ?o to Fo. Hence we define a topology on Q by identifying ?, with -Q under of a closed Riemann d. Therefore, a topology of the Teichmiiller space ?(.R) rest of this book, we In the g that of ?n. from genus is induced surface -R of topologies. with these are equipped "(.R) assume that Tn and Theorern

2.25. The Fricke coorilinalesfo: To ---+R6'-6 is injectiae.

= (or,"t,dv,...,a's-r, Proof. We need to show that every point fo(lR,t]) system canonical the uniquely determines Co-r,ils-1) in F, {oi,0i } of genpoint E)eTo. the Jfor model F\rchsian normalized eiatoriof the [8, i s o b t a i n e df r o m t h e r e l a t i o na i d i - b i c i : t F o r e a c hj ( j = I , 2 , . . . , 9 - l ) , 6 i BV the same with cr' ) 0, and hence oi is determined uniquely by fo(1R,4)' argument, 0i U = 1,2,. . ., g - 1) is also determined.

n

'(ls'al)U rq,u"

paururalap a^eq ellt.'aro;araq; 'ZZ'Z ureroeqtr slcrperluoc srql ureSy 'cqoqered sr ,L acuaq pu€ '6 = p D pug e,lr 'I = cq - pD uorlelel aq1 uror; 'snq; + leql ' V 6 ' Z e : U u . U u i e l f q 6 c a s n e c a q ' I = q + p a ^ e q e ^ , ru e q ? ' I = p a a OI ;r'are11

(erz)

.onP-I 'I-p+c

-0"

o-L I-q+D

san€ (2'6) otq (UI'Z) pu€ (II'Z) Jo uorlnlrlsqns

, D _ I - ocq -

(rrz)

.-P\ -al - -ro ^

(zrd_

'(Ot'Z) pue (9'6) uro.rg 1aBaan '16l pauru.ra?ap e^eqa^,recuag '(p - i/0 - o) - y pue 'I 'ZZ'ZvrarceqJq?rpsrluoc sSlJ 'cqoqeredu ,l * p'l * e ryql s^rolloJll snql qclq/'{'} - (f)zrl ecuaqpue'I = p ueq}'I = DJI'(p - I)y = I - p 1eq1sarldurr ol€q a^\ (O'Z) p* (9'6) uro.g 'qsrue,rlou saop 6c ro 6o Jo euo lseal le aculs

(orz) (o'e) (s'a)

.0=tp(I-p)+6qc '0=ta(r_y-p)*6ec '0=6?q+6e(t-o)

:ploq suorlenba 3uu*o11o;aql 1eql aurnsse .[eur 'p- po* '"- 'q- 'r- ,(q p pue 'a 'g 'o Surceldag arrr 'fressacau ;t '.,f1arr1cadser ' I - ? q- p D

=

,L 3ur11nd 'pl

=

'g)

, P + z c_ Q ) L 9tzo

p'e'q'o

'lld'fall=[U IeS'rldo6po6g= 6noLaltsqe/$ '.raq1rng lfd'!plr=!6lJ uolleler pluatuepunJ aql urorJ 'opa6c=6q+6o

Q'z)

eeuaq pue'I 1e lutod pexu e^rlcertle sll mq to'J roJ (ll) uorlrpuoc uorl€zllsruJou aql {g'I y q}l^{ zy = (z)6! el"q a.&t'J.loJ (r) uorlrpuoc uorl€zrl€ruroueq} /tg < '(g'Ul)ot,{q paururralap ers 6d p* to qloq }sq} q ^roqs ol sureuer leq1\ aredg e:1crrg'g'g

6'

50

2. Fricke Space

Notes For historical and expository accounts of the uniformization theorem, we refer to Abikoff [2], and Bers [29] and [36]. The original idea of using universal covering surfaces is due to H. A. Schwarz (cf. Bers [29], pp.264-265).Complete details of covering surfaces are contained in the books on Riemann surfaces listed in the notes of Chapter 1. The notion of a F\rchsiangroup was first introduced by Fuchs in the study of analytic continuation ofsolutions ofcertain ordinary differential equations ofthe second order (cf. Ford [A-31], Chapter XI). See also Yoshida [A-113]. For more details on F\rchsian groups, we refer to Jones and Singerman [A-48], and Lehner groups, [A-66] and [A-67]. Discrete subgroups of PSL(2,C) are called Kletnian which are intimbtely related to the theory of Teichmiiller spaces. It is most regrettable that this interesting subject cannot be covered. Concerning Kleinian groups, see Beardon [A-11], Berset al. [A-15], Ford [A-31], Krushkal" Apanasov and Gusevskil [A-61], Lehner [A-66], Magnus [A-70], and Maskit [A-71]. For relation between Kleinia"n gloups and 3-manifolds, we also refer to Epstein [A25] and [A-26], Fathi, Laudenbach and Po6naru [A-29], Morgan and Bass [A-76], McMullen [154], and Thurston [231]. Poinca"r6[A-90] is his collected works on Fuchsian groups and automorphic functions. For the interaction between ergodic theory and discrete groups, see Nicholls [A-86], Bowen and Series [47], Morosawa [158], Series [195], and Velling and Matsuzaki [241]. Fricke spaces first appeared in Fricke and Klein [A-33]. For modern treatments, see Abikoff [A-1], Goldman and Magid [A-36], Bers and Gardiner [42], Keen [110], Saito [186], and Weil [243]. For a representation of a Riemann surface' we can use a .9cfioltky group instead of a F\rchsian group, and we obtain a schottky space instead of a Teichmiiller space. This topic is discussed in Bers [35], Hejhal [98], and Sato [188] and [18e].

: t |utddoru ctryl"toutoloyfr"taag (wlo ru:al s6{crd-z.re,*qcg)

sa{s4os V V '1'g uorlrsodor;

'zz pup- Iz uee/r leq ??uDlslp?rDourod eq1 (zz'rz)d llet e^\.aclr€lslp Jo $uorxe aql seuslles d ryqI u^roqs sr.Il'zz pue rz ?ceuuoct{clq,$ 7 ut se^rnf, elqeul}leJ II€ sa^ou 3 'arag , l 'zi l' - t 'o i'

rf c / - l u' = l (zz'tz)d

lzplz J

'drlauroaS ueaprlcng-uou

'V tlz slu-tod orr,r.1fue log ) zr las alrlr eJo lapotu e lrnrlsuoc ol clrletu stt{l pesn ?J€culod 'H

. z Q l '-l i = "tP "1'P1tr

xuleu 9rv?utod eq} sr euo luelrodrur leqloue pue 'rftp * "*p = ,ltpl -- es'pclrletu ueepqcng aql sl ureqlJo auo'srrrla{u ..letrnleu,,Iere^es seq {I > lrl I C ) zI = 7 }tslP }Iun eqtr crrlatr l ?rBcurod 'T'I'8

l(llaruoag

rrloqrod/tH

pue rlrlatr tr ?rerutod

'I'g

'uolsrnql 'A\ fq pasodorddlluacars€^rqcrq/!\'acedsrellnurq)Ial aqt uotlecyrlceduroe Jo q)le{s e e,rr3 aaa.'p uorlcag ut 'fleutg elq€1oue uorlf,nrlsuot eql Jo Jo 'sq1Eua1 'urely pup a{?tJd suorle3tlsa,rul Jo Ieclsselcut ur3trosl! seq qctq,lr crsepoe3go sueeru,(q acedsueaprlcngue olur acedsrallntuqclel eql JoSutppequa ue ssncsrpa,n 'g uorlces uI 'e?€Jrnsuu€ureru pesoll e 3o ecedsrallnuq?Ial 'saleurprooc uralsdse auuaPe1'r aql uo 'saleurp.roolueslerNleq?uad ;o Pallec 'd.rlauroaE esoql fletcadsa crloqradfq 3urs11'scrsapoe3 Sutu.recuof, 6 uorlces ur 'serlredordcrseq{pn1spue elrlau ereculodaql eugepaiu.'1uorlcagut'1s.rtg '{srp uuetuelg }run eql uo crr}eruersculodaql dq pecnpulfl tlclq^rseceJrns uo frleuroa3 cqoqredfq aq1;o slcedseaurosssnc$pII€qsa^\ 'reldeqc sql uI

salBurProoc uoslarN-Iaqruad puB rt.rlauroa.D rrloq.radfll t raldBrlc

3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

52

lf'.(:)l

=. - l " r I - l f ( r ) l ' = 1+,

z € a.

Moroaer, if the equality holds at one point in A, then f is a biholomorphic aulornorphism of A, and the equalitg holds at any point of A. Proof. Fix a point z in A arbitrarily, and set w*z .tt\w) =TlZw,

/ \ w-f(z) lztwt=, -fz1w' Then 71 and 72 belong to Aut(A), and ,F(to) = J2 o f " lr(w) is a holomorphic mapping of 4 into 4. Since .F(0) = g attd 'l - l,l2

F,(o)= ffif'(r), tr

we have the assertion by Schwarz'lemma.

When we denote by /-(ds2) the pull-back of the Poincard metric ds2 = aldzl2/(l - lrl')' by /, Proposition 3.1 implies that f* (ds2) < ds2 and that I.(dt2) Corollary.

- dsz if and only if belongs ro Aut(A). /

Eaery holomorphic mapping f : A ------A satisfies p(f (zt), f (zzD 1 p(21, z2),

21,22 € A.

Remark. In general, the Gaussian curaalure /{(h) h ( z ) 2 l d z l z( h ( r ) > o ) i s g i v e n b y

of a Riemannian metric

4 fl2logh

,h(h)=-Fd

A simple computation shows that the Gaussian curvature of the Poincar6 metric is identically equal to -1 on 4. Moreover, we can see that, when a metric h(z)2ldzl2 is invariant under the action by Aut(A), it is coincideni with the Poincar6 metric, up to a constant factor.

'y otuo H lo (! + z)/(? - ,) = (z)1,uorleurroJsuert snlqgl i eqt ,,(qv uo zspcrrleru ?r"curod eql Jo {req11nd aql 1nq3mq1ouq qclq,tr

, z(lu'l) = ,rrp

"l'Pl

3ur11as,(qpaugap sr g aueld-geq .raddnaql uo {sp crrlaur ar"curod eq&'tlrout?[ ''V LV uo slurod oall i(ue Eurlcauuoc crsapoe3fre,ra'6'9 Jo rr"qns e q uorlrsodor4 ,tq 'teql a?ou eJaH ',0 fq uorlce eql rapun luerrslur sr f,y uxe aq; 'L p'V sDr€eql palpc sr Ve oI 1euo3oq1.ro sr pue slurod asaql q3no.rq1sassed qcrqa,llluaur3as auq eql ro alcrr? eql Jo y u1 lred aql l"ql Ip?eU 'Vg uo Lo prte L.r, slurod paxg Irurlsrp otrl seq l, 'crloqredfq 4 (VhnV 3 ,L uaqal '1eq1 lpcag

'lzz'o)

tr -

7 luaur3es euq eq? qlr^r lueprcurof, sr Cyr fluo pueJI (rh .2, I xp7,

= (zz'1)d acuag

,W"[ ",ol

eleq era 'zz pue g Surlcauuoo trts pesolc frala ro;'r".II C 'U f d elq€1lns qYal. (V)7nY P zlz -t

#eP

= Q) L

'0 1 zz pue = rz luauela ue fq tuaql Sururrogsuert /tq 1eq1 etunsse deur 0 en'(V)1ny fq uorlce ar{l repun lu€rrelur sr crrleru ar€curod eq1 acurg /oo.l4,

'v lo

sl puD zz puo rz q0norqt sassodt1cn1m7uau,6as Vg fi.topunoqeql oI 1ouo0or17.to euq eql ro el?Jr?eyyto ctoqns o s, puD anbtun st 7r |teaoanory'V ul zz puo rz |utTcauuoc crsapoa0o slsNaeere1l 'V ) zz'rz fi^to4tq.toro,I 'Z'g uorlrsodor6 '(C)l = (zz'rr)d eleq e^r JI '9z ul zz pve Iz $ullcauuoc (cr.r1eru gr€oulod aq1 o1 lcadsar q1ra,r)ctsapoe0e 'V ul zz pue rz Surlcauuot'g cre pasolc elq€Urlcar€ IIef, e \'V ) zr 'rz s?ulod orrr1fue rod'(r)/ fq 1r elouap pu€'C p y76ua1cqoqtadfr,tlaql sp "[ lV, ell.'V ur , ]re pasolc alqegrlcer fre,ra rog

scrsaPoaD 'z'T't ,{r1auroag cuoqradifll pu" f,rrlel{ gr"f,u-Iod 'I't

t9

3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

54 3.L.3. Hyperbolic

Metric

on a Riemann

Surface

Let it be a Riemann surface whose universal covering surface is biholomorphically equivalent to 4. Consider a Fhchsian model I of ,R acting on /. Let r: A r? be the projection of 4 onto R = A/f. Since the Poincar6 metric ds2 is invariant under the action by .l-, we obtain a Riemannian metric ds2pon R which satisfies r- (dszp)= d,s2. We call this dsft the Poincar6 metric, or the hyperbolic metric on R. Now, every 1 e f corresponds to an element [Cr] of the fundamental group r{R,po) of ,R (Theorem 2.5). In particular, 7 determines the free homotopy class of C7, where C,, is a representative of the class [Cr]. We say that 7 coaers the closedcurue Cr. When j € f is hyperbolic, it is seen that the closed curve -t, - A-, I 1l ), the image on .R of the axis A, by zr, is the unique geodesic(with respect to the hyperbolic metric on ,R ) belonging to the free homotopy class of Ct. We call L-, the closed geodesiccorresponding to 7, or to C.,. Proposition 3.3. Lel R be a Riemann surface with uniaersal coueringsurface H , and 11 be a Fuchsian model of R acting on H. Let

tk) =

az*b cz*d'

a , b , c , de P - ,

a d - b c= 7 ,

on R corresponding elemenlof 11, and L, bethe closedgeodesic be a hyperbolic to 7. Then the hyperboliclenglhI(Lr) of L, satisfies

t.'(r) - @+d)2= 4cosh2 e) Prool. Since t(L-r) and tr2(7) are invariant under the conjugation of 7 by an element ol Aut(H), we may assume that 7(z) - )z (.\ > 1). We may also this case,we have assumethat o - t5, b = c =0, and d = I/\5.In

((L.t) = Ir^ + = log) = 2log a. Hence we have the assertion.

!

3.1.4, Pants Consider cutting a Riemann surface r? which admits the hyperbolic metric by a family of mutually disjoint simple closed geodesicson R. Let P be a relatively compact connected component of the resulting union of subsurfaces.If P contains no more simple closed geodesic of .r?,then P should be triply connected, i.e., homeomorphic to a planar region, say

'd louoNeuelseueslerNaql Pellet sr d Pu€'d Jolaweq ueslerN erll Pall€l 'd st 2r 'f11en1rq€H'-dJ fq paurunalap flanbiun ,, j '.pto^ reqto uI Jo slued ',,(pee13 '2' 3o aee;rnsqns € s'e pereplsuoc sl d ;o rred anbrun eql s! plq^{ 'pelcauuoc ,{1dr.r1 '(t'g '31.{ eas) ; Jo lapour uelsqcqil e q d.7 pue 'acue11'lueuodtu6r ,t.repunoq qcee Suop uorSar palceuuof, .{1qnop ure3e sr 2' paul€tqo areJrns€ sI uaql 'a,t/V - d las elq€lrns e Surqcelp fq dr 4'uro.r; ql/d d pue'suotleurrogsuerl crloq.rad,tqom1fq pele.reue3dno.r3aa.ge s1 d.7 uaql 'd = ({)t }€tI} q?ns J Jo L dnor3qns eqt dJ fq alouaq '(d) r -! Jo lueuoduroc sluetuele yo Surlsrsuoc J Jo 1e = U aq1 aq pelceuuoce aQ V i )L pve'7 uo 3ur1ce JIV d 1e1'uorlcelord 'fpre.r1rqr" g 3o 7 slued ;o .rted e xtg U Jo lapour uersqcqE e aq J 1e1

'r'8'tIJ

4J/V:d

'U uo rlsapoe3 pasolc elduns € sl 2I ul d Jo frepunoq e^rleler eqt Jo luauoduroc palrauuoc frarra 3r Pue Palceuuoc i(1dt.r1q d JI g' 'g e IIef, e^\ '.re1;eara11 1o sTuod;o .rted e U Jo d eD€Jrnsqnslcedtuoc f1errr1e1e.r Surppnqa.r.ro; secerdlseilerus eql Jo auo s€ pereplsuoc aq ue? d et"Jrnsqns e qcns

({i t rr- ",}^{i >rr+,r})- { z > l z l } = 0 4 frlauroa.g rqoqraddll

cc

'I't Pu€ f,rrlel{ grsf,urod

3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

3.1.5. Existence

and Uniqueness of P'-ts

We shall discuss the relationship between the complex structure of a triply connected domain J? and the hyperbolic structure of P, the unique pair of pants of O, induced by the hyperbolic metric on O. Let L1,L2, and..L3 be the boundary components, which a.re simple closed geodesics, of the pair of pants P. Let J-e be a Fuchsian model of the domain O acting on A. Then'i-s is a free group generated by two hyperbolic transformations, say, 7r and. 72. We may assume that 71 and 72 cover .L1 and L2, respectively. Theorem 3.4. For an arbitrarily giaen triple (ayaz,as) of positiae numbers, lhere erists a triply connected planar Riemann surfoce Q such that t(Li)=a1,

i-1,2,3.

Proof. We prove it by constructing O explicitly. Let Cr be the part of the imaginary axis in A. Fix another geodesic, say C2, on 4 such that the Poincar6 distance between C1 and C2 is equal to a1f 2. On the other hand, geodesics on 4 from which the Poincar6 distance to C1 are equal to oBf2 form a real one-parameter family (i.e., the family of circular arcs Cl tangent to the broken circular arc in Fig. 3.2-i)). Hence there exists a geodesic, say Cs, in this family such that the Poincard distance between Cz and Cg is equal to a2f2.

Fig.3.2.

Next, let 21 arrd z2 be the points in 4 uniquely determined by the condition

p(rr,r) =

?,

zr e Ct, zze Cz.

Let L\ be the geodesic connecting 21 and 22. Similarly, let {z3,za} and {rs,re]1 be the pairs of points uniquely determined by the conditions

feur e,r. 'r(.ressacau.l uorleEntuot-(g)7ny ue 3ur:1et 'asodrnd srql lsrll eutrrnssp rog'r={{{o},(q peururralap,{lanbrun are zL pue I,L }eql ^\oqs o} seclsns U 'e? 'I zL) eL sraloc o teql pue (e = 1) 17 s.rairoc{1, ?sqt arunsss feur aa,r ,_(tf 'ara11'0.7 srolereuaS;o rualsfs e aq 'I.L} 'g eueld-y1eqraddn eq} uo leT Jo {zt 3ur1ce Jo lepour uersqr\{ e eq 0J pu€ 'd Jo uorsuelxe ueslarN eql eq d ?c,-I'd d Jo (g'Z'l = f) f7 lueuoduroc frepunoq aq1 ;o q1Eua1crloqlad,(q eql eq lp p"I 'fy.rer1rq.reuaarEsr sluedgo ned s 'too.r4 teql esoddng ('IIt'[Ott] uaay'93) 2, '4 perepro aq1lo st176ua7 cqoqtedfrq aqy fiq lo sTuauoduoefi".topunoq p?aunrepp fryanbrunsa 4 syuodto .ttorl o to ernlrtuls aelilutoc ?ttJ .g.g ruoJoaql '(g'g '81.{ eas) ace;rns peilsap " sr snqtr '0J ,tq uorlce eql rapun (O)tbnO tr {Ji go ,trepunoq eq1 Surf;rluapl fq paurclqo ?as eql Jo rorrelur eql sl (J ;o 2, slued yo rpd anbrun arl? lstll pue 'pelceuuoc f1du1 sr.oJ /V = U leq+ realc s-rlt

'8'8'ttJ

pele.rauaEdno.rEaq1 aq o.ir1e1 '(V)l"V

.z,L pue rl, aseqt fq t" sluauele arloqrad,tu are zl, pue rl, ueq,L

'Vtoeb-zL

tebsV)-rl-

'eslrlurod fg Eur,rraserd les 3 go ursrqdrour ''e'l'f^2 -o1ne crqd,rouroloq-rlu" aqt qlr^r o1 uorl?auer eql;q (g'Z't = f) laadsar 't=[{!,1'!c} lh p1 pepunoq uo3exaq cqoqrad,(q pesolt eq? eq o p"r fq ('(fa'g 'ft9 eag) 'ez Pue ez Pue'?z pue 8z Eurlceuuoc scrsapoaSeq1 'f1a,rr1cadse.r'!7 pve 1I ,{q alouaq 'flelrlcedser

'rC)sz'eC)sz

, 7 , = (sz(s2)i tp

' 8 8 8 8 8 C ) v z ( z C ) ,E zz - ( v z ' e s ) 6 Z9

L9

i(rlauroag ruoqradi(g

pu" rrrlel l ?r"3ulod 'I't

58

3. Hyperbolic Geometry and Penchel-Nielsen Coordinates It(z)=\22, 0<)<1, / \ az*b .r2(z)

=;ii,,

ad- bc- I,c ) 0,

and that 1 is the attractive fixed point of 12, or equivalently,

alb=sai, O<-!
Then we see that fz(m) = a/c ) 0 and a + d > 0, since the middle-point (a - d)/(2c) of two fixed points of 72 has a value less than Zz(oo). Next, write dz*b (, 7 3\ -) 1- ',( ,z=) - u u A ,

ad-bz- 1-

Since (73)-l = j2o 7r, we may assumethat

6,=a\, 6=b1\, E-c\, d=a1>,. In particular, d > 0. Moreover,the middle-point(6 - a11pe1ofthe fixed points of (73)-1 has a valuegreaterthan (73)-r(x) = d/8. Hence,a + d < O. On the other hand, by Proposition3.3, we have

() + l/r)' =4cooh2 (+) , (a+ d)2=4cosh2 f\ +) 2 / ', (+) @+ J:2=4cosh2 \ 2 / Therefore,'y1 and j2 are uniquely determined by {or, az,as}.

tr

We have proved that, for any triple of positive numbers, there exists a pair of pants admitting a reflection (induced, for example, by rlr) such that the hyperbolic lengths of the ordered boundary components are the given triple (Theorem 3.4), and that it is uniquely determined by the given triple (Theorem 3.5). Thus we have the following corollary (see also Fig. 3.4). Corollary. Eaerg pair of pants P has an anti-holornorphic automorphism Jp of order two. M o r e o a e r , L h es e t F r r = { z e P l J p ( r ) = z } o f a l l f i x e d p o i n l s o f J p c o n s i s t s of thrce geodesics{Di}|=, in P satisfyingthe following condition: For eueryj (j = L,2,3), Di has the endpointson, and is orThogonalto, both L i a n d L 1 ; r , w h e r eL + = L t . We call "Ip described in this corollary the rc,fl,ectionof P.

.6J uo selDuzpron ueslely-leq?ury pelle;. sI sel€ulprooc;o rua1s.{s€ qcns 'sacerdaql an13o1 pasn sralatuered 3ut1stall pellec-os arll Jo las aql pue slued olur uorysod -ruocep e^oqe eql ur pesn scrsapoe311e;o sqlSualJo les aq1;o.ued aq1 'rg aceds rellnurqcral aql roJ sel€urproo?;o ue1s.{s e se '.reptsuocue, a^\ 'ecue11'f1qe1tns secard3ur11nsarIIe 3urn13fq palcnrlsuocer sI U leql r€elc sI lI 'g'g ura.roaq; ,{q 1r Jo scrsepoe3,{.repunoqyo sqlEual cqoqradfq eql Jo a1dtr1aq1 fq paunurelep .{lanbrun sl g. slued ;o rred qcea Jo ernlf,nrls xelduroc eq} }eql II€reU Jo '1as uado Surureurereql ul 'p. yo qued 3o .rrede eq plnoqs ecard drarraueql paureluoc U Jo s)rsepoa3pasolc alduns eroru ou are areql uaq1yg, uo {sp f,Ir}aur flenlnur 3uo1ey crloqrad.rtqaq1 o1 laadser qllr* srtsepoa3pesolc aldurts 1u1ofs1p 3ur11ncraprsuoc'eroyaq sy'(e {) f snua3Jo af,eJrnsuu"ruerg pesol?e aq Ur lerl uorlrsodtuocaq

slusd'1''Z'e,

'.re1deqc$r{} Jo rapulqurer aq1 ut fleeq suollJasseeql esn 'g .ra1deq3 lr]un rueroer.{lsqtJo;oord e Sur,rr3auodlsod a6 [eqs e,lr q3noql 'g-fgll o7 ctr1d.r,ou.to?uo?! puD e-oell a! uzD'tuopD s! 6l acods aqu.r,treqJ (g1'g uraroaqa) 'uraroaql s.rallnurqtraJ '.{1e,rr1rn1ur reql€r 1nq 'o3e aurrl 3uo1e pelrecuof, sehruaJoaql Euro,o11o; eq1 're,roarohtr'e-6etlJolasqnse sl tdr ecedse4crrgaql'flsnoue.rd pa1e1ssy '.{rlaruoa3crloqred,tq Sursn ,,lq 6g o1 saleurproo) Jo ad.{1raqloue acnpor}ur a,r. 'uotlces slt{l uI 'saceJrnsJo slepotu uelsqcnd Sursn ,{q 'eceds ueeprpng tg lasqns l€uorsueurp-(S-0g) learJo (eceds e{clq aq1 parueu) (Z ?) f snua3 ;o tg aceds rellnuqtlal aql peluasardar am,'6 reldeqS u1

* *

sa +B ur P r o o cu a sla r N{ a q r u a J' z' 8 '?'8'ttJ

sat"urProoc uaslarNlaqruad'z'8

60

3. Hyperbolic Geometry and Fenchel-NielsenCoordinates

Now,. we grve a precise definition of these coordinates, and verify that they give a system of global coordinates on To. For this purpose, first fix a point [.R,.D] of ?r. A set 4 of mutually disjoint simple closed geodesics on .R is termed matirnal if there is no set 4' which includes 4 properly. We call a maximal set .C = {fi}j!, of mutually disjoint simple closed geodesics on rt a syslem of decomposing curaes, and the family consisting of all connected components of R - UltLi the pants P = {PxW' decompositionof R corresponding to L. Emmple. When g = 2, there are the two kinds of pants decompositions shown in Fig. 3.5.

Fig.3.5. be an arbitrary system of decomposing Proposition 3.6. Let L = {li}l=, curaes on a closed Riemann surface R of genas C (>-2), and letP = lPxlf=t be lhe pants decomposition of R concsponding to t,. Then M and N satisfy N=3g-3

and M=2g-2.

Proof. Cut r? along an element Ly of L. Let n1 be the number of connected components of .R- .t1, and 91 be the sum of genera of all connected components of R - tr1. Then we have 9r-nr=(g-1)-1. Clearly, the number of boundary components of ,R - -t1 is two. Moreover, we can see inductively that, whenever we add a cut along a new element of 4, the number of boundary components increases by two, and the sum of genera of all connected components minus the number of connected by one. Hence, we have components decrea^ses

gM __2N and 0_ M =(g_1)_N,

.l.r!t!"^ -= (7)!6 (t)!t-"

1ag'(1ou rc (7)17 Jo 1eql qlrar alqrleduroc s1 (3)fuJo uorteluerro eql raqleqa ol Eurproace a.,rrleEauro a,rrlrsod sg (3).r.r.1eq1 os) (l)l,l,l" (l)f, qt3n"t cqoqred,tq pau3rs eql eusap ust e^l 'lI lo l"ql uro+ peururalep uorleluarro l€rnl"u eql seq (l)fZ acurg'(3)z'fc o1 (3)t'fa urory (3)f7 uo f,trepatuerro eqt aq (t)fap1 .(ic'ta Jo uorlcagar aqlgo lurod pexss sl (U'I = t) (1)t'!c qcee uaql'(l)f7 uo Ot't1;o lurod aq1 (ic'ft fq pue '{t3ua1 prurunu qtyn (1)r'12,ur (7)t'!7 pue (1)f7 Eururof crsapoa3 aq1 (1)r'fc, i(q alouaq 't'17 ol Eurpuodsarro) (i't'!d;o luauodruoc frepunoq eq1 eq (7)t'!7 p"l'1'14 u1 tr'!7 fes 'luauoduroc f.repunoq raqloue pue !7 Sururof r'f6. crsepoe3 aq1 yo f7 uo lurod pua a{} sl (Z,I - q) t'fc qcee }€ql IIeceU 'f1a,rr1cadsa.r 'z'!4 pue r'14 o1 Surpuodsa.uo?(tU;o slued a.re qcrq,u,)(l)fZt=Jn - ? Jo sluauoduroc palcauuo? aql aq (iz'fa pue (1)t'f4' 1a1'f, pue l fra,re rog ''d.rl 'a.ro;aq ';;o sy I {rara.rog 1o1 Eurpuodserroc lurod aql aq l,3'rA) 1a1 '(g'g'3t.{ aas) f7 uo uol}sluerro us osls t1 "g'r'fa,tq = {) r't, qcea rc1 !7 uo Y Jo lurod paxg " e{"I 'g'g ureroaqJ ^q 'flarrrlcadse.r 'zf pue r/ uorlcager aql lrurpe z'!4 pue t'14 - r'.r2'a.raq,r,r es€l aql ,&roll"e^r areH'lueuoduroc drepunoq e ul slued;o s.rred oiu.1aq z'!4 pue r'fd 1a1'/ fra,re roy 'lxaN

alouep pu" '(e'I o1 fre11oro3 eql 'z'f4 leql II€lsU g" f7 Eul\eq d

sralaruerBd Eullqar;'g'Z'g

'6d uo cr1fi1ouo1oa.r, st (l)h uo4cunt y76ua7nsapoa| fi.taag 'Z'g BuruxaT

serTdur ueqy t7 rc1uot1cun! 6'9uorlrsodord vtfuq ?*?po;fflttftTopH iU

uo 'flluelerrrnbe.ro) 6g uo uorlcunJ € se (1)f7 raprsuoc eM'(ib fq fldrurs Ol7 p qfual cqoqrad{q eq} elouep eiu'f l(ra,repue td ur 3 itre,re ro;'aro11 ((l)!l)l 'f, fra,ra q (!1)tg sreloc qcq/rt ,J Jo luatuala u€ Jo slx€ eq1 ;o uotlcafo,rd aq1 sr (1)f7 leq? eloN'1fq paluesardarrgrJo lepow u"rsqcr\{ eqt aq tJ ta1 .rar uo salrnc Sursodruocap;o uralsfs e sr t--;f{11;t1\ = rj'eoue11 ',1 + | uaqar lurofsrp .r(11en1nur ere (7),17 pue (3)f7 l€rl? pue 'eldurrs sr (1)f7 ?€rll aorls ot llnclgrp tou '? uo (!7)r1 a^rne pasole arlt q ss€p fdolouroq eerJaql ur crsapoe3pesolc lI Jo enbrun aql eq (l)fZ f"t 'J ul !7 {ra,re rog '(y'1 uraroaq; 'Jc) tU A : tt 'tgr uo selrnf, Sutsodurooap ursrqdrouroauroq Eur,rrasard-Eur4retue alel tr(1arue11 I=J{(t) l?} = ,J urelsfs e ,tlanbrun aunurelap uec er$'t".II '? o1 Eurpuodserroc Jo t; ur "U] ,tq alouep e$'6,tr a*ds aqcrq4eqt ul t fra,ra rog '3. uo aqt lurod lt3' '6J selrnc Eursodurocepf" tlf{fZ} 7 ure1s.{se pue lo [g'g,] lurod e xld

suorlcur\{ q1Eua1 crsapoaD 'Z'Z't

'uorlrasse aql ,(1dun qcqaa u3,{'z't seleuProoc uaslarN-Iaqf,

I9

62

3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

Dt,z

D;,r

(P.;,1: Pi,z)

Fig.3.6. Then 0i(f) is well-defined modulo hr. We caII01(t) the twisting parameler with respect to L1. Lemma 3.8. For eueryj, exp(id1(t)) is well-definedand real-analylic on Fn. Proof. Ftx 1. For every f in .Fr, let fi be the Fuchsian group represented by t. Take an element of 4 which covers tr;(l), and denote it by 71(t). Next, for each &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&(= 1,2),Iet be the element of which covers L1,x(t) and [ 7i,x(t)

satisfes

that

the geodesicDi,n(t), connecting ,41(f) and A1,*Q) with the minimal length, is projected onto Di,*(t), where,4i(t) and A1,r(t) are the axes of 7i(l) and T,x(t), respectively(see Fig. 3.7). Here, we may assumethat the fixed points of 7i(t), ti,t(t), and 7i,2(t) move real-analytically on d. Hence, when we take aconjugation of ,Q by an element o f A u t ( H ) s o t h a t 7 1 ( t ) g o e st o f u ( t ) ( z ) = . \ j ( r ) . z ( \ ( t ) ) 1), the fixed points of fu,*(t) corresponding to 7i,1(t) move also real-analytically on Fo for each #. Now, c1,1(t) is the projection of the end point Zi,r(t) "t fu,x(t) to A1Q). Ifence, if we show that 6i,x(t) moves real-analytically on Fr, the assertion follows by the definition of 11(t) and Lemma 3.7. To show this, fix /c, and let p1 and p2 be the fixed points of ii,i(l). Set c13(t) = iv* (v* > 0). Since

, '7*(ry)' =(o'to')' we see real-analyticity of ci,r(l).

.Uo{lqy

';c) ursrqd.rouroe$lpe sl g_ogtl x ,_rg(ag)

-

.([tqz] pue ,h-yl 1.rad1o44 6l : 'dgenlcy 'IrDur?A 4i

'(sa,r.rncEursodurocep;o uralsfs eql qtyr\ 'to) d uotltsodtuocap slued aql 7 rl?-rrA paler)osse t;;o selouxprooxueslery-Ieq)uef ^seler;rprooc asaql II€r elA .6a uo uI 'e-oellX nuaq puD'6,I uo sa?Durproo? 7oqo16 lo ue7sfrro saat64i'.to7nc4.r,od 'OI'8 tuaroaql 6g to rusttld;oruoeuoq e s? 4 6utddnu, s?ttJ e-re(+U) oTuo 'le,roelotr41 'uaroaql 3uralo1o; eq1 erro.rd,r.ou ilBqs aru 'uf uo ctTfipuo-pat st ( ( l ) " - u " a ' . . . , ( t ) r0 , ( t ) e - u e T. . . , ( t ) r t ) = ( t )r t ueqJ '! tuaaa ut 6g uo (7)lg nTatuo.tod 6ut7stm7aqyto qcun"tqsnonurluo? panlna-e16utsD ottr '6'g eurtrrarl :3ur,r.lo11o; eql e^€r{ e,lr snqtr '(uraroaql ftuo.rpouour eql) t/ uo r{ou€rq snonurluoc panp,r,-e13urs € s€rl (3){6 frarra',{13urp.roccy'uleurop paltauuof, .{ldurrs (pueq raqlo aqt uO e s-rtJ l"q1 sale?s(91'g ura.roaql) ura.roaqls(rellnuqclatr

'{f = Itl I C > t} = rS pue tO < t lu>

t} - +lI 13seA\ereg

' (((r)t-rtOl)dxa'. . .' ((1)t6r)dxa'(t)e-aq . .'' (t)rt) = (t)'t :e-re(rS) x e-oe(+tI) -

6tr :

e Paugepeleq ax\ '.re;o5 4i Surddeu cr1.,{1eue-1ear salBurPlooc

uaslalN-IaqcuoJ'v'z'

I

't'8'ttd

Q|'r,

(t)z''g

(t1z''t

U7t'rt

sal"urprooc uaslarN-leqf,uaJ'z't

t9

64

3. Ilyperbolic

Geometry and Fenchel-Nielsen Coordinates

Prool. First, we show thatitrr is injective. Suppose that fr(tr) =rir(t) for some t1 and t2 in Fo. Let Ra, be the Riemann surface represented by t; (with the natural marking), and A = {P}(i)};s=;" b" th" pa^ntsdecomposition of &, correspondingtoP, for each f (= 1,2).BV Theorem 3.5 and the assumption, there is a conformal mapping, say g&, of P7,(1) onto P1(2) which respects the boundary correspondencefor every /c.Moreover, the proof of rheorem B.b implies that

dsl = (si4.@sl). Here, dsl (f = 1,2) is the hyperbolic metric on &,.In pa.rticular,every 9r is a hyperbolic isometry of the closure of Pi(l) onto the closure of Pp(2). Since |j(tr) = ?i(tr),

j = 1,...,39- 3,

all g1 can be glued together into a ma^rking-preservinghomeomorphism, say h, of rt1, onto -R1r. Since lr is holomorphic on r?1, except for a finite number of analytic curves, so is h on the entire .R1, by Painlev6's theorem. Hence, h is a biholo-morphic mapping of -R1,,which implies that t1 = lz. Thus we have proved that P is injective. Next, we show that f is surjective. For this purpose, we begin with fixing a .. . ,eas-s,e1t. . . ,ass-z) of (R+;sc-a x R3s-3 arbitra.rily.For every point (4a1, P3 in the pants decomposition P of R, denote by {Il,i}i=, (C .C) the boundary components of Pp. From $1.5, there is a unique pair of pants, say Pto,such that the triple of the hyperbolic lengths of the boundary components of Pf is equal pil and pi,2 be to the given triple {ou,i}i=r.SetPt= {P;}ir=lt.As before, let the elements of P neighboring each other along .Li for every j, and let pj,xbe the element of P/ correspondingto Pip (l = I,2). Let Ci,2be the point oii th" boundary of Pjp correspondingto ci,t for every j and l.Now, by gluing Pj,1 and P/,2 suitablV along curves corresponding to ,Li for every j, we obtain a Riemann surface, say r?'. We need to choose a suitable gluing (and a-suitable marking of .R') so that R/ corresponds to a point t, of F , s u c ht h a t f ( t / ) i s e q u a lt o t h e g i v e n( a 1 , . . . , a 1 s _ B , d l , . . . , a s s _ a ) . T h i s c a n be achieved by gluing Pr{,1and Pj,2so that the twisting parameter becomes the given ai for every j. We shall explain this procedure more rigorously by using Fuchsian models. In the rest of this proof, we consider only the case where Pil t' pi,z, for the other case can be considered similarly. Fix j, and let 4,r be a Fuchsian model of the Nielsen extension P|,r "t fj,t for each &. Here we assume that every 4.,r acts on the upper half-ptanl f , aird that the transformation l(z)=\2,

)-expai)L

belongs to both 4,r and li,z, and 7 coversthe boundary component, say Ll1r, of Pj,r corresponding to.Li for each & (& = 1,2). We also assume that the nilural orientation of the axis ,4 = {z € Hl, - 'iu,y ) 0} of 7 corresponds to the prescribed orientation of .ti, and that the point i € /l lies over cl,x with respect to li,* for each /c.

'Z-6?,'...'I = {

"la =(qa)rl

'eroJeq "'fa r'{" 3urn13 * pue l€tll q?ns ,U oluo A p rt tusqd.rouroauoqe xrJ fq paurclqoec"Jrnsuueuerg eql sq ,A 1eI"tr;o dlrrrrlcaf.rns;oyoord er{l Jod ('r 'dtq3 '[tz-f] tl{se4 acu€tsulroJeas'sruaroaql uorl€urquo) .rog'(saaeg.rns om1Surlaauuor.rog)uuoeql uorlDurquocs(uNelNse ulrou{ flpcrsselc q slqtr) '!/A Jo uorsuelxeueslarNeql Jo Iapou u€rsqcr\{ e sr I'lJr {q paleraueEdnorSuersqf,ndaq1'sprornraqlo uI'{ rl"€a roJ ,-gz'!J9, l ' pue ! , tt'!4 uo '?"p ,It!^ {.raaaEuole'}'I$ ul s (a,unaf.repunoqEurureura.r lueprcuror r'la p uorsuelxe uaslarN aqlJo uorlrnrlsuor eql ur pasnse^rqf,rq^\'ureuropEurr elq"lrns e 3urqce11e fq lnt urorJpeurelqoec"Jrnsuusuarg eql ''e'l) fr14ureurop ?ql uorsuelxeueslerNeql uo f,rrlerucqoqred,(q"ql'(H)?nV 3 9 acurg 3ur11nsar Jo 'fo r'fa uorJ fp3o1 olnpou ot lenba sr z'!p oI {fua1 uor}elsusrlarlt }erll qcns t'l,l z'!4 pue t'fa p 3urn13e e^eqe^{ueqJ to uorlecgrluapleql ,(q) 1z'!,7pue '8'8'EtJ

|;e

'nJ

('g'g'q.{ aag) 't'1Jo tuetuelerrc se pareprsuoo,LJoy srxe aq} uo (")g qft^ z'!tr p lueutueleue se pereprsuoc,LJo y srx€ eql uo z f.tete,tg11uap1'(n)WV lo

0 < lp

'ztp - (z)g

luauale eql raplsuoopue'(r,6f lofo)dxa = fp 1eg sal"urProoc uaslarN-Iaqf,uad'z't

99

66

3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

Then [rR',h-(t)] determinesa point of To.Let l'be the correspondingpoint of .Fo. From the preceding construction, it is clear that

l i ( t ' ) = a t , i = I , " ' , 3 9- 3 , and that ).t

0 1 ( t ' 1 : ^ : I " e d 1 p o 1 ) ( m o d2 r ) , uj

j = t , . . ' , 3 9- 3 .

(Note that, at preseni, we cannot say that 01(t') = ai, because the choice of branch of 0i is not unique.) Hence, letting Ti; R' ------R'be the Dehnlwist with respectto trl , the curve on.R'corresponding to.ti, we can find integers ftrt.'. ,fl3s-3 such that

[R :,(T i 'o...orti :i" o h) .( t) ] corresponds to a point, say ttt, which satisfies , i r 1 t ,= , 1( o r , . . . , a s s _ s , o t r . . . , o s c _ s ) . Thus we have shown that fr is surjective. Here, the Dehn twist Q with respect to.Lj is, by definition, a homeomorphism of R' onto ,? corresponding to the following surgery: cut R' along L! a.nd reglue after a rotation of 2zr (see Fig. 3.9). Note that applying Ti, we can make the value of di increase by 2r while every other 0i U'+j) remains unchanged.

o0 o twist Fig.3.9.

Now, we have proved that fr : Fo -------+ (R+)sr-a x R3r-3 is bijective. By Lemma 3.9, tit is also continuous. On the other hand, Teichmiiller's theorem (Theorem 5.15) states that Fo is homeomorphic to R6c-0. Hence the following theorem, Brouwer's theorem on invariance of domains, implies thatV is actually n a homeomorphism.

aldurrs (anbrun) aql aq lV m 'raqtrr\{ 'IV tq tr alouap pue '17 s])asre]ur qcrq^r fr14ur crsapoaEpasolc aldurs e xrg 'z'f4, n lI nt'!4 - lr14 ps'f fre,ra rog as"r eq} ^{olls e,lr'ure3y) 'luauoduroc,trepunoq e se lI Eur,teq ('7"{d = I'fd teqf '7 o1 Surpuodserroc 'J > lI {rarra rog dJo slueuele eq1 a'f4, pue I'ld' fq elouep U Jo uorlrsodurocapslued aql aq d +aI'U uo selrnc Sursodurocapgo 7 uelsds e pue (6 f' snua3 ersJrns uu€ualu e xg 'uorlces snor,rerd eql q sY l) Ur Jo '21'9 uorlrs 'uorl€zrJlerue.red;o pur{ raqloue arrrS leqs aira. -odo.r4 ur 'ra1e1 'suorlrunJ {lEua1 crsapoa3o^\} qll^r sa}eurproof,ueslerN-leqrueg paxg q relaurered 3ur1sral1qcea Surcelde.rfq '6g go slurod aleredas suoll?unJ 'a.re11 {}8ua1 esor{^\ scrsapoa3 pesolc aldurrs 6 - d6 Jo }es e lcn.rlsuoc aa,r '(FOt] IIBAToSpue eleddag ';c) elqrssodurr q slql '.rala,no11'oJuo selsurprooc pqo13 a,rrEsuorlcuny qfual asoql( srrsapoa3 posolt aldurs g - 69 Jo les € 6?ilrererll ;t elqsrrsap lsoru eq plnoa lI 'areJrns eql eururalep sqfual crloqradfq asoq^\ k ?) 0 snuaS;o af,eJrnsuuetuarg pasol, e uo scrsapoe3pesop eldurrs;o 1ese Surpug;o uralqo.rdeql replsuor e,n 'uotlaas qql uI

tu r p p e q tu g u la lx- a { r l4 4 ' g ' g ') o1 lcadsar {1a r_4;o flmurluor;o;oo.rd }cerrp € arlr3 osle lleqs a1ys3urddeu IeuroJuocrsenb Sursn fq g reldeq3 ur adfl $rll Jo uorleruroJep eq1 ele3rlsaaut Upqs a1yuotTotu.totap u?slerN-Ieycuadre pelpr sr elrnf, " qf,ns ,tq pelueselder ac€Jrns uu"tuerg e Jo uorlerJel eql 'tJ ur eAJnf,snonulluot e o1 spuodsauoc ( { U > I | ( € - 6 s o t. . . r r * l n , J ' t - ! n , . . . , I r 2 , 8 - 6 t D. ,. . , I p ) } ) r _ 4 eqt yo fg e3eurre.rdeql '0I'g ueroeqtr dg 'sralaure.red3ur1sr,ra1 e u oe { € l p u e ' n - r s ? I x e - n e ( + u ) ; o ( e - E e o ' . . . ' r D ' s - 6 8 D' . ' . ' 1 o ) l u r o d e x r g 'uollJasse aql a^eq arrr "{rerlrq.re ! sr f erurg'IO uo snonurluocsl r-d pue'O;o lurod rorJelur ue sr f '.re1nct1.red uI'IO qlr^r luaprf,uroceq plnoqs d dq g Jo g lul rorrelur eqf Jo (g' 1u1)d e3eur 'pueq retllo eq} uO -,lI aq1 'acue11'pelrauuoc s\ u^roqs eq u"? ler{} ll fI '/s' Jo uleluop rolrelxe eql pue ureruop rorJalur eql 'fla^llredsar 'zO pue IO {q alouaq 'slueuoduroc pelcauuoc o,r.l s€q ,S - .rll (uraloeql s(u€pJof leuorsueurp-, "ql fq 'acue11 'rll ul 'fle.rrrlcedser'ereqds pcrSolodol e pu€ IIeq pesol) 1ecr3o1odo1 e erc (gg)dt - rS pue,g're1ncr1redu1 '(g)dt = ,g oluo €r go ursrqdrouroeuoq € s.r g uo o1yo uorlculsar aq1 'lceduot sr Br aculs 'r reluar qll/rl - o pue'f1t.re.r1tqre pesop e f1tre.r1tq.re xg'(f)r-d las 5' ileq '[ZI-y] sreg 'y3) '(qc1aqs y) too.r4 O q n lurod e xIJ ('[98-y] ueu^\eN pue 'o oluo 'utotuop o s.t ("g)dt ueqJ ' iy olu! uE O ;g to tustrliltoutoeuoU o st d) puo 'onl uDlI ssq lou ta,a\ut. uo to uo4catut snonur?uo?p eg uE {_ ull : 6 7a7 eq u pI (sureurop Jo acuBrJBlur uo uraroaql s6rar*no.rg) 'tt'B uraroatlJ L9

turppaqurg uralx-a{f,rrJ't't

68

3. Hyperbolic Geometryand Fenchel-Nielsen Coordinates

closed geodesicwhich is freely homotopic to the simple curve obtained from 4! by applying the Dehn twist with respect to,ti (seeFig.3.10).

(Pi.r: Pi,z)

Fig.3.10.

For every t e Fc,let [it1,&] b" the correspondingpoint of ?0. For every closed geodesicL on R, we express as I(t) the corresponding closed geodesicon ft1, and denote by l(L(t)) the hyperbolic length of I(t). Set

ti(t) = t(Lj(t)), tuo_"+i(t)= t(al(t)),

= t(Aj(t)) tao-a+i(t)

for every j, and set

I 1 t 1= ( r ( t ) , . . . , / g o - s ( r ) ) . Wehavethe following: -s. Theorem 3.L2. The mapping L it o proper embedding of Fo inlo (n+;se and the preimage of any (That is, I is a homeomorphism onto the irnage t(F), compactsel in (R+)ee-s under L is compact.) To prove this theorem, first we fix a point ts of Fs arbitrarily, and write = ( o r , . . . , e 3 s - J , e r t . . . , o e g - s )€ ( R + ; a s - t x R 3 c - 3 . fr1to1 Fix j, and for every s € R, define a point t(s) of d t(s) = fr-'(or,"'

by

,a3s-s,Qrt"' ,a!-rtoi + s' o.i+r,"' ,asc-e).

Then we have the following Proposition.

alsq era acueg 'f1a,rr1cadse.r ',? pue og ol pue 0z spues pue '? slt<eeq? qll,r{ crloq.redfq e s rLoeL luetuele ,z uorltsoduror aql uaqf 'slurod paxg se'f1a^rrrleadse.r'0rn pue oz er.ie-q qcrrl^it,olrr1 raPro Jo {H)?nV Jo slueruala arldqla eql eq zl pue Il, }al 'pueq reqto aql uO '(,V'

, m ) d| ( , m ' , z ) d ) ( , 7 ' , z ) A

'(,n',?)d - (n'z)d 'suorlrugapeql ,(g teql r"elc s! ll prr€

.II'8'IIJ "I m om

aunss€ puts {srp }run aql fq p aceldar era eraq^a,11.g.3rg aag) ('0 = or leql '(rtrlatuoa3 cqoq.radfq erll ,f1e,rr1cadse.l ,02 Jo asues eql W) orn o1 lcadsar qlral pue',2'z o1 crrlatutufs slutod aQ 0z pue '?',V Ie.I '0o1pu€ oz r{SnorqtSurssedg uo crsapoe3aqt aq I Ie.I'F f z ueqll. dlqenbeur aq? ilroqs ol seclgns 11'loo-t4

\lln)

'tm=ffLpuo - z fi1uopuD spyoyfrp1onba ay1t.taaoa.to141 lt fi F 'fi4ao4ceilsat'(acuoTstpcqoqtadf,tl ayl o7 Tcadsa"t p puo m to puo F puo z to sTutod-aypptut?Ul ^tD om pup oz ?reqn 'splo1 (m'z)d * (,m',z)d

) (on'o116

f r T q o n b a u?tU l ' z I ) , 0 1 , ' t fnr . t a a a puo rI ) ,z'z fi,.taae.tot 'frpualoatnbg'{"1> m'r7 sr (m,z)d uat17 ) z I zC) (n'z)} uo &?auo?fi17cVr1s 'uaa$ aq .7I.g BurrrraT H q zI puo rI eusepoa| yu.to[sppQpnTnut onl prl 'ur.ro; SurmolloJ eqt ur pe?"ls sr urn+ ur qcrqaa 'd ecu"lsrp ?Jecurod eq1;o {1xa,ruoc Eursn fq pa,rr.rapaq ileqs uorlrsodo.rd srq; 'urnunutur sp sulnllo { t1cryn 7o s '.r,o1nc4,toil uI '(E o7 taulilout o so) g uo ..tailo..til puo to aqoa anbtun o fl ?r?Vl uauor fi17cg"r1s q (((s)l)jill = G)t uotTcun! aatTtsoil?ttJ .tT.e uog11sodo.r4 tutppaqurg u-ralx-a{rlrd't't

70

3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

2p(zs,ws)= p(zo,2s)< p(z', 2'). Thus we obtain

2p(ro,*o)1 p(r', w') + p(z,w).

tr Proofof Proposition3.19.Weconsider the casewhere P1,r * Pi,, and ^4! intersects ,Li at two points. The other cines can be treated similarly. Let l's be the Fuchsian group represented by ts, and 7i be an element of l-e which covers Li(to). On the axis,4; of 1, fix a point zs which is projected to an intersection point, say p, of Li(t6) and al(to).Let 6l be the element of l-o which covers Al(til and whose axis passesthiough zs, ar'd Bi be the ax-isof 6f .By the assumption, the projection of the geodesic.I contained in Bi and connecting zs to zto= 6l (zo) should intersect Li(to) at some point g other than p. Let z1 be the lift of qbn 1, a.nd,4j be the lift of Ii(ts) passingthrough z1 (seeFig.3.12).

Fig.3.12.

For any point z on an oriented geodesic -t on I/ and any a € R", let z(a)r be the point on tr obtained by translating z in the positive direction along .L bY

Eurdderu € eleq elvrueql ', o1 Surpuodsa.r.rortgr uo sselc fdoloruoq ea+ eql ul t, crsapoaSanbrun oql Jo (t2)l 'S g {raaa rog '1 dq peluaserder ec€Jrns r{l3ue1crloqrad{q aql aq (d!)-l I }el uueutrrarlleq? eq ,A tel'6,4:l l r(rar'aJod '(6 {) f snue3Jo Ar er"Jrns pesolc € uo selrn? pasolcelduns sassel?,fdolouoq ae.r;II€ Jo Surlsrsuoc1asa{} g ,(q alouaq Jo 'srrlolloJsp pegrpour q t1r;o Surppequraue '1srrg UI'g rueroaql ur sp euo qcns '[OZ-V] nreueod pue qcequepn"T 'lq1€d 'acuelsur 'punoy 's;oord 'uolsrnrlf ul ro; eq ueo qcrqrrr 1p lsourp lpro all 'A{ o? enp sr qclq^a '(e tg aceds rallnurrlrrel eq} roJ f.repunoq ?) d snua3 3o (te"pt) InJasne Jo uorl?nJlsuoc aql Jo auqlno ue aar8 llsr{s a^\ 'uorlcas slq} uI

uollBrullredruoc s6uolsrnqJ .7.s

'gl't tueroeql uroq s^rolloJuorlresss eq1 'drc.r1rqre u_f aourg ! 'a(+tI) olur U Jo Surppaquraradord e sr lr acueg 'snonurluo) pue 'aarlcafur 'redo.rd sr zll otur ?I Jo ("2 + s)rr'(s)rr) --+ s Surdderuaqt'tI't

uorysodor4 fq 11 uo.redo.rdpue xeluor.{11cr.r1s sr (s)/ eeurg

'@z+s)/ = (((16 +s)i[v)t = (((")r)fv)a e^€q e1'r'f, .,(.ra,ra ry (7)17 Euole 1srm1uqeq eqt Surfldde dq (l)jy uo.ry paurelqo e^rn, eql o1 crdolouroq flaery q (l){V &urs'AI'6 uruoeqJ lo !oo.r,4 'uorlress€ eql aPnlcuo) a^{ 'eroJeq peuorsuetusB tr (g)- = (s)rf acurg '11 uo .rado.rdpue xaruoc ,{11cr,r1s st (g)u, ?eql ^\oqs o1 f,sea sr 1r 'arourJeqlrng '1urod auo fllcexa te (g)u/ urnturunu eq? su.rep€ ll 'relncrlred uI'l,V y ly uo rado.rdpu" xaluo? rf11cr.l1s sl (g'rn'r),I'g paxg,{ue roJ snql 'g yo drepunoq aql o1 spuel ol ro z raqlra s" oo+ (rr"'z)d Wql tceJ aql ruorJ uaes.{lsea eq u€f, s€ 'rado.rd sr g 'os1y 'xeluoc ,t11crr1s$ J teql aas uec ar\{'tl't eurural ,{g .

( r r v r l e l s ; 1 2 ) j e, . ) a q ( , v 1 s - 1 m , 2 ) d- ( q , ^ ' r ) , 4

3ur11as,tq U x |V *

.

lVuo (s'rn'z)g uorlcun; ? eugap alr 'acue11

o ,,y]i"iu't"' { ( t r r r f r l s ) 1 2 7 [ s , i v ( s 1 n* )@

= (r)! leql

aas ue? a/r{

'g = s((07)t7)7!3 " r ,(q 0? fq palueserder e?eJrns {fua1 crloqladfq fq (01)f7 3uo1e3ur1sr,r,r1 eql ruo.t; peur€lqo er€Jrns pe{reur aq1 sluasardar (s)1 acurs 'la q13ua1cqoqraddq uorleryrpeduroC s.uolsrnqJ'?'t

TL

3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

72

/.(1):S-R* for every t e Fs, or equivalently, we have a mapping [:

Fo -'--* (R+)t

of ,F'ointo (R+)s. Ilence Theorem 3.12 implies the following: Corollary.

The mapping I*: Fo -

(R+)s is a proper embedding.

Moreover, this mapping remains an embedding' even when we take the quotient space P(n+;s = (R+)s/R+ as the target space. In fact, letting Pl* be the composed mapping of /* with the projection n of (R+)t into the projective space P(R+)s, we have the following theorem. Theorem 3.15. (cf. [A-29], Expos6 7) The mapping Pl, an ernbedding.

: Fo -

P(R+)s

is

which is defined by Next, there is another natural mapping of 5 into (F;s, using the geometric intersection numbers of curves. Here and in the remainder of this section, we set [T = {o € R I e 2 0}. For any two o1 and o2 € S, the geometric interseclion number f(o1,42) of a1 and o2 is, by definition, the infimum of the number of intersection points of .L1 and tr2, where ,Li moves in the free homotopy cla^ssof ai for each j. In particular, i(ay,a2) = i(oz,ar). 4lro, note that f(o'o) = 0 for every o € S. Define a mapping i* :.9 -* (F)" {0} bV setting i*(oX.)-i(o,.),

o€S,

where we denote {0}s simply by 0. Then we can show (cf. [A-29], Expos6 3) that Pi* - zro i* ;.9 -----*P(F)s

" i:3:""T: setting

to a mappingof R+ x 5 into (R+)t - {0} bv that i* is extended i - ( c ,a ) ( . ) - a . i ( a , - )

for every (o, a) € R+ x S. It is clear that

" (am;O- {o})= PLIs). Now, we know the following: Theorem 3.16. (cf. [A-29],Expos64) The subsetFr](S omorphicb SGs-7- {s € R6s-6 | lol = l}.

o/P(R+)s is home-

- {0} can be identifiedwith (and is written hereRemark.The set I.IFD after as) the set Mf of all Whitehead equivalenceclasses(or more precisely,

'uorlJesse aq1 saqdurr 0 qarqn 'r > (l)rt leql qcns ? a rnc pasolo alduns € eleq e^\ acuaH ',| 1e spua pue q}-I/'astlsls qcrq^r,Jr Jo J3el 3 Jo cr€qns e sr areql 'uraroaql ecuerrncer s (L)rl ryq1 qcns I cr€ l"srelsrrerl e a{el'0 < I frane.ro;'t:eIul'JW >(rl',1)-dfra.rlaroJ0otBurEra,ruocecuanbasesureluoc{S>ol["]rl]]esaql 'uraroaql acuerrncers(gr"curod Sursn ,(q (pu€q Jaqlo eq} uO +eql aoqs u?c e^{ (3 uo Sutpuedap) luelsuoc e,ttlrsod e sanlel aql 't4 3 3 {ra,re rcg.too.r4

fq rrolaq uorJ pepunoq ars (S 3 o) (o)(l)-/ '"(tg)

ul lW

u-ro.{7arctnp q (6[).1a6our

ay1 'zr'g

uor+rsodor4

'g-rsll o1 atqdlouroeuoq eq ol uaou{ $ urnl uI q?Iq^,r '(e.rn1cnr1s xalduroc ua,rrE aq1 o1 lcedsar q?l,rl) U uo sl€rlueraJrp crle.rpenb erqd -rourolorlJo aceds eq1 ql!/{ pagrluepr aq rrcc uo arnlcn.rls xalduroc {0}nJW'g ua,rr3 fue .rog'pq1 pe^roqs [191] lnsery pus preqqnH 'pueq reqlo eqt uO

'lfutv4l - rt arns?erul"sralsu"rl lecruouec aql qlpt paddrnba '(7 raldeq3Jo U$ 'Jc) e,rrlrsod fl U uo d prlua.legrp crle.rpenb crqd.rouroloq orez-uou paqtrasard e qrrqra Jo J"?l f.raaa Euop 'uorleqo; e $ 'arnlcnrls xalduroc e qty( paddtnbe fl U uaq^{ 'g uo uorlerloJ parns?eu e yo aldurexa pcrdr(1 v 'A IJoueA ' lW ) (rt',i) = d leraua3 e rog ua,ra (g ) d) @)(d)., se uatlrra{ s\ [d]d 'uo ereq uroq{ '(7)r/ d)'Ilur = ldlrl areq^r

'g>d

'@)("'D).!-ldlrl

uollelloJperns€eru " qlr^irpeglueprq (s 3 p,+lr f p) (,,,r;-, .rffi.t:;y;f] 'g to1 rl arn6?er.ul"sraAsrr"rl e pue 'sarlrreln3urs pelslo$ q?ns qll^ar uo uorl€rloJ€ .rrede 'uorlrugap ,fq 'q (r/'d) uo4otlol p?Jnsoeu y U dr Jo '(!)d = (n)r1 uaql ,4 yo 'ddolosr ue fq ;ea1 a13urs€ ur paurcluoc sr qcrqar Jo llqro qtea ,/ euo Jerllou€ ol '1eql sauslles qclq/'\ pe^oru aq u3? u dr Jo sJ€al h'0] , la cJ? lesre^susrl e;1 Jo scJ"qns lesrelsuerl II€ Jo las aql uo arnsseru e sl 3, .ro; r/ e.rnseatulesJelsu€r? e '1xa11 '0 = z .{lrreln3urs eq} r"eu '1 re3elur a,rrlrsod etuos q}ua errllrsod sr '{O} C uo uorlelloJ e Jo teq} s? eJn}cnJ}s zzp,tz q)qM Jo Jeel ,f.razra3uo1e elqellueraJlp l€ool atu€s eql e eq ppoqs qolq^rJo qcea 'sarlr.repSurs pelelosr qlr^r '(g '[OZ-V] ';c) g gsodxg eceyrns lceduroc e uo uorlerloJ A' uo uo.rlsgoJe eq d la1 'ereH 'e?uarue^uoc Perns"au s Jo uorlluuep aql lle?er e^\ Jo e{es eql roJ

'Gw)" -- Jwd

se (S)-la ssardxaoE" uec e^{ ecueg ('g gsodxg'[OZ-V]aes)'U uo suorlsrloJ parnseatu;o (suorleredos,peaqalrq1\pu€ fdolosr repun sesselcecuep,rrnba uorleogrlaeduroC s.uolsrnqJ't't

TL

3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

74

Now, the crucial fact to construct Thurston's compactification is the following theorem. Theorem 3.18. (cf. [A-29], Exposd 8) For eaery system L = {Lj}?s-13 of d'ecomposingcurues of R, there is a nalural homeomorphism qx : l*(F.) -.

U(L) C Mf

,

where U(L)= {p=(F,p,)e MF(c

(R+)") lp(Li)>0

for every Li e L}.

The construction of qa is as follows. Here and in the remainder of this discussion, we use the same notation both for a cutve and for the free homotopy class of it. For every t € Fs, we can construct a measured foliation Pt = (fi,p'r1 such that F1 is transversal to Li for every j, and that l,(t)(Lj)=i.(p1)(L1), i = 1,"''3s -3, or equivalently, l(Lj(t))= ttrlLil, i =1," ',39-3, where,Ll(f) is the geodesic, on the marked Riemann surface represented by t, which corresponds to Li as before. We set q2(1.(t)) = Pr' It is known that this mapping ga is actually a homeomorphism onLo U(L). "projections" qL, *u can derive the following: Using these natural Theorem 3.19. (cf. [A-29], Expos6 8) The subsel Pt.(Fs)U PMT with the relatiue topology is a compacl manifold wilh boundary. o/P(FF)s is homeomorphicto the real (69-l)-dimensional Moreoaer,Pt.(Fs)UPMf closed ball {c € Rog-o I l"l S l}, and the boundary is coinciilent wilh PMf (which is homeomorphicto Soc-7). we shall show how to construct local coordinates in a neighborhood of an arbitrary point of the boundary PMf . Fix p[' e PMF and ps e "-t(p[.) arbitrarily. Then there is a system I = of decomposing curves of ,R such that lfij|5" f . ( p s ) ( . L t )> 0 ,

i = I, "' ,39 - 3.

(See [A-29], Expos6 6.) Fromthis 4, we construct afamily {Li,A?,A}}}n=1" "t simple closed curves which gives an embeddingof Fo into (n+;s0-e as in $3.3. For every e ) 0, set

ad{1 ;o ef,€Jrns uu"rualg e ;o areds rellnuqcrel aq? l€q} ^{ou{ a^it ,.re1ncr1.red ut ('[t-V] .Uo{lqv 'acuelsur ro; 'aag) '(g)g aceds rellnuqcratr eql yo s3urppequra s(ulaly-a{?rrd pue sa?surprool uaslarNlaqcuad eqt 'g$ pu" U$ ul suorlrugep ol dl.repurrs 'augap uer ear 'arlleur cqoq.radfq aql slFupe U ereJrns e q?ns Jl '(u'0) adfiy aTru{ fr11ocryfr1ouo uI'(ur'u'0) ailfiy Jo eq o? pres sr (0'u'6) edfl;o aceJrnsuueruerg e'.re1ncr1.red 'srlsrp pesolc u, pu€ slurod u lo acoltns uuourerq e lurolsrp ,(1en1nur3ur1a1apfq f snua3 Jo ar€Jrns uueruerg pasolc s ruoq peur€lqo 'ace;lns uu€tuerll e IIel e \

saloN

'r-o1 tuor; peurclqo sl @Jo asre^ul aqJ tr 'Surddeur-;1es flrll repun ?uerrelur f1.rea1c arc IWd pun (61)*ld'a.reg '"(+U)2, ;o p Surdderu-Jlassnonurluo? s sacnpur ef,uaqpue J1as1rotuo StJo uorlce I€rnleu e sa?npur Jleslr o?uo A Jo d ursrqd.rouroatuoqSurrrraserd-uorleluarro fra,rg '{oo.t4

-ut

'l1asp oTuo D saonp n(6g).ru = (i)*ld IWd lo u.rsnld.tou.toau,oq .f.tu11otog t1asyzopo A lo utstrltLtnuoeuoq fun.taserd-uorlDlueuo fi^teag

:3ur,uo1o3aql ,(q u^roqs aq feur uorlecgrlcedruoc $ql yo ecuelrodurl eql '6a p fi"topunoqspolsrn?J pell€c s\ ,(repunoq slr pue '6a 1o uorToc{9ycod lWd -ntoo s.uo?srnqJ peIV) sr tg aceds rallnuqclel erll Jo uorlecgrlceduroe srq; 'od Jo pooqroqq3rau e uI seleurproor l€col se,u3 F ecuaq pue 'a3eurr slr oluo ursrqdrotuoauoq € sr d Surddeur slql leql a\oqs ue) e^r '(t'7)p)

a

'H+(o)((c)7b).!)

(")r > (o)((c)20).r

l€q} qcns y ?uelsuo) e sr eraql 'S g ,c ,(rale ro; '1eq1 sa1e1sqcrqu, '.{lqenbaur pluaur€punJ e Sursn fq 'rn.{I

'(4'n",)= (ivx(')'b)'t + (D(@)qb)*r) ((rtl"lrr 47b)*t+ 3 -) o*" ((a)t)d '(t'7)n 1nd e,rll,

'M)q

3 c ,(rarra .rog

'(o'd)=(4)o1

se(t'ol x l4 olur(r'f,)2a n /A lool Eurddeur e eugep "'1urf'"j;f""tfi:i[i $srr, l"ql 3ul1ecag '(ol)Va Jo ernsop eql ur uado sr (r'l)fla

n1t4 lerlt raoqs

'f1rea13 osl€uec eM'.1A= (Q'7)n)tb o I pue '(u,I)*ld ur uados\ Q'J)t)a '(0)d" - rA pue(("1)n)" = Q'7)nd '{g-fg, ...,1= !

, t < ( f 7 ) a l ( u , t ) . 1 fc } = ( r , 7 ) 1 saloN

76

3. Hyperbolic Geometry and Fenchel-NielsenCoordinates

(g,n,^) is homeomorphic to g6s-6*2n*3m. Abo, Thurston's compactification is considered for such a surface (cf. Fathi, Laudenbach and Po6naru [A-29]). Relating to $3, we recall the inverse problem, of whether we can determine a closed Riemann surface by the length spectra, i.e., the set of the hyperbolic lengths, of all simple closed geodesics on the surface. This problem is equivalent to the problem of M. Gel'fand: whether a closed surface is determined by the eigenvalue spectra, i.e., the set of all eigenvalues,of the Laplace-Beltrami operator on the surface with respect to the hyperbolic metric. On this problem, see Venkov [A-110], McKean [152], Sunada [218], Vign6ras 12421,and,Wolpert [246]. For Fricke-Klein's embeddings, we further refer to Keen [110] and Okumura [173]. The argument in $3 follows that in Fathi, Laudenbach and Pc€naru [A-29], Expos6 7, which is due to A. Douady. As for more advanced investigations on convexity of geodesiclength functions, see Kerckhotr [112] and Wolpert [256]. A survey on Thurston's compactification by Thurston himself is given in Thurston [234]. Fathi, Laudenbach and Po6naru [A-29] is a good introduction to measured foliations and Thurston's bounda,ry. See also Gardiner [A-34] Chapter 11, Strebel [A-102], Hubbard and Masur [101], Marden and Strebel [137], and Masur [144]. As a generalization of Fenchel-Nielsen deformations, the earthquake deformations have been considered.See Kerckhoff [112] and [114], and Thurston [233]. We also cite Bonahon [a5]. Finally, there are many proofs of Teichmiiller's theorem stated in this chapter. For example, see Chapter 5 and Notes of that chapter. There are also proofs by Wolpert [256] using geodesic length functions, and by Fischer and tomba [74] from a differential geometric viewpoint.

'[g'r] >cfraralsorul" .rog uo snonurluocflelnlosqe sr (/i'*)t 'fr Jo uor]cunJe s€ pue 'ftt'cl ) fi [p'a] f.ra,ra1sour1e.rol [g'o] uo snonurluorflalnlosqe x (fi'a)l'c;o uorlcunge sy :srxe f.reurEeurreql ol ro srxe I"eJ eqt ol reqlle lalered ar" seprsasoq^{ ul '"] x = e13ue1car f.ra,re.rogsploq uorlrpuocSuraaol O A W [g'rf loJ eql y (saut1uo snonurluocfi1a7n1osqo) nV aq o1 pr"s $ O ur€ruop.reueld e uo (rt'r)l = (z)/ uorlcunge 'araq :lCy q / ter{t Eururnsse ,(q 'eldurexaro; 'paalue.ren3 'a'e alqerlua q s1ql'uretuop q pereprsuoc eql (areqal{rarra lsorule) 'esuasalqe?rns -ragrp i(ler1red '1sea11e'aq ppoqs 'luaurarrnberqqt 3f Jo esn€)eg aruosur ztrl - z1 uorlenbenueJtleg eql segql"s fpressacau 1nq'alqerlueresrp ue reprsuor o? lu"^r eM lou $ qtrqa 3[ ursrqdrouroauoqSur,,rraserd-uorlelueuo sturddel4l luurroJxrootsen$ ;o y uorlrugaq

c11,(puy .l.t?

serlredor4 r(.reluawalg pu€ suorl.ruuac 'I'? 'qooq 'sJoord lxal PJsPU€lsuI PunoJ eq lil/'l qcrqrn 1noq1t^/l, p.r3alur anEsaqal ;o froaql aql ul $uaroaql plueurepunJ lereles esn e \ 'sJaB '-I pu" sJoJIqy 'rI ol anp 's3urddeur roy elnuroJ ltsuroJuoersenb 'g uorlcas uI 'uorl"nba Ieuoll"rrel l"lueurepunJ e arrr3 aal lerlua.ragrp rurerllag erlt Jo uorlnlos aql Jo ruaroeql ecuel$xe eql errordean'7 uorlcag uI 'suor?ruUepaql ruo.r; fpsea Suurrollogsll€J crs?q elels pue 'suorlrugap esorll Jo ecuele,rrnba aloqs 'sSutddeur '1 leuroJuocrs?nb p.raua3Jo suor?ruUapIeJaAasalr 3 ea,r uorlceg u1 'sur"urop .reueld uaearlaq s3urddeur 'raldeqc qq} ul 'f13urp.roccy IeruroJuocrsenb;o asea ar{t ol sellesrno lcrrlser am 'sur€ruop .reueld uee/\{}aq esoql Jo esec eq} ol sareJrns uustuerg uaarralaqs3urd -deur go esrc aql ecnpar uec e^\ 'uraroaql uorl"zrurroJrun eql fq '1eq1 elop 'aser alqerluere$p eql ur se erues aql sureruar s3urddeur qcns Jo lueul€arl 'suorl"nlrs I€tuJoJ ayqal IeJeueE arou flqe.raprsuoc w Iool e sr sEurddetu Ietu 'uorlruUep eql ul -roguocrsenb asn ol sn sluolle luauralordurr Iscruqcel " qcns uorlrpuos ,(lrpqerluaragrp eql ua{ee^r e,n 'era11 's3urddeu leruroJuocrstsnba1qer1 -ueJeJIp paugap a^eq ell.r'1 raldeq3 u1 's.raldeqcJa1eIaql ur pepaeu arc q)Iq^r sEurddeu leuroJuocrsenb;o sarl.radordcrseq ureldxa II€qs e^{ '.ra1deqcslq} uI

sturddetr I ler.rrroJuorrsunb ? raldBrlc

4. QuasiconformalMappings

?8

Remark .1. An absolutely continuous function .F(t) on an interval 1 is differentiable at almost every t € /. Hence, when a function f (z) on a domain D is ACL, the partial derivatives /, and fz of f are well-defined and finite at almost every z € D. It is not difficult to show that they are measurable. Now, as a natural generalization of the notion of conformal mappings, we make the following definition: mappings. Let f(z) be an definition A of quasiconformal Analytic orientation-preserving homeomorphism of a planar domain D into the complex plane. We say that f (z) is quasiconformal (qc) on D if / satisfies the following conditions: (i) / is ACL on D. (ii) There exists a constant & with 0 < & < I such that

lfd S klf"l a.e. on D. SettingX = (l+k)/(L-/c), wesaythat f is K-qcon D. Wecalltheinfimum of 1((> 1) such that / is K-qc the marimal dilatation of /, and denote it by I(y

or K(f). Example -1.A conformal mapping of a domain D is quasiconformal on D. Since we can take 0 as I in the above condition (ii), a conformal mapping / is l-qc and K1 - 1. Example 2. An affine mapping f(r) = az *bz * c (a,b,c € C,lbl < lol) is quasiconformal.We can tal<elDl/lol as &, and hence Ks = (al+ l0l)/(ldl- lbl). Emrnple 3. For a given lc, we set

f (t)=

( z.

ze H

\ r + i K v , z = x * i y e c - H ,K > ! .

Then / is a quasiconformal mapping of C, and Kt = I{. Remark 2. Set

I(,)=T:Ep, z€A.

Then / is an orientation-preserving diffeomorphism of the unit disk .4 onto C, but not quasiconformal. Actually, there are no quasiconformal mappings of 4 onto C. See Proposition 4.32 later in this chapter. Remark 3. Let f be a K-qc mapping of a domain D onto another domain D', and g be a conformal mapping of D'. Then the composite mapping I o / is K-qc. In fact, it is easy to see from the definition that g o / satisfies (i) and (ii), and that KsoT - I{y.

'g n! pue 9 > lrl qlp z fra,rerc1lzlh > (t)t ) g rell€urse3ur1e1'luarun3re.repturse,,(g leql osl€arunss deurerrr',,(lessacauJr 'g ) n pue > to1lzll"t> (z)t leql q)ns 9 e,rrlrsode sr araql 9 lrl qtt^ z .,l.ra,re

'l ((o )u/ - ( *)u t)n+l l (s)'tx- ( o ) /- ( ") / l - @)l - @p+ n)tl ) Q)1 l@)utn ef,urs

'l ( o) o /n -ft),l a -(o) / - ( r ) { l=( z) t 1eg'dperlrqre I > L I 6 q1r,ul,l xrg '0- 0z ?eq1eunss€ ar*'flrcqdursJo a{es eq} rod'02 }e elqelluareJrp,tlelol sl / ?eql ^roqs Iprls e \ pue 'lfi! + oa = 0z lurod " qrns xlJ ('"? ;o slurod dlrsuep eugap e^\ 'fgelnurg 'ofrg lo uorlcunJ orlsrralr€req, eql fl onal araq,u

'*#,i" 'r - *p(r)oor, ,:*"""'[T J\ '{g

ong

'flerrrlcedse.r Jo lurod flrsuap e sr 0r 1eq1 ,{es ealr,e.reg)

> fi!+0r I u ) fr) =o"g

pue

0n!+o u ) n} = ong {s ) I

slesqns aq1 go slurod ,(lrsuep a.re 0f pue 0r 'g ) ont * 0e dre,ra lsorule '1s.rrg ro; 'pq1 dldun uraroaql s.lurqng pue rualoeql ,,{lrsuap s,an3saqel 'f.rerlrqre sr ro;'g uo r 'e'e elqetlueragrp 'uorlresse ,t1e1o1 sl ^roqs ol secgns aq1 a,rord o5 leqf 1r / 'snonurluoc sr .rog'g uo ;l' snonurluoc erc nt pue t/ 1eq1sarldur ecueS.ranuoc ruroJrun aql leql aloN .Ar ue q)ns xU em Joord srqlJo 1sa.ratll uI'0 - q se fl uo.,(1uro;run (r)uI o+ se3ra,ruoc VI (?) I - @! + z) l) teql etunsseosle {eu aar'1uarun3.rer€lrrurs€ /tg'0 1- q e g uo flurrogrun (r)'l ol sa3raruoc,t/(@)l -(rt+t)l) teq? pue r u€q? ssel$ rr-O Jo €are er{} }?ql qrns o Jo ar lasqns elqerns"eeue puu uer e,r,r't}{"6} eeuenbas eql ol ueroeql s,go.ro3g Surflddy 'oo + u se (I uo 'o'e 0 * uf ?sq? aloN

It't't Q)l - ? t+,)l

l u ) q ' u l r > l q l >- o dns (r)"0 I I

'u 'f1tre.r1rqr€, 1as draae .rog 1u€lsuoc e,r111sod e xld 'eer" alrug e s€q O ueq^\ uorlJass€eql a,rord o1 sargns ?;''/ oo.t4 'O uo 'e'D alqn4uaufitp fi11o7o7 st t uaqT'g uo 'a'o ut pu, 'l saatToau,ap eql sDq C ory! e urDtuopDIo I tustyiltotuoauoq Dfi .tV uorlrsodo.r4 1ory.r,od 'peurtslqo uaeq seq olqarl pue Sur.rqag o? enp llnser elqe:lreurar Suro,olloJ eqt 'stusrqdrouroeuroq Jo ese) aql ur '.rale.tro11'uorle3rlselur JaqlrnJ o1 alqecrldde ;l';o sarl.radordpoo3 aalue.ren3 o1 qEnoua 1ou sr a/ pue ,t salrlelrrep lerlred ar{} Jo ef,uelsrxe 'leraue3 u1 sarlrador4 ,(reluauralg pu" suorlrug:aq 'I't

4. QuasiconformalMappings

80

Further, taking a smaller 6 if necessary, we may assume that, for every z = a * iy with c, y ) 0 and lzl < 6/2, there exist points x1t r 2, iy1 , and iy2 of E such that Q - rt), < nl < e < x2< (1 +4)c, (L-q)a (er < alUz<(I+q)y. (Note that oo = 0 and ys = 0 a.redensity points of Euo and.8ro, respectively.) Then, 1(z*) < qlz* | for every z* on the boundary of the rectangle ,R = lx1,x2)x tyr,yz). On the other hand, the maximal principle holds for /, since / is a homeomorphism, and hence is an open mapping. Therefore, for some suitable z* on the boundary of .R, we have

I(r) < lf (r.) - /(0) - 0f,(0) - y/y(0)l < I(r') + l, - r.l (l/,(0)l+ l/y(0)l). that,for everyz with c,y ) 0 andlzl<6/2, Sincelz - z*l
A(E) l"t,p1a,a,s

(4.1)

for every measurable subset E of D. On the other hand, by Proposition 4.1 we see that / is totally differentiable at almost every z € D, and at such a point z we can show that

JyQ)= l f" (r)l ' - lf,( ' ) l' . Since the condition (ii) on / implies that

lfA' slil' s *rt the assertionfollowsby (a.1).

a.e. on D, tr

Remarky'. Actually, the set function.A in the aboveproof is absolutely continuous,and hencethe equalityholdsin (4.1).SeeLemma4'12 in $1.3.

el€q a^r uaql '(f)zd(t),d *toJ qlr^1,uollrunJ e a{€}'((oo)tl)"$CJo ue sv'r o1 qg,rtr/Jo elrleArrep 1er1.red }uetuala leadsar 'W'")xlor'of = (or)U f"r pue ,[g,o] uo 0c Fuorlnqrrtsrp eql ["/] ,(q aloueq 'fgrerlrqJe e xg ,esod.rndsrqt roJ lurod e xrg O ur W'rlx [g'r] = g, e13ue1ca.r '1CY sl ./ l€rl? /roqs ol secgns t! 'y uorlrugep eql Jo esues aql ur leuJoJuocrsenb sr 3l' leql ^\oqs oL' ,V uorlrusap eql Jo esuaseql ul O uo leuroJuocrsenb aq / la1 '/H uorlrugap aql Jo esueseql ur I€ruroJuocrsenbsr 'eaua11'r(r) uorlrpuoc arll sess-rpsy uorlrugep aql Jo esuesaq? ur / Surddeur / .loo.t4 IeruroJuo?rsenbe 1eq1 t'7 pue 6'p suorlrsodord ss uiroqs fpee.qe a^"q a \ 'Tuapatnba fr11on7nu.t a.to s0utddou loru.totuoctsonblo rV puo y suotytu{ag .7.? uraroaql

'O uo'e'€ l'tlq)lttl ? s q l q r n s I > { t 0 q ? l i a{ } u e l s u o c s s } s f f e a r a q l ( r r ) '6r uo'f1a,rr1cadsa.r'rt pnt "/ suorlcun; alqer3alur r(1eco1fq paluasa.rda.r aq u€? Z pue z o1 laadser qll,lr / Jo salrlelrrep 1er1.redlsuorlnqrrlsrp aql /(l) :suorllpuoc oall Eur,rnollo;eql segsrlesI y O uo lout"loluocrconb sl / leql ,(es e711'uorl"luarJo se,r.resardq?!q^a C olq O uretuop e yo ursrqdrour .sturddBru -oeuoq s eq / 1a1 [BturoJuocrsenb Jo ,v uol+rugop c1+rtpuy :s^{,olloJqrrqa auo a{t se qcns 'sEurdderu leruroJuocrsenb go y 'g'? pue uoltlugep eqt Jo uorlecglpour e raprsuocu?c a/r4, 6'7 suorlrsodo.r43ur1o11 s8urddel4l lBruroJuocrsen$ go /I/ uolllugaq

c11,tpuv

.2.T.?

's1rcd ,tq uorler3alur z(q .no11o3 O suollrasse aqt 'TCV st ;f aaurg 'sp.r3elur paleeder e{l se ue}lrr!\er eq uec (g'p) pue (Z'7)Jo saprsprrerl Ual eql(ureroaql s,lulqnJ pue Z,'Vuorlrsodor4 Ag.loo.r,4

(e'r) (z'v)

'fi,pxpz61 'ttf ttil - - fi,papdt

I

puD

'frprpzdt t"il -=ftpapdt,t"lI

'sq.toililns Toqysnollol 7t Tcndu.tocypm .tot 'frputo7J 0 uo suotl?unt qToorus11oto 7aseW'(O)JC to dt Tuauala fi.r,aaa 'uorryqpprp zt saatToauappty.toil to asuaseUIur ?soql Vpn lu?pN?uroJ?JD,l puo urDurop o lo I Futrldoutlouttotuoctsonbfuaaa JoI .g.V uorlrsoilo.r6

?t#'O T8

sarlrador6 freluauralg pu" suorlrugtag'I'?

4. Quasiconformal Mappings

82

=v)(p r)' (c)e 2(y)dxdy. f v)et@)e2(v)dtdv | l ro., @, | | *outt'lte' Here, let rp2(y) tend monotonously to the characteristic function of (c' a) (4 € [c, d]), and we have fn lxo =f@,0(pr)'(x)ddv. lf"l(r,y)e{x)dxdv J" J, J" J, fq

lso

Since 4 is arbitrary, we conclude that

" liltr, y)ev(x)dad,v= - I "' f (r, v)(pr)'(x)dr dv | "" for almost every y on [c, dl. Next, for every sufficiently large integer n, take as gt = gr,n in the above equality a suitable function which is identically 1 on nl, and is monotonouslyincreasingand decreasing,respectively, la*If n,rs-lf -I/n,rs]. Letting n + oo' then by the above equality, and on [o,o +l/n] [cq we have fto

I

Ja

lLl@,y)dxdy= f (xo,il - f @,v) almosteveryy e1",4.

e.4)

Here, the exceptional set of y depends on 00. To get rid of this dependence, consider the set, say -8, of all rational numbers in [a,b]. Since,E is countable, (4.4) holds a.e. on [c, d] for every o0 in E. since both sides of (a.a) are continuous with respect to rs, and since .E is denseon [o, D],we conclude that (a.a) holds a.e. on [c, d] for every cs in [o, b]. Note that for every y where (4.4) holds for every xo, f(x,y) is absolutely continuous with respect to o, and that [/'] is coincident with the usual pa,rtial derivative f, a.e. on [o,6]. As for the partial derivative of / with respect to y, we can confirm a similar assertion. Thus we conclude that / is ACL on D and that the distributional tr derivatives are coincident with the usual ones. corollary L. Let g be a confonnal mapping of a domain D onto another Dt , and f be a K-qc mapping of Dt. Then f og is a l{'qc mapping of D. Proof. Let u) = rtr* lu be the variable on D' . By the a.ssumption,there exist the distributional derivatives /. and /,5, and they are locally integrable on D' . Since rpo g-1 belongs to Cff(D') for every I € Cf (D), it is easily seen that there exist the distributional derivatives (/og), and (f "c)r, and that they are coincident with locally integrable functions (f. o g).g'

and (f* " g)'7,

respectively, on D. Moreover, the condition (ii) for / o g clearly holds with the tr same /c as in that for f. Thus we have the assertion. Corollary

2. A l-qc mapping is conformal.

'0 - npnp dl"Ut) - "H)lt [ [ T;lli J J

'g uo t"ql (g'?) 3ur1ou ,tq ,uoqs u?c eilr ecurs pue

zt = z(lb) pu€ ,I = "(tb) acurs 'oo ts u se g uo fprloJrun 3f o1 saSrerr '.reloe.rotr41 -uoc u/ pue 'u a3rel rflluarcgns fraaa lof (O)JC o1 sEuolaq $ (g'f)

.t]h)

* "dt = z("1)

pu€

"(ttt) * "d = "("t) eleq eilr 'u fleaa roJ uerlJ

'))

n

'npxp(z)!(z)t(, -*)"d,"[[

=@)(!u)*udt

- (m)u!

J J

Pu€ C)z

'(zu)6.ru-(z)"dt 1es

'u .re3alure,ulrsod fre,re ro; 'reqlrng

vf f '1=npxp(z)dt | | JJ '{srp 'e.rag lu"lsuo, e esooqC }run aq1 sr y

leql os ,

'v-c)z

v>z

'0)

,("+-)

d x ae

]=?)d)

les eA\ 'O vo o1 3uo1eq pue lsrxa a7 'd z(lL) pue z(/lr) 'raq1.rnd 'O ul l.roddns lceduroc e seq ll-t uaql Jo poor{roq -q3reu euros ur 1 o1 lenba {pecrluapr sr qctq^r (O)"$.e ) lr luaurela ue ng'too.t4

's= nprpolz! -,("!)l"lf *tf (q'')

'o - npap - ,('!)1"[[ *tf al"!

'oo + u s0 tDq?puD f uo fiyl^tolmn t "{ p1t o7 safi.taauoe u? r-;j{"!} acuanbasD s, areql ,O lo I lasqns Vcns (q)}g tuana.tot ueUJ'CI uo a1qo$ayur fi,11oco7 ato fils1,7os Taodu.toc alzllpuD alzllll?t? 'fr,pu.rou'e uo "t fr11oco1 zt ^to puD senqDtuuap pqlod puorlnquqtp esoqm aI 'uaafi eq .g,V Bururarl O uxprilopn uo uotTcunlsnonuNlu@D ?q t 7a1 l
4. Quasiconformal Mappings

and

ardy= o, lim I I l$"), - Qtf)rlo J JD

N-6

tr

we obtain (4.5). Hereafter, we call such a sequence {/.} seqaencefor / with respect to F.

*

in Lemma 4.5 an LP-srnoothing

Lemma 4.6. (Weyl's lemma) Let f be a continuous function on D whose distribational ileriaatioe f2 is locally integmble on D. If fz =0 in lhe sense of distributions on D, then f is holomorphic on D. Proof. Fu a relatively compact subdomain Dr of D arbitrarily, and construct an .tl-smoothing sequencefor / with respect to 4 as in the proof of Lemma 4.5. Flom the construction there, we see thaf (nf)z = 0 in some neighborhood of E[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[. By (a.6), we see thar (f")7 = 0 on D1 for every sufciently large n. Thus, on D1 for every sufficiently large n. Since /' converges to / holomorphic /,. is n --+ oo, / is holomorphic on D1 . Since D1 is arbitrary, we on as uniformly I obtain the assertion. 4.L.3. Geometric

Definition

G of Quasiconformal

Mappings

We give a geometric definition of quasiconformality. For this purpose, we first introduce the notion of quadrilaterals. The closure of a domain bounded by a Jordan curve is called a Jordan closed domain. A, quadrilateralis, by definition, a pair (Q;qt,9z,Qs,Qq) of a Jordan closed domain Q and four points 8r, Q2,93, q4 on the boundary 0Q of Q which are mutually distinct and located in this order with respect to the positive orientation of 0Q. We call each q1 a aertexof the quadrilateral. If there is no confusion, we denote a quadrilateral (Q; er,8z,Qe,qa)simply by Q. Proposition 4.7. For euery quadrilateral (Q;Cr,Q2,QB,qs),lhere is a homeo' morphism h of Q onto some reclangle R = [0,4] x [0,b] (a,b ) 0) which is conforntal in the interiorlntQ of Q, and satisfies h(gr) : 0,

h ( q 2 )= a ,

h(c") = a*ib,

a n d h ( q a )= i b .

t f h. M o r o u e r , a f b i s i n d e p e n d e no We call the value alb the module of the quadrilateral (Q;qt,!2,Q3,9+)' and d e n o t e i t b y M ( Q ; 8 r , Q z , e e , Q e )o, r s i m p l y b V M ( Q ) . Proof. First, by Riemann's mapping theorem, there exists a conformal mapping h1 of Int Q onto the upper half-plane H. By^ Carath6odory's theorem, h1 is extended to a homeomorphism of Q onto fl U R. Here, by composing a suitable Miibius transformation, we may assume that

sa{stqosO uroulopo lo { tutililoru cb-51fr.nog'8'? BurtnaT 'erutuel 3uuro11o;aql e^"q eir ueqtr, '(vb)!' ...'(ID)/ rllllrl Plel"Urpenbe t" (O)f lePlsuotaru'3 olut @;o seclllel ro; '.rary:earag e pu" (tD,ab'zbtrb1fl) le.Ialeyrpenb,(.ra,ra /' ursrqdrouroauoq 'l'? uollls tr -odo.r4Jo uorlress" puoceserll se[du4 qclqa '0 = p pu€ g 4 a ]e{} aPnlf,uocert.r '0 < (b)q '0 = (ID)tl= (b)t! '0 < (zb)y Pu€ A?UIS

'P '(O f"I ) z) P * Q)t1c = (z\rt eleq e,rr Pu" t sreqrunu xalduoc elqellns qlr^1' 'flpaleedar 'acue11'(c)pV to lueruela u€' o1 papue1xeaq uea , -V o V leqt ees a,u '2'7 uI se suolllPuoc aures eql uotltsodord aldrcur.rd uorlcegar (zJs^rq?SEutflddy 'lxap sessrles qcrqa'rEurddeu raqlou" aq [g'o] x [g'o] = v *@ : q le1 .I'?'EIJ :

s

T

I

'

!

a T

ffi

"q

x+'l

<_ I't

'(t'f

'b)z

'EU) Eutddeur P?rrsaPe sI

'x*Q)tq"zr1-@)rt '0] x .(0 leql eesel'aaruaH < ,y,x) l,x [y'>l-]

a13ue1cer auos Jo rolralul eql o?uo g ;o Sutddetu lsruroJuoc 3 sr zV ueqJ

'H)z

,Qt"rt--lG,

-il

z

| =e)",

P

z

Pue'(sr;tq7l { ?aS 't < (vb)rq- = (80)It1 pu€ '1 = (zb)tq 'I- = (ID)rtl sarlrador4,{rcluauralg

98

Pu" suorlruyeq'I'}

4. Quasiconformal

60

Mappings

I

KM(Q)<MU@))
R -

F=-hofoh-r is a quasiconformal mapping of the interior of R onto the interior of E which m a p s 0 , a , i b , a n d a * i b t o 0 , A , i b , a n d 6 + i D , r e s p e c t i v e l(yc f . R e m a r k3 i n $ 1 . 1 and Corollary 1 to Theorem 4.4). In particular, .F(z) is ACL on Ii, and hence for almost every y € [0,D], we have

a < lF(a + iv) - F(iy)l =

iv)drl= 1,"F"t+ l.zt)dr l,' #o +

Since /J* Jpdrdy S ,4(n) = dD (as was stated in the proof of Proposition 4.2), integrating both sides of the above inequality over [0,b], we obtain

-< ( [ [ "Ur,l+l+Da"av) @b)' / \"r.h

- JJ^,1r1:77d'oa'JJo," '' -.

[ [ I ( d x d- y . [ [ t o o , o y ! K ( a b ) @ b ) . JJn' JJn'

haveM(f@)) < KM(Q). Here,we setl? = {tr,'e ,RlIF"(*)l I 0}. Thus^we Next, replacingf' = h o f o h-r by (ih) o f o (ih)-r (or considering (Q;qr,qs,q+,qr)),we can showby the sameargumentthat I

K

W@Ds Ma) tr

Thus we have the assertion.

Now, by noting Lemma 4.8, we give another definition of quasiconformal mappings without using partial derivatives. Geometric definition G of quasiconformal mappings. Let / be a homeomorphism of a domain D into C which preservesorientation. We say that / is quasiconfonnal on D if / satisfies the following condition: (iii) There is a constant K > 1 such that

MU@)) S IiM(Q) holds for every quadrilateral Q in D.

'a.rolaqso ftpnp{ilt11

Q'v)

= @)V alaqm 'g lo g ?esqnselglrnsoelrz fri,aaa.ro! sf f

- ,l"lD (s)v= nprpQl'll IJ sa{sr7os CI utvluop Dlo I |utddoru Totu.to{uocrsonb fuaag 'ZT'V r.urrrroa

'oraz ear? seq ((A)/)r -t = g D '91'7 ureroaqJ les aqt wq+ zI'v eurtuerl {q ureSeees e/lt Jo (l) fq pu.ro;uocrsenb osle sI r-l eculs 'orez ear€ s€r{ (ar)/ }as eql '71'V eluuo.urc1 }xeu eql dq acuag 'A uo'e'e "t 0 = zt pu€ elqernseeu $ A ueqJ'{0 = I O ) z} = g pg'too"r4

'o uo 'a'o 0+ "l ueql'(I utDu.topo uo lout"toluoctsonb s! I lI 'II'7

uolllsodo.r4

',) uollluuecl ruoq r€al) sI PJIqI eql tr '(2'7 uorlrsodo.r4go yoo.rd aq? aag) 'appour ar{} Jo e)u€rJelur leurJoJuocaql ,(q ueas sI puo?asaqtr'{reureg Eurpace.rdeq} urorJ s^{olloJuorlresse lsrg aqa /oo.l2, 'cb-eXrX '(O)l st t o 6 0utddout pasodruoceq? .tog (t$) {o 6 0utddout cb-zX fi.taaapun e urotilop D Io I |utildout ab-rX fr".taaa 'ab-y oslo s? ?q?'e oluo O to t |uzrldout t_\o I oU 0utddou.tpasodu.t,oc cb-y fr.tana.tol puo'fr1aai7tadsa.t'epuo q suwu,6p lo tq puo t1 |urildotu 'fi1atuo111 'Tuottiaut fi11ow.to{uoc st fiTtfoutoluoct,sonb-y Tout^totuocfi.taaa.tot 'cb-y oslo st Outildou cb-y o to as.taauraytr

(r) (r)

'0I'7

urorooql

( '''t ureroaql o1 1 frego.roC pu€ I'I$ q I {reureg se perlord.{pearle ueaq s€q gl'f ureroeqtr Surmolo; erll Jo (11)wut II"csU) '6'? ureroaqJ prre 5l uorlrugao Jo sarrsllorot 'a.rag 'y'1$ ol Ierelas e,rr3 e.r,r. 6'? ruaroeql yo goord aq1 auodlsod geqs all 'guapamba Q7on7 '6'V urarooql -nur, etD s0utddou.t loru.totuoctsonblo g puo V suotTru{ap onJ

'O ul roJ sploq 0l€replrrpenb .,(.re,ra

@ ) w > r > ( } i l w > @ wI ! l€rll qcns I < >I lu€lsuoc e st a.req; ,(ttt) :euo eql ol lualerlrnba fl uorlrugep Suro3e.ro;eql ul ($) uorltpuoC 'qrDuaq Surr*o11o; 't to t X uorlet"llp lerurxetu eql ol lenbe sr jyr qlns Jo turuugur aq? 1eql aasa^{'ra1e1pa,rord$ qrrqa gI'? €ruurarl pue 8'} eurural {g sarlrador4 dreluarualg pu€ suorlluu:aq'I't

L8

4. Quasiconformal MaPPings

88

Prool. First, we consider the case that E is a rectangle contained in D and that / is absolutely continuous on the boundary 0E of E. In this case, in view of Proposition 4.2, we find an tr2-smoothing sequence put /' = un* ian. {/"}Lpr for / with respect to E' (cf. Lemma 4.5). For every n' By Green's formula, we have tt - (.,-)r('")"} I I {@^),(an)v J Jn

dxdv -

I u^d'an J| , E

for every n and.m. Let m * oo. Next, to the right hand side, apply the formula for integration by parts for the Stieltjes integral. Letting 7r *'+ o9, we obtain t[ t [n

- uuo,)dtdyUf"P- lf1\ara,- = [ [ (u,oc [ uda. Jan JJn'

Here, we write / = u+ia. The right hand side of the above equality is interpreted as the line integral of uda along the Jordan curve df(E) on the w(- u*ia)-plane. By the assumption, Af@) is rectifiable. Hence, we can show that t

tl

= A(E)' = Ju""o' JJ,r"rdudo Thus we have the assertion in the case stated at the beginning of the proof. Since / is ACL, every recta,nglecontained in D can be approximated by such rectangles. Hence, (4.7) holds for every rectangle contained in D. By a routine argument, it is proved that (a.7) holds for every measurable subset E of D. tr Now, by Propiosition 4.11, for every quasiconformal mapping f of a domain we can consider a quantity D,

f, rtt=T

a.e. on D. This pt b rbounded measurable function on D, and satisfies

< r. ess.sup lptQ)l< 9+ tfy*L zED We call prythe compler dilot'ationoff on D. proposition 4.13. For eoery qaasiconfortnal moppingf and g of o domainD, mappingg o I-r is giaenby composed pso!-t of the dilatation complex the . f, Fo -!!_, lrsol-r"I=TI_-trrlrn

a.e. on D.

(4.8)

proof.By (i) and (iii) of Theorem 4.10, gof-L is quasiconformal on /(D). Hence by proposition 4.1, g o f-r is totally differentiable on /(D) except for a subset .O of ,rr"""rrre zero. Applying Lemma 4.12 to the quasiconformal mapping /-1, is also of measure zero. Hence by Proposition 4.1, both / we see that f-r(E)

'1purs dlluatr leqt atunss€ feur aaa 'U -gns eql uo snonurluoc ,(pr.ro;tun sr 3f acuts 'alo11 & 3ul{"t }as lceduroc ('e't 'St,f aeg) 'oo - ll uaqn tlt Aq .{lrlenbeur eq} Jo eprs pueq lq3rr aq1 acelda.re^\ pue 'llq'lr) = {J 1es ein 'era11 I={

'Z/,-!,1?lr-q)-{)13 Pu3

(4+!q)I =2

, ( o f r ?!+e ) l = o )

1€rl1q?ns({on} x ft)/ uo slurod r=l{{)} }esalruge sr areql ueq; 'f1.re.r}lqre (0 <) r pue f xrg Jo 'd1e.rr1cadse.r'({on} x ft)/ pue f7 1osqfual eqt eq !,1pu, ll t"j '{ f.rer.aro; (A)t = lO pue fft(ofrlx !7 - lg '[g 'r] r--,I{!il e a{el ,tlF tJ ellug les pu€ Jo sle^ralurqns 1u1ofs1pf11en1ntuto 'uo a.requrorg '6 < & leql erunss? (orf- /t e qcns xlJ'[p'af 'on - n - & ) ff frale lsotule l€ alq€ItuereJlp sI 1aspue e sr (f)g ecurg '[p'a] uo /t f.ra,re.roy([n'a] x [g'r])/ Jo g 'uor1cun3Sursea.rcap-uou exr.g'too.t4 eer€ eql (n).I fq aloua(I'fp,re.ryq.re0 qW '"1x [g'rf =A e18ue1cer 'O uo ICV sr g uorTtu{ap aq7 ''I'' Bruurarl to asuas?W u? CI utDtuop D lo t |ut,rldout Toutlotuoctsonbfinag '(g) p"" (r) suorlrpuoe aql segslles gr uollluuep eql Jo asuesaql ur Suldderu leuroJuocrsenb e lsql ^ oqs gI'? pue ?I't s€ururaT 3ur,uo11o;eq1 'dlesre,ruoC'($) uorlrpuoc eqt segsltes y uolllugap aql Jo esuas aql q Sutddeu leuroJuoclsenb ts 1eql 8'' eururerl uI u^roqs fpeerp e^eq el6 6'7 tuaroat{I

tr

Jo Joord

'v'T'v

'uorlresse aq1 fldurr sarlrpnba e^oqe eql snqtr, 'O J z ,'hala lsorul€ roJ

O*1"^G-t"6) pu€ '0*'6 '0#'l leql fq ,raoqs uet e^t 'luaurn3rerelltuls e 3ursl Zl'V evutej pup II'f uo11tsodo.r4 'g). -'0 lo t(r_/o f) *'l' I o'(,_1o 6) Pu€ ^G-t o6)= z6 "!' "t(r-l" 6)+'t' I t o eAeIIa/rr'(r)I - ^ 'r{lEurproccy'p[e^ q elnr urcq?eql (z lulod e qcns ?V 'O uo z frale Eur1r.r,n 'flarrlcadsar'(z)rf ?s pus z le elqellueraslpf1p1o1a.re1-./ o d pue lsourle .ro; sarlrador4 dreluauralg pu" suorlruyaq'I't

90

4. Quasiconformal Mappings

Ir

Ir r5 0 r> 1 Fi9.4.2.

lf(ro+ i(yo+€)) - /("0 +;vil S * for every cs on [o, D] and every { with 0 < € < 4. Take any curve L in Ri connecting two sides of )Ri which a"reparallel to the y-axis. Then we can see that the length ol f(L) is not less than

I,=il(*-(*-,1i. &=1

On the other hand, let f, be a homeomorphismof somerectanglefr.1= la1,6il x l\i,dil onto the quadrilateralQi which is conformalin the interior of Ei and respectsthe vertices suitably. Then we have /

"

,6;

Ii s | |

\/a;

\2

,6;

tlt'ta,)s tai- ai).t |'ai li,'t'a,. /

Integratingboth sideswith respectto y on lei,dil, we obtain

S s u{oi).4i. Here, we denote by li the area of Qi. Now, suppose that / is K-qc in the senseof the definition G. Since

u(Q)
(l-)
'(t)o +ffi

'ft)"lq j 6)tl

'f11ua1e,rrnbe ro

,$),t - 6)'l < (oETl6)"/ )/ '6 'snq6 13ql epnlcuoc a,ll o+ puel r 3ur11a1

:x ? (dDw 7 ('a)wx

ul"tqo r '(g'' f '[Og-V]ueuelrr1 pue olqerl'ecuelsurro; '';a) flrpnbaur s,1a3uag fq'acua11'g?*e = (0)/ eraq!\

' [, ((o)"/- (o),/)+ s's]x P((o)"/+ (o)"/)+ D'Df elEuelce.raq1 flaleurxo.rdde sr ('g,)/ .t"ql 'a elrlrsod frarr.aro; [r'0] * [r'0] = 'g a13ue1rere raprsuoC 'ur3rro sqlJo pooqroqq3rau e ur

(lrl)o + z'(il'l + z'(o)"1+(o)/= ?)t se papuedxa q ./ l€rll elou 'asec srql uI

o < (o)"/ teqt 'r?elc$ uorlrassB aurnsss arrr'acue11 eql ueql'o = (o)"/JI'(0 <) $)tt 7(il"t

'6'7 uorlrsodordJo;oord eq1 ur pelou e \ sy'anrleEau-uou a.re(g)z/ pu€ (0)"/ (20 pve Id sraqurnu leer alqelrns qq^a (z . "sre)t . ,6re l€ql eurnsspJeqlrnJ feur an Sutreptsuoc ,(g '0 = oz Ie-qI'flqelaua! Jo ssol lnoqtl^t 'eurnss€ feur er* ara11 'oz = z e q?ns xld 'O 'acua11 ) z f.rcrlo lsorul€ le elqerluaragp f11e1o1sr 3f '1'7 uorlrsodord 'loo.r4 Jo uorldurnsse aql segsrles / teqt saqdur yl't €ruural

'0+x)10-x)=qer?qn 'O uo'e'o l'llq > Itll uayl 'N uor?Dlolrplourroou eUl ypn g uotTru{apaqy 'St'V BtutuaT {o asuas ?qI u? CI urvulop o to 0utddoru Tou.totuottsonb p sr I lt 'O uo rIcY sr D '[g'r] > 0c .frerralsotup ro; d ;o uorlcunJ e se [p'a] uo / reqr epnl)uof, ar'l snqtr, snonurluoc {1e1n1osqe* (fi'oa)l }€q} ^\oqs uec ea\ 'luaurn3re eures aq1 fg 'o Jo uorlrunJ * * [g'o] uo snonurluoc * llt--,t'3se 0 - !,lt--,j31€rll apnlruo? flalnlosqe 4(fr'r)!'flluanbesuoC'0 '6 e.!r', - lA? sr f7 r(.ra,ra l€ql ees rr€c ea\'relncrlred u1 <- & se \e?urs'ellug anl€ etlug e ol spuet (ofr - q/((on)a - (n)A) '0n te alqeltuereJlP sI dr e)uts sarlrador6,(rcluauralg pup suorlluya('I'p

I6

92

4. Quasiconformal Mappings

4.L.5, Other

F\rndamental

Properties

of Quasiconformal

Mappings

We state here, without proofs, two of the fundamental and important properties on continuity of quasiconformal mappings. Theorem 4.16. (Mori's theorem ll57D A f is a K-qc mapping of the unit disk A onto itself with f (0) = 0, then - f (rz)l < t6lz1- ,rltlK , lf (rt)

2 1 , 2 2€ A , z 1 f

22.

Theorem 4.17. Eaery sequenceof K-qc mappings of C onto itself firing 0 and I conlains a subsequencewhich conaergesunifonnly with respectto the spherical d,istance. Moreoaer, the limil function of such a subsequence is again I{-qc. For proofs of these theorems and further information on quasiconformal mappings, see, for instance, Ahlfors [A-2], and Lehto and Virtanen [A-69].

4.2. Existence Theorem on Quasiconformal Mappings We have seen that a quasiconformal mapping / of a domain D induces a bounded measurable function pJ on D which satisfies ess.sup362lttt!)l < L. In this section, we shall show the converse.Namely, for every measurable p with €ss.supz(DlpQ)l < 1, we construct a quasiconformal mapping whose complex dilatation is equal to p.

4.2.1. Preliminary

Considerations

the complex Banach space of all bounded measurable We denote by I-(D) functions on a domain D. Here, the norm is given by

p e L*(D). llpll- = ess.supz€Dlp!)|, Let B(D)1betheunit openball {p e L*(D) | llpll- < 1} of L*(D), andcall any element of B(D)1 a Beltrami coeficient on D. First, we note that a quasiconformal mapping with the prescribed complex dilatation is essentially unique. More precisely, we have the following: Proposition 4.18. Let pr be an arbitrvry elernentof B(D)1. Supposethatthere erists a quasiconformal mapping f wilh the cornpler dilatation pJ = p. Then for eaery confortnal mapping h of f (D), the cornposedmapping ho f has the same complex dilatotion p. Conaersely,for euery quasiconformal mapping g with Fg = H, the composed mapping g o f-L is a conformal mapping "f f (D).

( o ' r ) ' c r ) ' ( c ) )atzt ' o p (, p 1 -+ ) e t u " f+l -=o ) q a 3ur11es,tq (C)aZ uo d roleredo reeutl € augap eA\ tr '@me u se 'flartlcadsa.r '(g)a7 ul zt *

'uotllesse eql aAeq e^r "("/) pt" 'g uo ttpuroyun / +- u/ aautg

,Pv,Pffi"il+-ffi"f +=e)"! se,rr3elnurro; s(uearp 'u fla,ra rog ',tpe.r1rqre g ) ) lutod e xg aauanbesSutqloours-o7 ue a4e1 '1xa1i pu"'g o1 lcadsar qgi'a / roJ I*{"/} 'flqenbaur s(raploH fq s.uo11o; uorlresse aq? 'g uo alqe.rEalulsI rl() - z)/11 acurg 'trsJ u1 'alqer3alur f1a1n1osq€sI apls 'I = b/I +dlI fq paugep aq (Z >)D lal '1sq1g pueq lq3rr eql uo turel puo?es arll Jo puer8alut eqt l"ql a?ou erlr /oor2r 'C u! g tlnp uado tueaa .tot

a)),nP,pfi'ff +_#"'[,+=o)/

sa{s4nst uaqJ '(dn

Io

tt puo lt s7uau;1afr,qpeTuesaul^teJo searry)auap1otq.tod PuorlnquqrP esoqn C uo uotTeunt cnonutluoz o ?q I pI '@ > d > ?, qfn d qI 'At'V uollrsodorg 'uralqord-g aql e^los ol Frluasse x opru"tot s,ntadtuo4 letrsselc Euu,ro11o;eqa

'(r)ar),t

= oil,ltl ot,(np,por:[f)

,(q uarrrEfl rurou aql eJeqA{'C uo alqerSelur a.re alll 1etll qcns c uo / suollcunJ elq€rns?au 1e 3o aceds qc€ueg xalduroc eq+ eq (C) n lel 'oo > d ; I qll^{ d f.ra,ra rog 'uolleruroJsuerl fqcne3 'pueq Jaqlo eql uo aql m ua\our1iflecrsselc u e/ urog / lanrlsuooar o1 fem e 'uorlenba aql p ((rl)grl)9 = t ua,u3 ruerlleg ''l pue r/ uaaallaq uorlnlos e ur?lqo a,r,r,'(r/)g = zt tjd'rol aql u-r 11 3ut1t.ra,rag

'(trt)C='el)g=zt uoll€lar e e^sq aruue{} ',t1en1ry 'zJ' ue^€ eq? urorJ 'et)C / / uorleluaserdarelqellns e 1eBa,n3r elqelrnse Surpug''a'r 'tualqold-g..If Eunlos 1srgrePlsuocea.'asodlnduq1 rog '"lrt - zs uorlenba1er1ue -regrprtu€rlleg eql e^losol lr.roqreplsuoceaa'I(C)B 3 r/ ua,r€ {ue ro;',tlo1q 'rb-1 sr 't'? ureroeql o1 f.re1oro3 ,(q tr 6 leurro;uoc8I etueq pue '-Io61€rll s&olloJtl'gl't uotltsodor4fq (O)/ uo'e'e g ol lenbas.r'-lotrl acu-ts '1xatr1'rl = trl - {ottil e^eq e/{'tI'} uo11rsodor43o;oord aql uI s€'1s.rrg/ool2' stu-rddery l?uroJuof,rsen$

uo uraroaql

eluetsD(g 'Z'?

94

4. QuasiconformalMappings

Then we have the following: L e m m a 4 . 2 O . F o r e a e r yp w i t h 2 < p < q a n d f o r e u e r y h e L p ( C ) , P h is a uniformly Hiilder continuousfunction on C, wilh erponenl (7-2/p), and satisfiesPh(0) = 0. Moreouer, Pf satisfies

(Ph)a = h on C in lhe senseof distribution. Prool. First, as in the proof of Proposition4.19,we shall show that the integral on the right hand side of (4.9) is well-defined.For this purpose,defineq by the equation\/p+I/q = 1. Since 1

1

C

z-C--r=r1r-q belongsto Lq(C), Hcilder'sinequalityimplies that 1 a I P A ( C );ll
lo,

d,rda. II l--:--- l axay=lCl2-z'II l--;----l J J c l z ( z- C ) l J J g l z ( z- L ) l

Hence, there is a constant I(o depending only on p such that

l P h ( ( )Sl K r l l h l l o ' l c l ' - ' t(o€, c , e + 0 .

(4.10)

SincePh(0) - 0 by the definition,(4.10)is valid evenwhen ( = 0. Next, set hr(r) - h(z + Ct).Then we obtain

phlCz(,)=-+il"h(z+c,)(4+_6- l) o,o, 1

f f

/

1

zrJJc "\z-(z = Ph(cz)- Ph((r).

1

\

z-Q/

Combiningthis with (4.10),we concludethat

'lc,- crlt-'to, cr,czQc, lph((r)- pn4)l S Kollhllo

(4.11)

or equivalently, that Ph is a uniformly Holder continuous function with exponent | - 2/P' in Cf (C) such that To show the second assertion, take a sequence{h"}Lr ---+oo. (Such a sequenceis constructed, for example, as in llh h"llo * 0 as n the proof of Lemma 4.5.) Then for every h., we have

'(C)"3C q ,trerraroJ (C)-C o1 sEuolaqqd IeqI ) leql pu€ '(tt'l) paglre^,(pea.rpe^"q eM'too.t4 (gt't) p"* (Zt'f),(q aasu?c aa,r'1xa11

(qrr) ftr'v)

'C

'zllqll = zllutll puD

uo rtJ = "(Ud) sa{s4os(C)JC ) t1tuaag 'TZ'7 Bururarl

'(c)"3c >,1' {**fft*rt-4t11i-}"i =o),tt 1er?e1urrelnEurs eql ,iq 'acua11'lerluasse $ lurel peugap ; roleredo r€eull aql ol uorluelle .rno fed am puores aql '0 e , sB 0 ol sa3rarruocapls pueq lr{3tl aq1 uo tural }srg aq} eculs

2 . { r r u , r z ) - z )t " t t - " l \ [ [ ? " + -r' o ( z1,)'ut , = l ) - z l [r y g ] n i = (4,1 I Y) lJ | f.-' )-zcffz:t6 = zPvzPGfi o))('1a) ll ;

(tn)

sarrr3elntu.ro; s(ueerC'asec stql uf (C)"3C o1 s3uolaq q eJaq^l esec aql eulurexa aal.'asod.rnd elqsllns e ul€lqo ol Peou ear'1xa11 slql rod '"(qa) roJ uollsluasa.rder1e.r3a1ul 'uollresse Puoces aq1 pe,rord e^eq e/rl snqJ

tr

cf f

)f

f

'(c)Jc>a'fipxpz41''ld -=npxpdq II || J J I J

seat3 ,(1t1enbaa^oqe aql '(Ot'l),tq 'oo + u lal e^r ueq/t'acue11 C (lo slesqns lceduroc fue uo r(luro;tun ''a'r) uo .{pr.royrunrtlpcol t12,o1 seS.rar'uotut14'oo s u se 0 F oll"U - qll ecutg

'frPaPz4''' "'ld"[

- fiPxPdtu;t [ t J J

J J

'(C)JC 1eBa,ra.

'relnctlred u1 lt d .{rerraro3

.()),q= rpY*{=tt-zt! = (illeltd) ' (z)"tl +t;ri t I se,rr3elnurro; s.uaalg ecueg cff v )-z 'np,pG)1",1 Jlt-=

lzt'v) 96

(^0.r6{;s"ll +)# =eFyta) s8urddel4l l"uroluof,rspn$

uo ruaroaqJ af,ualsrxg 'Z''

96

4. Quasiconformal Mappings

-* AdZ 1rh17= ll;ro>"r-rolzdz = AdZ + Il"ph.(-ph),,d2 =

^dz * ll"ph (h)zdz

= -+

Ad2= llhll1. ll"nenl,dz

Thus we haveproved (4.15).

tr

Lemma 4.21 implies, in particular, that the operator 7 is extendedto a boundedlinear operator on L2(C) into itself with norm 1. Sincewe have consideredthe operator P as that on LP(C) with p ) 2, we consider? also as such an operator on .tr(C). Then we see by the following classicalCalder6nZygmund'stheoremthat ? givesa boundedlinear operator on U(C) (p > 2) into itself. Proposition 4.22. (Calder6n and Zygmund) For eaeryp with 2<-p 1crc, Cp -

sup ll"hllo n€c8p(c),llallr=l

is finite. Hence,the operalorT is edendedto a boandedlinearoperatorof Lr (C) into itself with norm Co. Moreoaer,Co is continaoaswilh respecllo p. In parlicular,Co satisfies lim C. = 1.

P-2

(4.16)

In $4, we shall include a proof of this basic result for the sake of convenience. Here, assuming this proposition, we solve the Beltrami equation. Note that Proposition 4.22 gives the following: Proposition

4.23. For an arbitrarily giaen p (> 2) and euery h e U(C), (Ph), = Th

on C in the sense of dislribulion. Proof. Take a sequence {h"}8, in Cff(C) approximating h in Lp(C) (cf. the proof of Lemma 4.20). For every n, (4.14) implies that f I tl rn".gdxdy= - I I pn".g,dady, p€Cf(C). I I JJc JJc

Here,Ph,, - Ph locally uniformly on C by (a.10)and Thn - 11, in 7r(C) bV Proposition4.22,respectively,as rl + oo. Hence,we obtain

epnltuoc a^r eunual s,1fa6 r(q ure3e'ecua11'C uo'e'e zD = zt salrE osle uorlenbe rurerlleg eqt uaqJ'C uo'e'" "6 =,! 1eBaar'uorldtunsseaq1 fq 1 ) dg1 ecurg

' oll'0 -' I Cq ) dll(" 6,t),t- $ rt),tll= dll,6 -, tll llo ul€lqo e$'ZT,'V uorlrsodor6 fg 'l*('6rl)a="0 aA?r.l arrr'arroqe se 'ueq; 'f uorlnlos l€rurou Jeqloue sr areql 1eq1 asoddns 'uorlnlos '(gf 'f) uorlenba Sursn '1xa11 Ier.urou eql Jo ssauenbrun er{} ^roqs n"qs eAr

(st'r) fleug

'l+('tilh= "t

uorlenba eql ulelqo all^'tC'V uorlrsodor4 Eurlou pue z ol lcadsar qlr^{ sa^r}e rrep eql Eur4ea

'C)z

(trv)

' z+(z)(zt)d=?)t

- (r),,tr e c u e qp u e ' 0 = o a A " r Ia ^ { ' 0 = ( 0 ) / a c u r g ' ( 3 ) D ) D + z - ( z ) I ' ' " ' ! ' I 'I '(C)al o1 Euolaq (rt)l = '(t)a l"eql epnlcuoc usc a^r snql d saop os pu€ ecurs (pu"q raqlo eql uO 'C elor{A{ eq} uo crqdrouroloq q (z),9 I J 'acua11'uorlnqrJlstp eql 1eq1 sarldun (9'7 eururel) eurural s,1fa7y1 Jo esues eql ul 0 = cdr rraloerol tr 'O = (O),9 pu" snonulluoc sr (z)g '96'7 eurural fq ueqa

'c)z

'(r)(l)a-Q)t=Q)t tes

'(Da reprsuor ue? e,r,rsnqJ '(C)d? o1 sEuolaq osle zt '(C)al o1 sEuolaq 1 f eculs pu" 'lroddns lceduroc e seq ztd - ?'J'acurs 'i(;sr1esplnoqs rl rc1 uorlnlos I '1s.ng "/ elllellrap 1er1.redaqt qcrr{irr uorlrpuo? € aArrap aaa /oo.l2'

I€Lurou eql Jo

purrou eqrv.''iluaroeqtr,r,lrl#rlojii"o" I*Iluara'rprurertag aqrrouorwlos 'suor,ppuo?asaqTfrq frqanbtunpeurulJepp sy, uo. qcns 'taaoa.to14J I 'uotlnqu?s'P esueseql u' J uo to - z1 lrt sa{n1os t puo '(3)67 o7 sfuo1aq - 't '0 = (0)/ uotTcunl snonuNluo) D slsrr,eanqT 'l,.toildns lDUl I Urns t 'I > dCtt Tcoilutocq?tn puv { > -llt/ll ql4rr"r(C)g ) rt tuaaa .tot uaqa UWn o ?tloJ'fr1g.to"r.7gy,n I > { t 0 toql qrns q a!.{'VZ'V urarooql k <) a aarTtsod, 'tueJoeql

'mo1q lelueuepunJ Eurmolloy aq1 alo.rd o1 {pea.r are alr

suol+nlos [BurroN aql Jo acualsrxg 'z'z',

' ( c ) J cr a ' f t p r p z d t , " " [ - - f i p x p d t , t " i l f sturddul4l l"ruroJuorrs"nS uo uraroeqJ af,ualsFg 'Z't

L6

98

4. QuasiconformalMappings

that /-gand u r . h o l o m o r p h i c o nC , w h i c h i n t u r n i m p l i e s t h a t f - g l-g should be a constant. Since /(0) = S(0) = 0, we conclude that / - 9, which implies the uniqueness of the normal solution. Finally, the existence of the normal solution follows also from (4.18). In fact, repeat substituting the whole right hand side for /, on the right hand side. Then, we have the following formal series for f" - l:

f" - | = Tp * TjtrD + T(p,r@Tp))+ . .. . This series actually converges in ,Lp(C), since the linear operator which sends h € Lp(C) to T(p.h) € ,p(C) has the operator norm not greater than &Co(< 1).

We set

h=Tp+rfurfi+....

(4.1e)

Then h belongs to trp(C). We shall show that

f(')=P(p(h+I))(z)+z is a desired solution. In fact, tt(h+ 1) belongs b Lr(C), for p has a compact support. Hence, Lemma4.20 implies that / is continuous,"f(0) = 0, and 7, = p(h-t 1). Moreover, Proposition 4.23 irnplies that

f"=T(p(h+1))+1=h+1. Hence, / satisfies the Beltrami equation fz = pf,, and f" - 1 belongs to Ip(C). ! 4.2.3. Basic Properties

of Normal

Solutions

From the construction of the normal solution in the proof of Theorem 4.24, we have immediately the following: Corollary l, Underlhe same circumstancesas in Theorem/.2l,lhe inequalitieshold:

following

I

ll|illp S *L - / I ; V p llpllp,

(4.20)

and,

l/((r)- f&z)ls fi^lrllpl(r for euery Cr,Cz€C.

- czlt-ztp+ l(r - (zl

(4.2r)

Herv, I{o is the conslanlgiaen in Lheproof of Lemma y'.20.

Proof. Let h be as in the proof of Theorem 4.24. Since h = T(prh)*Tpby we have

+ collpllo. llhlloS kcpllhllp Since f/; = p(h * 1), we obtain (4.20). Next, by (4.17) we have

(4.19),

'(c uo 'a'o) trt rl sa{sqos puo'5to |urddou o st rl .tol Tottt"totuoetsonb I uo\nlos 'fA'f uteroeyJ u, sD snuolsurnrrn eutDseV?repun .gU.' uraJoaqJ

IDttltou eW

'3;o Surddeu 'g1as1r oluo C Jo leuroJuocrsenb e sr erueq pue rusrqdrouroauoq e '1ce; u1 'sr uorlnlos l€urrou eqt teql ^\oqs ileqs e,u,,inolq 'g uo fprro;tun tr / o1 sa3ra,ruor Y fnqf epnlf,uoc aan'u {ra.re roJ oo pooqloqqftau paxg e ur crqdrouroloq sr ,f - $ acurg Jo 'oo <- u se .C "! I ) {ra,le ro; C uo rlprroyun f1eco1 t * lsql aes eilr snqJ

"l) - "Illt + dll' l(,t - ,t)ll}ox } * -rl)l {dll'(

l()X'(V)-'l)al = l())Y- O)/l

urclqo e,rn'(0I'?) pue (ZI't) fq 'txaN '(ZZ'V) 'oo 1aBa,r,r, * s se 3 uo 'a'e r/ url ecurspue 'pepunoq,tpr.rogrunetr- url o1 seS.ra,ruoc 1e yo slroddns eql ecurs

.dllz!(uil - ql:"T; t,

dll,('!)- "!ll al"q ell 'acue11

'd llz l(d- ,t )l l o c+ d l l '(" 1-)' lllo cq ) oll("led-,t)).r,ll+ oll(('(V)- "t)",t)l;ll, oll'el) - "!ll selr3 (91'7) 'u .,(.rarre ro3 '1srrg 3foo.r4'

'0= dll,!- "(Y)ll?i,r"

kz'v) .rt t ot uo,4n ros * *

puo 'a + u so 3 uo fiTnttotr,unt +- uI ueqJ

r.J :, :, :, "r:" : " ::,, :;" ;r: : ::," : r,:^::," :,

rr,t

puo 'u to Tuapuadapu!W luolsuo) elqo?rnsD y?!n {W > lrl I C ) z} u? pauwtuo? T"toddnso soy url tuaaa (n)

'u tuaaarol tt > *ll"rlll (t)

ur acuanbaso aq r/{ur'}

nf

:suot?xpuo? 6utmo17ol aq70utfitstTosr(C)g 'fa'f uero?qJ u, sD eq d puo t1pI .Z it.relloaog

'sllrolloJ s" sluarrlgeoc rtupJ]leg eq] uo suorlnlos leturou eql Jo ecuapuedap eleq a^\ 'arourJeq]rnd '$6V) pu" (II'p) fq s,rao11o3 (16'y) acueg

!

'lz)- r)l+l(z))('l)a- ())el)dl t l(z))/- (r))/l s8urddey4i l"urroluof,rssn$

uo uraroaql

af,ualsrxg 'Z't

100

4. Quasiconformal Mappings

To prove this theorem, we need the following generalization of Weyl's lemma (Lemma 4.6). Lemrna 4.26. Let u and o be continuous functions on a simplg connecleil domain D whose distributional padial derioatiaes can be reprc.sented, by locally = FurTher, suppose that u2 u". integrablefunctions. Then there edsts a funclion f which is continuously differcntiable (i.e., of classCr ) and satisfiesfz = u anil

fr=u. Proof. Ftx a rectangle R in D a^rbitra"rily. Take an .Ll-smoothing sequence and {or}f;=, on rR for u and o, respectively, with respect to r? such {,rr}Lt that (u")7 = (u,)" for every n. (This is possible. See the proof of Lemma 4.5.) Then for every n, Green's formula gives

[ 6 ^ a r * u ^ d z ) =J[J[R {("")t - (o^),} dz A dz 0.

JAR

Hence, letting n --+ oo, we have

= o. I pa, * udz)

J8R

Since ,R is arbitrary, we conclude that the indefinite integral of udz * udZ is tr well-defined, and gives a desired function. Returning to the proof of Theorem 4.25, we take a sufficiently large M so that {z e C I lzl < M} contains the support of p, and fix it in the rest of this proof. Fix also a sequence {p"}flr in Cf (C) with llp"ll- ( * such that the support of p, is contained in {z € C I l"l < M} for every n, and that p.n - 1t a.e. on C as n --* oo. We denote by /. the normal solution for ptnfor every n. Lemrra 4.27. In the foregoing situalion, fr: C ------C is a honteornorphism belonging to Cr(C) for euery n. In particular, euery fn is quasiconformal. Proof. Consider a function g with 9z - pngz. Set tt = gz

and

a = gz = I,tnt-r,

To show that 9 belongs to C1(C), it suffices by Lemma 4.26 to see that u is we set a = logu, this is a continuous function which satisfies u7 = (lt"u)r.If equivalent to proving that o is a continuous function which satisfies cz=Fncz*(tt")".

(4.23)

Now, differential equation (4.23) is solved in a similar way to the case of the Beltrami equation. In fact, first as in the case of (4.18), we can construct a solution h in IP(C) of the equation

i, =r1p^rt1+T((p),).

ftz'v)

'C

) e z t r z f r r e a er o t

l(zr)"1- (r)"ll + +r(4 - t) pz - r71 ) a1"-rl(zr)'!- (tz),!111'r11ft sa{s4os ut fuaag '62'} BrrruraT :3uraro11og eql pe?u e,u 'arourraqlrng

'ursrqdroruoauroq tr ''C " q./'snql eloq!\ eql uo qeu"rq panle^ e13urse *{ r_/ pq1 serldurr uraroaql {ruo.rpoirour aq1 'peleauuoc fldurrs sr m3 acurg 'nC uo eAJn? l"rrssrlt fraaa 3uo1eflecrlfpue panurluoc eq uec rl Jo qcuerq fue pue 'mg ;o 1u1od fue;o pooq.roqqSraue ur seqf,u?rqarqd.rouoloq wq r-.f 'uolldurnsse fq mop ('crqd.rouroeuroqf1pco1 s1 ,f roy 'pa8ueqcun sr z3 uo f3o1odo1 aq1 '1 1eq1a1op) raldeq3Jo I'7'I$ ul s€ '? uo ernlcnrls "q1 1c*q 3uq1ndfq 'C uo ernlcnrls xalduoc alau e Surcnporlur ,tci auop aq uec stlJ 'uorlcun; crqdrourbloq e se / raprsuoc o1 sr ursrqdrouroeuoq € sl / ltsqt aoqs o1 ferrr auo '1xa11 ''? = (e)t "? 1eql apnpuoc a,ra'(,p)/ sr os o?uaqpue'lcedtuoc sl ecurg'uado q ("?)/'Surddeur uado ue sr 3f acurg'flarrrlcadsal'/;o 1aEle1aql pus ulsuutopeql ere qcrqn sareqdsuu€tuet1l aqt 'C,tq pue '9 fq eloueq'too.t4 'C oluo g o fi11onyco lo tus.tyd.touro?uto! <- g :i uotTcunl o fi 'gZ'7 BururaT sr t uayT 'crqtuoruoatuotlfi11oco1 sl ? 'Jleslr oluo go ruslqdrouoeuoq s sl salldun eurual lxau aql ! 1eq1 3 V 'aro;e.reqa'C eloq^{ aql uo crqdrouroaruoqf11eco1osl€ sr ut (@ = z 1e alod aldurrs e wq V acrirg '3 uo crqdrouroeuroqflecol q V 'ereq,nfterla earlrsod sl V Jo

"eel.(el"dl- r) = "lr(I)l - "l"et)l uerqocef eql ef,urs'rl11eurg '1c sselcJo sl "/ - 6 t€q? apnlcuor ellr snqJ,'"1 +eql serldturuollnlos eqlJo ssauenbrunaql'acua11 '(C)al o1 s3uoleq I- r0 leturou 'g **"u41 pu€ oo - z leql aes aal 3o ecroqc eq1 ,(q 1 = @)t lo pooq.roqq3rau e ur crqdrouroloq sr t = zD aours '0 = (0)t l"ql ewnss? feru arn 'ara11 '(rurl -1z6url - z6 Surfgsrles(C)rC ] f uorlcun; e slsrxa ereqt leql sarldurr gA't surua'I snql '"(t"rl) - zt snonulluoc sI r uaql ' oe = ! 1as 'u,o11 Pue 'GZ'V) Jo uollnlos snonul?uoce flenlce sL slql '"("rt)

* qurl = zo

pu€

e ur crqdrouroloq sr r leql aloN)'0

rl = ((rl)

* qurl)a = ,o

"".rrrl'oo = z Jopooq.roqq3rau = (z)o -*"urll l€ql os C esooqc alu\ereg

'C+(eil)*q"rt)4

- o 1as'1xep

s8urddell FruroJuof,rsen$ uo uraroaql af,ualsrxg 'Z't

TOI

4. Quasiconformal Mappings

702

Proof. For a fixed n, we can see from the proof of Lemma 4.27 that (/,)-1 is of class Cl. Also, (/,)-r is quasiconformal by (i) in Theorem 4.10. By Proposition 4.13, we obtain tt(J.)-,ofn=-*'r,. lh)' In particular, putting un = p(I^)-t, we have lr"" f"l = ltr"l (a.e.).Thus we have tf ll - l(f"),12)axay = I I lp"lo(l(f")"1' lv"lpdxdy | | JJC, JJC

t [ | l,,l,l(f^)"l2drdy= [ [ U,^l'-'l(f^)zl2drdy JJc

JJC

/rt \'t' /rf \; dxdv dadv = (//" lu"lP \J J"l$"),lP ) ) Then (4.20)in Corollary 1 gives

S 0 - kc)-3 . lle"llo. llr"llo Thus, applying (4.21) with / = (.f")-t and (i = f"Qi) clude the assertion.

(, = 1,2), we con!

be as defined before Lemma Proof of Theorem 4.25.Let {p"}T=t and {/,}Lr on by 2 to Theorem 4.24. to uniformly C Corollary 4.27. Then /,. converges / --* * valid when we 4.29 is still oo), $.2\ in Lemma Since llp"llo llpllp (n ----r^eplace/, and pnby f and,p, respectively. Hence we conclude that /'0 C is a continuous bijection, and therefore is a homeomorphism. Next, since f , - | belongs to Lp (C), so does f z = Ff ,. Thus, / satisfies the assumptions in tr Definition At, and hence is quasiconformal. 4.2.4. Existence

Theorem

We have shown the existence of a quasiconformal mapping with complex dilatation p when p € B(C)l has a compact support. This conclusion is valid for a general p e B(C)r as well. Theorern 4,3O. For eaery Beltmmi coefficientp € B(C)1, lhere exisls a homeomorphism f "f e onb e which is a quasiconformalmappinC of C wilh compler dilatation pr. Moreouer, f is uniquely determined by the following normalization conditions: /(0) = 0, /(1) = 1, and /(oo) = oo. We call this /, uniquely^determined by the normalization conditions, the canonical p-qc mapping of C, or the canonical quasiconformalrnapping of C with compler dilalalion p, and denote itby f u.

'C Jo ,tt Surddeu ab-r/ lecruoues eqt slstxa araql'gg'7 uraroaqa .,(qacuag 'I(C)g Jo luauala ue se r/ pre8ar ue? a^r 'V - C uo 0 - r/ 3ur11es 'O + - r/ tg'lrl 1aspue V: 3f SurddeurIeuroJuocrsenbe qcns xyg'!oo.t4 'O oluo tusyldlouoauroy D ol pepue?se s? y, lo CI uxDttropuDprof .tt.7 uolllsodorg p oluo V qslp pun ayl to 0utddou.tlou.t.totuoctsonbfri,ang '(69'p uraroaq;) ureroaql eruelsfxa aql Jo suorlecqdde lereles elels ar\ 'uorlcas slql Jo pue aql ?y 'auo pansap aq1 s f 'acua11'suorlrpuoc uorlszrlerurou aql sagsrles O d'f1.ree13(rl = 6il 1"qt os zrl peuufapeler1ea\'g1'7 uorlrsodo.r43ur1ou fq'1oeg u1) '('a'e) d - 6il leq? eas u?? a.rapu" 'gl't tueroeql ul (H) fq leurro;uocrsenb 's?stxa osle 'lroddns sl ,r/ o ",tt = f 'ra,roarotr11 "n/ lreduroc e seq zrl s3u1g

{tz'v)

1_(.'rl)"(#H)=",

'slslxe rrrJ ler{} u^roqs e^€q e^\ uaql ?as osle aM

'v)z

$zv)

v-c)z

'{srp

',3\= u*

lrun aql fl 7 alarl/I{

'(r)rt

(es€) slql uI '?uarogeor rrrr€rtlag lerauaS e u r/ 1eq1 asoddns 'dleurg las e^r 'euo palrsep aql sl / leql epnl)uoc e^t eoueH 'suorlrpuoc uorlszrlerurou eql seg$les / 'r(pea13 /z\

-7

'C uo'e'€ (z)r= t ( = l l C- a = Q ) { i l rl/ zz eeq e.rrlsnqJ 'elnr ursrl?Pnsn eql fldde uec a^r leqt os elqerlueregp,t1e1o1osle sr

-?/r)rl ;=(z)l Surddeur leuroJuocrs"nb aq1 'zf 11ur.odqcns fra,re 1V 'C uo'e'e elqerlueregrp ,t11e1o1 sl il 'fV uorlrsodor4 ,tS '? go nrf Surddeut cb-rl l€?ruorr€c aql slsrxe araql 'aro;aq se acueg 'lroddns lceduroc e seq pue r(C)S o1 sSuolaq d uaql

{gz'v)

'5G)d=Q)!

'))z

+aseAr'r/l= (z),0 uorleuroJsrr€rl snlqgltr eql ,fq d laeq 3ur11nd,asec slqt uI 'ur3uo aql Jo pooqroqq3reu aruos ut ('"'r) O = r/ leql asoddns '1xag 'euo pensep e{f sl (I),r,4 /?)n,t teql salldur gZ'V l.uu'er 'asec srql u1 'lroddns -oaql ueql'r/ ro; uorlnlos lcedruoc lerurou aql aq a,iI pl e seq r/ reql asoddns '1srrg 'at Jo ecualsrxe eql .&oqs il"qs e^a 'acua11'suorlrp 'too.t4 -uoc uorlezrleurrou eql pue 8I'? uorlrsodo.r4 fq sArolloJssauenbrun eqa sturddery l"ruroluo)rssn$ uo uraroaqJarualsFg 'Z''

t0I

4. QuasiconformalMaPPings

104

-t . Then g is a quasiconformal mapping of D. By Proposition Set g - f F o f 4.13, we seethat Fg = 0 a.e. on D. Hence, Corollary 2 to Theorem 4'4 implies that g is a conformal mapping of D. Since /p(4) is a Jordan domain, Ca,rathdodory's lh"or"* gives the extension of g to a homeomorphism of D onto ;r'14;. Since

I = g-' o f t', we obtain the assertion. mappingsof A onlo C. Proposition 4.32. Thereexistno quasiconformal Proof. Suppose that there exists such a quasiconformal mapping f : A C. Then,f-1 is also quasiconformal.We set p = pJ-t;then there exists the c a n o n i c a l p - q c m a p p i n g f P o f e . I f * u s e t g = f P o f , t h e n p , = 0 a . e .o n 4 , and hence g is a conformal mapping of A. On the other hand, since g-1(C) = A, Liouville's theorem implies that g-I D should be a constant, a contradiction. Proposition 4.33. Let p be an arbitmry element of B(H)1. Then lhere exists a quasiconfonnal mapping w of H onto H wilh compler dilatation pt. Moroaer, such a mapping w (which can be edended to a homeomorphism of H = I/ U R onto itself by Proposition1.31) is uniquely delermined by lhe f oI Iowin g n o rrn ali zalio n conditi ons: tr(0) = g,

to(l) = 1,

and ur(oo) - m.

We call this unique to satisfying the normalization conditions the canonical p-qc mapping of H, and denote it by ut'. Proof.The uniquenessfollows by Proposition 4.18 and the normalization conditions as before. To show the existence. set

z€H t ( z )= { ; : ' '

I t(t,

z€R z€H*-C-8.

By the uniqueness theorem, the canonical ;l-qc mappi.tg .ft of e satisfies

fi(r)=74 p(H) -In particular, we see that /a(R) = A. Since /P preservesorientation, f tr I/. Hence, the restriction of /i onto l/ is the desired one.

.21t r7tu>l"l>z/rir'l

#,1'-##l{'"''*il*l l l = ( , t ) r n u

uorler3elur;oureuop eql apr,rrpeaa'asodrnds-rqtrod '?/t)al

)

:s^rolloJse /t = ?),r,tr eraq^,r

{ z > l " lrl r 'npxpalr-,Gd)l =ttlr

IJ

{1r1uenbaqt eterurtseo} secgnsq 'I .- (I),r,4 pue (r lt)r,l /fi)n2, = (z)n./ acurg

(ez.r)

.0*- -llr/ll = -lldll se g * fipapoll - ,G)ltz>vtrfI JJ

a/y1'0 poorlroqqfteu aruosur qsru€ d IIe teqf asoddng teqt aor{s Jo llerls '0 pooqroqq3raupexg aurosuo Jo rlsrueA ,{1uopue Jr papunoqfpr.ro;run a.rer/ 11ego slroddns eqt leqt e}oN d U* I '(gg'7 ura.roeql;o ;oo.rdaqf Jc) C lo il Eurddetueb-r/lecruoueceql u 7/ uaql '(72/"21' ?l:.)rt (4'!,t = (r)d

pue (r/;.)nl /t = (,)a! las

'lxeN

0 -- -llr/ll se O- Y'dllt-'Gt)ll 'VZ'Vruaroeql o1 fre11o.ro3 ,tq 1eq1sarldurr(ZZ'V) 6 -llr/ll $a I ts (1)ng ecurg'fi)ng/?)a,t = (z),tt a^eqa^. '69'7ura.roeq;,yo 0'rl qcns fre,ra rog ,/d uollnlos leturou er{l slsrxa araql ;oord eql ur palets seit.rsy '-llr/ll ,C. -llrlll acurg'fpre.r1rq.re (6 a) lpus {lluarcgns q}l^ar/ fre,ra roJ I > d xrg 'papunoq ,(prro;run er€ r/ 1e ;o sl.roddnseql 1"q1 etunss?'1srrg /oo.l2'

o,u: = v'dttqtl 0,,(oo,o I I) puor(C)g3danym

'0 * -llt/ll 80 - Y'dllt- "Gl)ll 0 '(Z <) d finaa tog 'tg'7

BurruaT

'eurual aq1 errord ?srg e^n 'n1l' lecruouec eql roJ 'lZ'' ureroeql ol 6 ifre11o.ro3 3ur,rao11o; ur r/ o1 lcadsa.r {lln ;^{ suol}nlos Isurou aq1 ;o ,(lmulluoc u^{oqs fpeaqe a,req ai!\ 'rl luerf,Ueoc nu"rlleg aql uo ;3f ;o acuapuadap ure?uo? q lo il Eurddeur I?ruroJuof,rspnb lecruouec aql uo st?eJ InJesn pue luelrodul tsotu eqt Jo aluos

sluerJsaoc

'8'?

ilrrBJlleg uo eruapuedeo sluarf,SaoC ur"rtleg

90t

uo acuapuadaq 'g'7

4. Quasiconformal MaPPings

106

isholomorphicon{z € C I lzl > R} Notethat,for asufficientlylargeR,everyFts = on z uniformly to Fo(z) and Fp converges {z e C I lzl ) i?}. Hence,we can * since the other hand, * 0. On seethat Iz 0 llpll. = tt

l z 2 1 1 r u 1 " 1 -z )1 ) ,

fI

J Jt'tr.t,t."tl-lFu(')r

i

z2

@t'(z)Y

,'ll P d x d y

lu

'

Corollary 2 to Theorem4.24showsthat 1r - 0 * llpll- - 0. Thus we obtain (4.28). Finally, for a generalp, set uQ)' u(z\=[ 10,

z€A z€C-4.

Letting /v be the canonicalz-qc mappingof C, we set gF = f Po(f')-r' Again, by Corollary 2 to Theorern4.24,tv '--+id uniformly on C as llpll- --- 0. Hence we may assumethat everyf'(A) is containedin {z € C I lrl < 2} and contains {z e C I lzl < t/Z}. Then by Proposition4.13,everyper vanisheson somefixed neighborhoodof 0. Further, since (f p)" = (gF)"o f, .(f,)" * (c\, o f' . (V),, and sinceeither (/'); = 0 or (gp)t" f' =0 a.e.on C, we have

- 1)" f' '(f')"llp,a+ ll(f'), - 1llp,a. S ll((su)" ll(f')" - 1llp,a Express the right hand side of this inequality as /3*/a. that .Ia * 0 as llpll- --* 0. As for -I3,we obtain

We have already shown

"l

ll ( t " ) os = : - I I l ( s u ) "- \ P l ( f ' ) , " ( f ' ) - L l P - 2 d x d v L_tc"JJy"1ay

(rt .- _-t t l l | kz

^ I x d Y ' l l ^ l t r l " l ' o - ' d]''dr v' |

l'." ( s,-r ) , - l l 2 p d . x d y . I l l ( f ' ) " | ' o - ' ' tJJtt,t.rt

with ft = llpll-. Hence, by using (4.28), where we replace iF and pby gts and 2p, respectively, we can show that .I3 * 0 as llpll- * 0. Thus we have the D assertion. To investigate dependence of. fp on;r, first we shall derive the following integral formula for f P. Lemrna 4.35. Fix p with p > 8 arbitrarily. Let p be an elemenl of B(C)1 satisfuingllpll- . Co 1 I. Then the canonicalp-qc mappins Ip of C satisfiesthe following integral formula:

- rl 112) ( g t ) o =l)zW T all

{ c = l " lrl

I

'arag 1eq1saqdrur(gg'p)

( ) z - 1 ) " ( ( z ) a { ) ' vff 'nnxn::----)?)-#: _ Jl ,,

_),

-t@)n!'onI

zp

z

I

leql 6I'? uorlrsodo.r4goyoo.rdeql ur se ees u€f, e,lr ,e1nurro;s(uearC Surfldde ,(g '{t (9 > l"l > g I C ) z} = ey tes enrysod geurs flluarcgns fra,re .ro;'irro11 'leus dlluarcgns sr -llr/ll * 3uo1 se r/;o luapuadepur $ u, t"ql alou ,.raq1.rng.(Og'l) sa,rr3qrrqal ,ur3r.roaq1 Jo pooqroqq3rauauros ur snonurluoc raplog-(d/Z- il.1 r_(,r1) leq+ epnlcuo? e^r snql 'ge't uraroeql;oSoord eql ur u^toqs s€^ s? ,ur3rro eqtJo pooqJoqq3reu ,prnq Jaqlo eq? uO eruos ur snonurluof, rep19g-(d/6 - 1),tpr.royrun sr ,_(21) 'utErro aql pooqroqqSrau aruos ur snonurluoc zlrqrsdrl fpr.royun a?ueq pue Jo 'leur.roguoo.l .ur8rro aq? Jo pooq.roqq8raua(uos ur r_(V) l€ql ^rou{ e,lr ueql saqsruel t{rl pue poddns laeduroc e serTc{rl ,f1aue11 .gg.t ureroeql go;oo.rd eql ul se fl.repurrs peugep aw lt eraqa.r,'rl ozt s (z)7/ esodurocap ,1ae; u1

(oe'r)

'A> z'

k _ q t d l z l tZul ( r ) n l l

/pooqroqqsreuers,xeareqlffilX'&'.T":':11..i':.1"T: lou se^eqeq (r)n! /r leql {raq? plnoqs ar* 's1q1op oI 'epturoJ s(ueer9 fldde aar '7 uo ?er€ ue fq aprs pueq aql uo aq1 acelde.ro1 ,,rno11 1e.r3a1ur lq3rr ler3alur 'sluelsuo? ar€ €r pu€ y'eraH

. ) z- t G ) a ! o n [ )*)s zP

- I ,o (), * r.

z vQr

I"

+v= onf

-

-,

onf

. \ , " ) " r zl \) (/ z.\ t J "l/= ;Qf ,f i t ) , l =_ ,_r^ )b ) , rJ

e^eqe^racuag'Q)nI /, o1 pnbe * n/(n)nl uaqa.zf 1- m 'asodrnd 'V ol z elq"rr"^ aq1 a3ueqc qq1 .16g eare elq€trnse dq C uo 1e.r3a1ur aprspu"q 1q3r.raq1uo 1e.r3a1ur plnoqsen ,uorlresseaq1anordo; tsrg eql acelda.r

o+=Q)a! v)) ,op,pffi"ff +-;e"'l serrr3(61'puorlrsodo.r6) "lnuroJ s,nreduro4'lsug'too.t4 'Q/t)nt

,V nt /t = Q)r! a.raqm ) ) fr^taaa

"-u(, op,p \'=l'!)" [[ . (' ,)--r- !;) ,") / (r)'G!) il r \ Gz'v)

n- n ,(e \zr _ 1r +- z; - ,) - z \ , . - . - . v f f y ) ( , ) ' G It I) t - ) = Q ) a t

r0I

sluerf,saoC ru"rllag

uo acuapuadag 'g'p

4. QuasiconformalMaPPings

108

as 6 - 0. Since p > 8, we can show by Htilder's inequality and (4.30) that the area integral on the right hand side converges absolutely as 6 -* 0. Hence we have

rrp((\=o!?c \ s / - 2 " i -!r J tt Ja

Gu),(')--L,d,ay U'),Q)0,0,,-c' t - ; l l a j[[ t'@yl-rc z-\

for every C g A.Moreover, since both sides of the above equality, considered as functions of (, are continuous on ^4, the equality still holds for every C e A. Hence, by using the normalization conditions that .f'(0) = 0 and /p(1) = 1, we

obtain the desiredformula. Using integral formula (4.29), we prove the following: Proposition 4.36. It 1t conuergesIo 0 in B(C)1, then the canonicalp-qc nxap' ping fP conaergesto the identity mapping locally anifonnlg on C. Proof. F:xp > 8. Since llpll- * 0, we may assume that, for every p considered, .Cp 11, and hence (-4.29)in Lemma 4.35 is valid. Writing the right hand llpll-

ria" or 14.20;as( + r11;+ i(C), weshallshowthat both 1(() anai(1; converse to 0. First, since (fp)r=p((fp)"_1)+p, we can find, by using H
l / ( OS l M ( l l ( f r )-"l l l p , a + l ) l l p l l - ,e e a for every p. Hence by Lemma 4.34, we see that .I(() converges to 0 uniformly on

4 asllpll* -* 0.

Next, recall that V and rn in (a.30) can be chosenuniformly with respect to p. Thus we can find a constant M such that

l i ( ( )< l M $ G \ " - l l l p , a + l ) l l i l l - ,e e a for every;r, where! is definedby(a.25) in the proof of Theorem4.30.Sinceir is the canonicalp-qc mappingof C, and sincellpll- = lllll-' we seeby Lemma 4.34 that i(C) - 0 uniformly on .4 as llpll- * O. Hence,we haveproved lhat f rt - id uniformly on 4 as llpll- -'- 0. Finally, for everypositive r (< 1), set p',(z)- tt|/r) (z eC), and consider the canonical,rr-qc mapping fl" of C. Set f,(r) = fP'(rz)l fP"(r). Since p1.(z) - F,Qz) = p(z) (a.e.), and since /" satisfiesthe normalization conditions,the uniquenesstheorem gives /" = fP.On the other hand, since fp" - fd uniformly on ^4 as shownbefore,so doesf, on {z e C I lzl < Ilr}. tr Sincer is arbitrary, we concludethal f tt ---+fd locally uniformly on C.

o1 saSrerruoo?/ Q)rl |€rll ees uef, ara'V ) ) frarra.ro; y uo alqerEalur dlalnlosqe s1lue.rofeu sql ecurs 'yg a?w1r(lluaragns elqelrns " qll,$ 3 fra,ra ro3

( l )-lr ) v > ) , l , - r l'ffa-qtdz-lzlw aloqs aq] go puer3alul lueroleur e seq 1e.l3a1ur aql teql sarldurr(gg'y) aleurrlsauroJrun eq1 'raql.l\{ '0 *- t s? y uo ,t1u.ro;run

t r

g+- \fipxp

(r-t ) r - r \ z ( @ ) a , r ! ) ""f _ _ \ | _ _ f ,r)) Q)t ttrJ O)'^/ \z)

eqru-rreqlol rerrurs "" fi1:Tfi:{Lt-?ffii:jl,il luaurnEre ;oo.rd 'Qz/rz1'Q/i"=Q)c Pu"

Qz/ "21'Q/i!),t = ()'!il = G)(t)d

'1xap 1as '0-ts"f,Zuor(luroyun

**(*+T-.)

(se'r)

@:[f+-

o1 seEreruoct/Q)\ leqf gg'? uorlrsodor4go;oo.rdeql ur leql ol relrrmsluaurnSreue,tq,noqs usc e,r\'0 - I se 0 + -ll, - l/(l)rtll e?urspu€

='(nl) (t)a+ $ - "QsrlD(r)rt

erurs 'lsJrJ

'O)'^r+())'r+)"' (3)r/rol (66'p) elnutoJJo aprspueq 1qEr.r eql ssa,rdxg'1frara.rol (3)r/- r/ qq^ p1e^ q (OZ'f) t"qt erunss?feur aar 'gg'7 uorlrsodor4 ;o ;oord eq1 u sy 'too.r4

et.v)

.cr ) ,oo,oWUy"[;-

=e)t^l!

:uotlvlu?seJdat,pliayut aqy soq lnlt '.teaoato141 '3 uo tu.tottun 'C) ) fi.r,aaa"tot sTnia Qpcol * acue6.taauo?e1l puo

t'ii'' = (:)t"lf =--*-----. , )-

(re'r)

O)r,l,r/

'0 *- s? ts -ll(l)rll r rour u?ns (c)-z 0 c ) z'(z)(1pr*

ueqJ ) n ?lqopncqnn

) (7)t puo (c)-z ( z ) ' t y= ( z ) ( 7 ) i l

''a'l'0nt^totaq1u, u?I1r.l,r.l" sl q G)rt tID elqv?tua.ta$ry Q)il 'TnTaunroil aalilu.roc o ro IDar tayt puo'0 *- l sD 0 {- -ll(l)rlll 7oq7asoililng o uo |utpuadapsTuarc$aoc runuH?glo fr,1gutol o aq {(7)d} pI .Le.? ruorooq;, uo acuapuadaq 'g'p

sluarf,Ig:eoC rurcrllag

60t

110

4. Quasiconformal Mappings

/,\ c , _ -fr1a'ao _ ! t f . . / 1 \ r ((r=

" JJ^"1;)t

G34)

locally uniformly on 4 as I * 0. Thus, changing the variable z to If z in (a.3a) and adding it to (4.33), we have (4.32) for every C € A. Finally, the same argument as in the last part of the proof of Proposition convergestothe right hand side of (4.32) locally 4.36 showsthat (/p(t)(()-./)/t

uniformlyonCast*0. Corollary. Let {p(t)} be a family of Beltramicoefficienls depending on a real lhat p(t) is differentiable or a complerparametert.Suppose att =0, i.e.,p(t) is written in theform p Q ) Q )= p ( z ) - t t v ( z ) + t e ( t ) ( z ) , z € C w i t hs u i t a b lpe€ B ( C ) 1 , v e L * ( C ) , a n de ( t )e . L - ( C ) s u c ht h a t l l e ( t ) l l -- 0 as t ---+ 0. Then

1 t,(r)1=1 ;Iu (0+tj t'l u l( O+o|r l) , ( € c locally unifor"rnly on C as t *

0, where

i'vc)=-+|1",v,ffi0*0 Proof. Set 1, - 7u(t) o(.fr)-t.Then

the complex dilatation ,\(t) of fl is given by

/ t'(t)-t'%) ' \ ( ' =t'r,= )

\i:Fmffi)u\r

' , r u/, - ' '

Hence, .\(t) is written as

. \ ( r ) = r i + o ( l r la) s r * 0 in.L-(C), where

^\ -=( ( , - tuP

( f P ) "\

(*:)o(ru)-'

Apply Theorem 4.37 to this family {/r}. Then we conclude that (fi(() - C)/t convergesto

(((-1) / t i l ( (=) - 17 tJ[J[c ^ t az \-; 1 ) k - C ) a x a v

locally uniformly on C as t * 0. Hence, changing the variable z in this integral to (f r')-r(z) and noting that (/r(t) - Ir)/t = {(/, - fo)/t} o f u, we have the assertion.

(qe'r)

) - r ['
f y o*,

/ t*r

= (?)dn

frq pau{ap dtto dtp uotTotu.totsuvr?peql?H aqt '(U)J,? ) d'tfitaaa.rol (g) puD 'I - zV puo 'd o7 TcadsatVpn snonutluoa st.dy (r) o s, ^reql'(2 1) d frtaaa.tog (zsar11) .gg.7 BrrruraT TuoTsuoc

?Dtlllons

dy

'u1(ou{ 'uorleuro;suerl lJaqlrH eql Jo sarlJado.rdssncsrplsrg 11eirr,tlecrsselc ere qcrrl/rr '(66'7 uorlrsodor4) ureroeql s,punur3,{Z-ugJepleCa.rrordo; [eqs a^r eroJalaqt 'uorleuroJsusrl ?reqlrH 'esec Ierlss€1f,eql sl J srrll of Surpuodsar.rocrolerado aq1 leuorsueurp-euo eql uI 'atl = g uo pr3alur reln3urs e s€ paugap ser uorlrsodo.r4ur 7 .role.redoeqa T,?,'V

tuoroaql punrutfz-ugrapl€C Jo Joord 'p.? 'o'"all .ll ur.roueql ol lradsar qll^{.{ll€rlr{d.rouroloqro f11ecr1.{1eue 'fllecrqdrourolor{ ro f1ecr1,{1eue -leer uo spuadap ueql ',t1arlr1cadsa.r ? (r),// 'elourraqtrnd -leer ? .relarue.redxelduroc € ro l€er e uo spuedap (l)rl uaqin

'o-

"''"llf..tp1

, - (torn.f) llL(o)Dl

r

llo*, -ll urr

uaql '? ralaurered xalduoc e ro I€ar e o1 lcadsar qll/tr 0 = ? le elqerlueraJrp st (l)r/ uaq,rl '1xe11

'0 *-

t se C uo f1u.ro;run

n'"slln! ,t11eco1 o1 sa3reauoc u1 'oo + u sB * ,73f '.re1ncr1red "nlll ;rf 0 ueql 'oo <- u s€ uo'e'e d <- url pue u.r(.ra.rre roJ { > -ll"r/ll teql C tel'reroarotr41 leql qcns s?uerrlgaoorurerlleg 3o acuanbase eq IT{"d}

e q {{ t -ll/ll | '''slln/ll} uaqyr > oCrtpq1 asoddn, p,* 'Tti"i"l';

tlll^{ { xlJ 'Z 4 d pue ) Jo I SurddeurIsuroJuorrsenb lecruouec ,tra,re roy a 5 l " l rr \

all"lr r\

l o p , p o l ' {r lrll l/ *

l o p , p o l , rllrll l/ *

/

/ a/r\

a/r\

r"r ,qr. / r_ 1; ",- ::) ,u ir e "f = , , "s1l/ ll l(zz)I- P)tl 'ar las luslsuoc erlrlrsod e xrg :([91] sreg pue sroJlqy ';c) s11nse.rEurmollo; aq1 e.rrr3suorle3rlse,rur radaap aurcS 'qJDuaA 'C f ) paxg fra,re roJ t ol lcadsar qlr,u crqd.rouroloqfl Q)fr>nt uaql 'l ralaurered xalduroc eql o? lcedsar qt-ra aleqa{Ja^e elqerlueraJlp q (l)r/ ''a'r 'fllerrqd.rour -olotl ? ralaurered xalduroc e uo spuedap (irt uaqm. '1eq1 palou eq ol sl 1I uraroeqJpunurt.{2-ugrepFCJo Joord 't'?

III

4. Quasiconformal MaPPings

ll2

salisfies

< Apllpllp,n, ll//pllp,n

wheru ll .llp,n meansthe LP'norm on R'. prool. Since the assertion for the complex-valued functions follows from that for real ones, we may a.ssumethat I is real-valued. First. set

( =€* itt,n> 0.

=+ iu(() r(()= u(C)+ I:*0,,

Since u(() is the Poisson integral of rp, it is harmonic on the upper half-pla^ne .F/, continuous on I/ U R, and coincident with rp ott A. Th" real part u(() is also harmonic on ,E[.Moreover, since

,(o=*l:d#e@)do P G + , ) - ' P G - o )^ " = d , x'+no

=! f

t

rJo

for every( = €+a4 with ? > 0, we seethat u((*f4) every( € R. Next, set

* neG) as ?--* 0 for

-f,t"t}Y. w(0=lr(c)lo

Then we obtain by simple computation that

azw ,*, = -2

l " ( ( ) l P - 2 ) l F>' (0c,) 1c2e H . ffi,G) f,tlrtc)lo-2Hence, applying Green's formula on the domain

D e , R = { ( ef f l n > r , l c l < f t } ' 0 < e < E ( o o , wehave

(

f

J,o.*(j,

i \ a l-uc y

o<>o'

Now it is easy to get a rough estimate , , , , _ 1 t (l(l-*). =o(cl-') lal law |

Hence, letting .R + oor we have

t

Yuso

J 1r1=eyotl

Note also that another easily obtained rough estimate

'eturuel Surirrolo; aq1 sarrr3 3 a1pue c I ) araq^\

gg'7 eururerl'g.ro1e.redoqql ol lcadsa.rqll1\'(C).35)

' ftztz f i Q + ' 1 4{t
,

'l

{;4lcllg3ar^

opey-,(*O;;ooe

Jt I \

+'*'i)""1|=,t,^t

,ro1erado aqr,.a..r "urr"oirT:i"*t:f?lHJ,[Y"1.::*t'T:[i ,,a3?*elerncrr),,

aql o1 U uo uorlsuroJsrrerl 1reqllH erllJo suorlezrl€raue3aq1 Jo euo sV'C uo uoll)unJ e se dtg p.re3ara,n'elqerrea xalduroc e s€ l Surraprsuo3'(qg'l) "t = q oI e alqerr"^ aq1 a3ueqc pu" (C)JC ur d uorlcun; e aqel'mop ? -,

'd ,{.ra^rra JoJlu€}suoc peJrsepe sr ("to\'-

/r-a\\ ot"\T))

-o-, 'acua11

-,,('-,r,(#)) t - -, tu'dlldflll

-!" .a'a114511zlax

sa,rr3eururel s(noleJ '0 ol puel r 1e1',{1eutg

'?par(,p + il^r*f "''(t

-'t'(#)) +?),r*[ t tu dre? l€q} aPnlruoc e^\ snql

+ tq ",,{,,"(rrot('r

* /)

*

+ ))nt :/) } ","(*ur'p

fq papunoq $ aprs pueq lJel eql leql aas er\ 'flqenbeur s(r{srtro{ur4 fq 'a.ra11

*-[ 'JpalQt+ t)nl + o J

-7 ltudllp+l).rl t

-

J

'p.r3e1ur ?€ql apnlcuocaar'(oo'r] uo lr o1 lcadsar qlrm Surler3alur'acua11 1se1 aql uI uollsllueres1p pu€ uor1e.r3a1ur Jo repro eq1 e3ueqc uec e/r pq1 sa11dur1 qcrqal, 'lr o1 lcedsar ql1,u ,(1turo;run ,t1eco1 sploq '.Ztl e3.re1dlluarcgns e qlr^r

'u > + l ' , ( I " l ? l ) w l ftr., r # r l uraroaqJ punurtfT-ugrappC Jo Joord '?'t

TII

4. Quasiconformal Mappings

ll4 Lemrna 4.39. For a giaen p (> 2), lhe inequality

llsell,< ioollollo holds for eaery I € Cf (C). In particular, lhe operator S is extendedlo a bounded,linear operator on LP(C) into ilself. Proof. For every 9, set gs(z) - 9Qei0). Then

llneellp. llsello.*z | e'uo lo,rl On the other hand, for every d, Lemma 4.38 gives

= [* ( [* Wrrloa,\av lneell!, ' J-o \J-o / f @ / 1 6

< AF l .JI- -

l\ .l/ - -

\

l e s l P/ d x l d y "

=allleelll = elllvlll. 0

Hence,the assertionfollows.

theoremis reducedto the following The first assertionof Calder6n-Zygmund's lemma. Lernrna 4.4O. For eaeryI € Cfl(C), Tp = -S(Sp). Prool. First, by Green'sformula we obtain 't tt a /-2\

sp(0= lg n

+(); JJrr,r,r,Pe \a )o,oo

t (

=n^11[[ .*o ur

e " e * c"' l l a , a- y +J [ lzl | tt

"l>,J ["/"/11 1 =;I t t

lPQ,\)or\

9"p,1 2 1

lzl

)

= * C) + JJ.e"(z C)^a'av ; I J.peQ ^dxdu = L ? ( [ [ e ? ) . d "" /d , ) . r0.,\JJclr-Cl

As in the case of the operator P, we replace the kernel Lll, - (l of the integral on the right hand side by

k ( t , C=)

1

l

l, - ..l- A'

z,(e c'

.(oo - u) ry-)

L

- z)g- = l*-a- tl of@- zLln o

P

u

(E;Y'n'uu [[) +=(W lsql alq"lr"^;o

"

t

a

I

*''^-n',,1 *

a3ueqc fq aas a,u,'urJa1lsrg eql roJ sY

-.)ll)- zl {u;t*-ttt ) --a' -,tt>l{a>ttt}S ze I l> I _ln SS's

t-rt;p

tt ni-u

ll Q--tplp

fq aprs pueq tqSy aq1 aceldag

[) + [?] *r =(,,wffi"[ {**F*{a'n-)|

'acue11'oo {1eBeaa U s€ C uo f,prroyun d1eco1 g o1 se3ra,ruoc

11_21{a
/

u/ e

f"etefu)t

**m"lI

tlsrtg tgq? eas uec arr,r

'ale11 1er3e1ureq1 yo f111q€Iluareglp prl.red eql {caq? o} Peau aal

(ze r) {oo,o (**F#"1[) 3 ,4^"[f]++- =(rn)(ars) 'c ) n dra,ra .ro; urelqo aa,r'e1nullo; s(ueerc Sursn ,(g

^"|[ ] ry+= (**F#"[u," [) {0,,0 "f ^"t (E#u," t)('')h[] ry+= {uo* (

)f f\

morL

I I I it=('xas)s \uotr@')ho)as 'ecue11'(C)JC > a 1eql ueroeqt s(lulqndSursn,(q easar\ ro; (C)al o1 s3uoleqdS tn,{t saqdur 68't €IuureT'ptreqraqlo eq} uO d.ra,re

;?l{"1'1 rrerr,a ro;pte^ ,"ur,Xll,t'lJ:l{;":ffi:Jffitffi? IIII'q(ge't) aq1 ;o ecuaEtreluoc et{1 aalu€r"n3 o1 sr uorlecgrpour srqa) I al frarra .ro; 1e.rEa1ur

' (owor|z)q(z)dt"il) o)ds ++=

(ge'r)

uaqr a^eq a^a

uaroaqtr punurtd2-uoroppC Io 1oord '?'t

9II

4. QuasiconformalMappings

116 Similarly, we obtain

!0z ( tt

:{4:)

l.,ll,- Cl) \J Juetsny

- - oz 1 m). \'-p * --/'

Hence. we conclude that

*(ll"f*"*) exists and is equal to z-r=-' Now, going back to (4.37), we have shown that

=* | l.ora(* - i) **\ s(s,p)(,o) {+ a ^ =-i;,rr(w)=-Te@).

tr

Thus the assertion follows.

Finally, the last assertion of Proposition 4.22 follows from the continuity of with respect top. Noting that Co ) 1, which is verified easily, the continuity, Co in turn, follows from the following Riesz-Thorin's convexity theorem. Lemma 4.4I. The funclionlogC,

is conuet wilh respecllo llp on (0,I12).

Proof. Fix p1 and p2 with pi ) 2 for each j. Set oi = llpi (j : 1,2). It sufficesto show that

and Ci = Cpi

llrfllu" S C,'-'.cl.llfllq. h o l d sf o r e v e r ya = ( 1 - r ) o r + t a 2 ( 0

< t S 1 ) a n d e v e r yf e t ' r 1 " 1 C 1 .S i n c e I t l

llrfllu'

' sdxdv s6Ltt(t-a1rllirn,r.,-., rrlJJ"Tf

by duality, we shall estimateil"f f .gdxdy. First, we assumethat / and g are step functionswith compactsupports.For every complex value (, we set

f, r(O = 177GY' t/l and

e G(O = ;r1(t-a(e))/(t-d

lcl'

whereo(O = (1 - C)ar + Caz.Clearly,,F'(O and G(() are alsostepfunctionsfor every(. With suitablereal constants.11, we can write

'y xtPuaddY;o p u e ' g r a l d e q 3 J o s e t o N a e s ' o s 1 y ' [ 9 1 6 ]u e a r t l n sp u e ' h O I ] e m e 3 e g ' [ 6 9 1 ]s a u o l '[gg] Surrqag '[99] ut.*tq 'lttl'ltZl s.rag '[61] sroJIqY '[gO-V] raqoqts '[tO-V] r€{eN pue orreg '[16-y] rauaqc,(g pue uuerulag '[Sg-V] zfztypve zcrr*oufrinel '[qg-V] Surrqag '[gt-V] ralqceqos pu€ uqof 'srag ees'aldurexa roJ 'suollnqlrlslp 'sace;rns uueruelg enl€A Jo pue 'suorlcunJ luel"Alun ;o 'sdnor3 u€Iulely ;o Jo salroaql aql s€ qtns elqelJe^ auo srsfleue xaldtuoc Jo splag snolrel uI osle lnq Jo 'sacedsra[nuq)IeJ;o ,t.loaq1aq] ur fluo paqdde pue '1oo1l€]uau€punJ pue 1ou 'luelrodurt '1n;asne se paztuSocar's{eperrrou'are s3urddeu 1eu.lo;uocrsen$ '[gSa].H.\t Pu€'[69I] !\otsotrl'[g] pre3y ol uolluel]e ll\€rP osle e/11'[60I-v] gpslg1 'actrelsut ro; 'aas 'sSutddeur leuroJuoJrsenb prleds rog '[Og-V] uauelrt1 Pue otqarl se qtns s3utddeur leturoJuo?Isenbuo s1xa1 ctseq Jerl?opue suolllugep asaql roJ 'pasn flluanbaq prepuels aas 'sar1.rado.rd '1 osl€ are 'q13ue1l€ruallxe eql ,(q euo Pue (7$ .ra1deq3jc) uorlel"llP lelncrl) eql '[6-y] fq auo 's3urddeur leuroJuocrs"nb;o suolllugep luale^Inbe reqlo 1o aurog alou ern??alpaterqelac (sroJlI{YJo A Pue 11 s.ra1deq3uo paseq st laldeqc stqa

saloN 'uollJass" aql sa^IraP luaun3re uolleunxo.rdde D eurlnor e 'slroddns lceduoa qt!^{ suol}?un; dals frerltq.re ere f pue / acutg '?=)W

. " / ' l l / .l \l c . , _ I cI l ( t ) o l

l€ql aPnlruot II€? era

,D/rlUll?ol - c7?o1j- r28"r(l - t) - lO)olsot zpl+tp(l-r) uorlcunJ truowreqqns aql ol ,fllcallP 'acua11 eldrcurld leurrx"ru aq1 turrtldde fq ro 'uraroaql sauq-?arql l€elss?lc eq] /tq

',

r,C I tlltll)zc> l())ol ulelqo "^'{I

= } | C ) )} uo'{gePu15

.,1,,(ttlvll),c> t'"-r)/rll())cll'"/'llo)a-rll t l())ol aeqaa{'{0=llC>)} uO'{I > ) > O I C > )} uo crqdrouoloqpu€ papunoqq ())O'relnrrlred u1

"f 'fiptp0.la | =r,t,

'(oo >) l"qt snol,rqosI lI 7,9,(ra,raro3 '{11uenbasuo3 {W > l}ll C > bp+ I = )} uo arqdrouroloqPrrePePunoqtI O)O

"[l '(tunsalruse) O).r.2 = OW 2,rara! = ftpap())g. seloN

LTI

118

4. QuasiconformalMappings

Recently, quasiconformal mappings have played a crucial role in new investigations in the complex dynamics. A new notable result, called the improaed \-lemma, has been proved, the statement of which, for the sake of convenience, we include here. The improved )-lemma. Let E be a subsetof C, and tet f (\,2): Ax E be admissible,i.e., let f satisfy the following condilions: (i) /(0, z) = z, z € E, (ii) for eueryfired \ e A, the map /(), .) : E ( i i i ) / o r e u e r yf i r e d z e E , t h e m a p f ( . , r ) , A -

e

A it on injection, and e is holomorphic.

Then there is an arlmissiblemap iQ,4 i Artsx e -----*A such that f - i on A t t e x E , w h e r eA, t t s = { } € C l l l l < 1 / 3 } . Moreoaer,for eaery fixed \ € Atls,l(f , .) is a quasiconforrnalhomeomorphism of A oilo itself. The second assertion is the contents of the so.called ),-lemma. For the proof, see Bers and Royden [43], and Sullivan and Thurstonl2lTl. See also Slodkowski

[20e].

As for related papers in this field, we further cite Blanchard [44], Douady [52],Douady and Hubbard [55],Mafr6,Sad and Sullivan[135],Shishikura[206] and [207],and Sullivanl2l4l,l2I5l, and [216].

'asodrnd qql 'ace;lns aseq 1U f11eeu s3urddeur IsruroJuocrs€n$ eql uo sernlcnrls xelduroc aqt Jo 1as a{l se aceds reilnuq?ral eq} reprsuoc e^r ueqin 'sa.rn1cn.r1s xelduroc eqt suorlJol$p eq? ol uollrpuo? sseupepunoqruroJ Jo -Iun aql esodurrol sI qqt op o1 ,teu aug 'sacedsrellnurqcral eql Jo uorlrugep eq? ur 'sauo pcrSolodol rrcql rer{?sr 'suorlrpuoc c1!t1eue pu€ e^r}f,rJ}seJeJoru reprs -uoc plnoqs em 'q3noua suorleErlsarr,u! eq} e{pru o1 'acuanbesuoc s sV InJtln{ 'C uo 'y uo lou op lnq lsrxa suorlcunJ crqdrouoloq pepunoq luelsuocuou pue suorl?unJuearC s? qcns suor??unJcrseq 'f11en1cv 'sarl -.rado.rdctytleue xalduroc luaregrp snorJ?Ae^€q lnq 'crqdrouoagrp f11en1mu are y {slp lrun eql pue C aueld xalduo? aql 'acuelsur .rog 'srs,tleue xelduoc 3o 'fpureg a13urse ur se?sJrns crqdrouroegrp flpnlnur lurodalar,r aql urorJ ts?al 1" uu"ruerll praue3Jo esecaq1 ur'ralemog'(I 1e eceld o1 q3no.root sl 11'secegrns raldeq3 yo g$ ees) sEurq.reurse sdno.r3 l"luauspunJ eql ueealeq sursrqdlourosr qlgm paddrnbe g, o1 crqdrouroagrp sac"Jrns uueruarg (paryeur) il€ Jo las aql 'g. 'pa.raprsuoceler{ eru'9, ace;rns uu€ruerg pasolc e rod ;o eceds rellnurq?ral aql se

sacedg rallnuqrlel

Jo uorlrnrlsuo3

cry(leuv 'I'g

's8utddeur railnuq?IeJ pa11ec'sSurddeur I€ruJoJuo?rsenbpurerlxa eql Jo sseuenbrun pue ecuelsrxe eq1 sr;oord eqt ;o ,ta4 eq; 'aceds ueapqcng l"uorsuaurp-(g - 0g) I€ar eql q II€q lrun uado eq1 o1 crqd -Joruoaruoq q (a f snuaS;o ec€Jrns uu€tuerg pasop e go eceds ralpuqcreJ eql ?) 'ruaroeql s(rallnuqcrel arrord plr€ '(Z () 6' snua3 }€q} sel€ls qcrqin Jo sat€Jrns uusruerl{ pesolc esec aq1 ele3rlselur ear 'g pue suor}ces u1 's3utddeu leur U Jo -.ro;uoarsenbEursn fq a?eJrns uuerueql frerlrq.re ue ;o aceds Jallntuqrle1, eql Jo uorlruuep areu e arrrSaal '1 uorlaaS ur '1srrg 's3urddeur l€ruJoJuocls€nbEutsn fq f1e,r 1eure11esaceds Jallnr.uqcral lcnrlsuoc II€rIs eiIr 'reldeqc $ql uI

saJBds Ja[[nuqJ.raI

g ra+dBqc

5. Teichmiller

120 5.1.1.

Teichmiiller

Space

of an Arbitrary

Riemann

Spaces

Surface

Fix an arbitrary, not necessarilyclosed,Riemann surface.R.For every quasiconformal mapping f of R onto anotherRiemannsurface.9,considera pair (^9,/). I We say that two pairs (S1,fi) and (Sz, fz) are equiaalentiff2o f , is homotopic classof to a conformalmappingof 51 onto,S2.DenotebV [S,/] the equivalence

(s,t).

We call the set of all such equivalence classes lhe Teichmiiller space of R, and denote it by "(r?). This "(,R) can be identified with the secalled reduced Teichmiiller space ?#(I) of a Fuchsian model I of R, as shall be seen in $1.2. Letting idbe the identity mappingof .R, we call [.R,id] the basepoint of "(A). The topology of "(.R) shall be introduced in $1.3 by means of the Teichmiiller distance. In this definition, we have used the notion of quasiconformal mappings between Riemann surfaces. Since quasiconformality is a local property and conformally invariant, we can naturally define quasiconformal mappings between Riemann surfaces. However, there is another way: namely, by the uniformization theorem in $1 of Chapter 2, we define them from quasiconformal mappings between plana,r domains as follows. For every Riemann surface E, take a universal-covering surface ,E of ,R. By the uniformization theorem, we may assume that I is one of C, C, or the upper half-plane I/. For every homeomorphism f of R onto another Riemann surface .9, by Theor em2.4,there is a homeomorphism /, a lift of /, oj E onto a universal covering surface .9 of 5 (which is also assumed to be one of C, C, or Il). We say that / is qaasiconformalor K-qcif the lift / is quasiconformalor /(-qc. (Here a quasiconformal mapping of d means a canonical quasiconformal mapping of e composed with a M
'7'I ureroeqJo1 spuodsarroc euruel 3urmo11o; aq; ' r.7 sserdxeosp e1y rJ / H = S t-! .t ! .n pue 'rJ dnor3 uersqrnJreqloue oluo J;o Jg ursrqdroruosr ue elsq e&ruaql 'J)L 'r-!oLo!=G)!g ^q PeuseP$ qcrq^r

'(a,'z)rsa * 1:!6 ursrqdrourouroq airrlcalur ue aA€q a r '/ UI Ie?ruoue, aq1 Sutsg 'tr o7 Tcadsat qryn I to l,ttlu lonuouw eql / qqt II€c a1l ('g ot g;o Surddeu leruroJuoersenb B Jo uorl?rJlser aql s.rprr€ 'flanbrun pau[uralap sr 3f ue qcns 'gg'7 uorlrsodor4 fS) '- pue 'I 'g;o qcea saxu qcrq^{ ,g *- U : ,f Surdderu l€ruJoJuocrs€nbe;o '{p!} 'asrmreqlo al"}s lou op e,raJI H +- II : / lJll eql reprsuoc s,{e,rnpam J Jo '0Jo rpso ?uarualaauiosSolurod paxgs sr oo pu"'I l"ql aurnssefeur ea,r'mo11 '02'Z eululerl ,tq tf Jo slurod paxg eqlJo rlc€e tuo.r; lcurl$p sl zlJo lurod paxg /tue uaq?'eqoq.radfq s-t'I,L fes '1uaua1aauosJI 't={{!Llyo flurtlelnuruo?-uou ,tq t=j{ fa} lcurlsrp fgenlnur are 'cqoqered f,L ere ?eql eas uec aal'f,LJo turod paxy enbrun aql eq fd Surllel ueql '(0I'A pue g'U ssrutueT '3c) aqoqered .ro cqoq.radfq raqlra sr f,L frerra letll IIe JI /r\ou{ eAr'oqy'e,rtlelntutuot-uou are t={{if1 Jo o!\1 fue uaql 'zLorL - ef }es '.reroero141'rL ozL * zLo t,l qtyn zl, Pge Il, sluaurale o11rlsul,gluoc .1"1ce3 u1 'slurod ee.rq1 ls€el 1€ sureluoc {pp} - J Jo slueuela yo slurod paxg il" Jo }es aq} l€q} a}ou 'esec srql uI'e^rlelnunuoc-uou sl J leql arunsse sfemle aar'I'I$ ur uorssnf,srp eq1 3ur,llo11og'g aueld-g1eq.raddn eql uo 3ur1ce U Jo J Iepour uersqo\{ e xrJ dno.rg

uursqrr\{

u 5o acudg roilnurqclal

'Z'1-'g

'(*+'t) Ie^ralur uedo eq1 qlr,lr pegrluepl q (U)J 'ecue11'lualerrrnba {1pru.ro;uoc flpnlnur lou are s luaragrp o1 Eurpuodsarrocsuretuop Eur.rlsq? apnlcuoc eu, 'aldrcurrd uotlceger eqf dq'ranoarotr4i'zf s +t z Eutddeur leturoJuoc eql o1 ropxo? crdolouroq sl .gJo fre,ra lsql pue'{" > lrl > I I C ) z} =S: uleuop Surddeur-;1as leuroJuocrsenb dleur.roguoc st Eutddeur leturoJuotlsenb e fq gr go 3ur.r raqloue o1 luapltnbe a3eun eq1 iraoqsuer e11\'{t > ) z} =U }"qt esoddns'1xa11 leql I C lrl> I '1urod alEurs" Jo slsrsuoc ({O} - V)1, p* (V),1 'pr o1 crdolouroq q Jo qlea eruag U;o Surddeur-;1esFr.uJoJuoctsenbfrarra 1eq1 pu€ 'U o1 lualerrrnba ,tlpurro;uoc sr Eurddeur leuroJuo?rs€nb e itq g;o a3eurl aql '6 leq? eas ot llneurP 1ou s! 1I'to) - v =a tol = u ?"ql asoddnS a\du'oag '{(* >), > l r l > I I C ) z } s u r c u r o pS u t r . r o ' { O } y ' ( q s r p l r u n aql) y Jo auo o1 luele,rmba flpur.ro;uoe sI U JI ,t1uo pue Jr u€Ileq" sr g ;o dnor3 'es?c slrll IelueruepunJ eIIl ?sqt 6 laldeqg Jo U''$ m I {retudg uI Palou aA"rI a^r u1 'aue1d-;1eq.reddn eql 'Il = U. ler{t arunsse sfea,lle e^a'uo araq uro.r;'snq;

tzr

saezdg re[lurqf,ral

'1'9 Jo uorltrnrlsuog rr1,{puy

I22

5. Teichmiiller Spaces

Lemma 5.L. Two points [Sr,.fr], lSz,fz) € "(n) satisfy [Sr,.fi] = [S2,f2l in f@) ,f and only if 0i, = 0i", where fi is lhe canonical lift of fi for each j (= I,2). Proof. Finst, suppose that [S1, ft] = lSz,/z]. By composing a suitable conformal mapping of ,S1onto 52, we may assumethat .S1= Sz, and fi is homotopicto f2. A homotopy between .fr "nd /2 is written as a l-parameter family {.fr}r5r5, of mappings of l? to 51 . Let /, U" th" canonical lift of fi with respect to f . Then the homotopy {ft} bas a unique continuous lift, say {F1}, under the condition that F1 - fi, and {fl1} gives a homotopy between fi and alift F2 of f2. Fix an element I e f and z € Il arbitrarily. Then both of the paths {f'1 o

{z) | ! < t < 2} and {it " t " i;r1rr1z71 | 1 < t ( 2} havethe sameinitial

point fi o7(z), and have the same projection {f, I L < t < 2} on 51. Hence, both paths actually coincide with each other. In p-arlicular, the terminal point F2"7Q) of the former is coincident with hoto ir'6'rQD. Since z is arbitrary, we conclude that

F 2 o 7 o F ; t= 0 i , 0 ) . Since 7 is also arbitrary, and since each of 0, 1, and oo is fixed by some element of f -{id}, we see that F2 fixes 0, 1, and oo. In fact, assume,for instance, that 0 is the attractive fixed point of a hyperbolic element 7s. Then F2"1so Flt = 0i,(to) is also hyperbolic, and has 0 as the attractive fixed point. Hence, we see that

r2(0) = 0. Thus we have shown that F2 is coincident with the canonical lift f2 of f2 with respect to f , and hence 0 = 0 i, i,. Conversely, assume that 0ir- 0ir= d. Then, for every 7 € f we obtain j=1,2.

iiol=0(t)"ii,

For every t in the interval [0, 1] and every z € fI, letting !" be the geodesic frith respect to the Poincard metric) connecting fi(z) and fz(z),we_denote by f ( r , t ) t h e p o i n t w h i c h d i v i d e sg , i n t h e r a t i o t : ( 1 - l ) . T h e n { f i = f ( z , t - l ) | 1 < t S 2] is a homotopy between fi and /2. From the above, we have ftot

=0(t)"

fr,

I e l, t € [1,2].

Hence, "n".y /, is projected to a continuous mapping fi of Rinto 51 - ^92,and E we have a homotopy between f1 and f2. Noting this lemma, we set f#Q)

= {ei l/ ir u .utronical quasiconformal mapping of C such that

eiQ) = if

i-t is a Fuchsiangroup).

We call this "#(l-) the reduced Teichmtiller space of l- . It can be also regarded as the set {d;(f) l0; e T*(f)} of Fuchsiangroups equipped with iso"marked" Fuchsian groups obtained the set of morphisms to l-, or lquivaleirtly, as deformations of f by canonical quasiconformal mappings of C.

€uruerl fq ar'r1cafutpue'paugap{le^r sl eql ul se / qcns fraaa ro;'lxaN'I'9 !0 ol (2I)J I U',g] tutoO e spues qerqn Surddeut aqa'!oo.t4 a^oqe se G)+,t> '(.7)t s? oslo sN rttp^ pa{t'7uapt (a)l,urrlt'Tcodutoc A fi'.t'aqTtng '(sles so) (,t)*Jyr?n pe{lwew s? eql a Io (d1 acods.ta11nu'w?eJ '8'g uollrsodor4 ueqJ 'A aao!.tnsuuDutaty o to Tapotuuocsq)nl D eq J pI aql apnl)uot e^\ snql :3ur,no11oy

t

r

'

uo c! - V r"qr aos e,rir'frerlrqre u ) acutg 'Q)"1 = ())!

U

wUt apnlcuoc utsc a^r

'(oz)!! ll o (oz)uLo ("L)e acuIS t?{",t} ') uorlrun; ecuanbas llr€lsuoc e ol I/ uo r(prro;run {1eco1 se3la,ruoc 'rerroa.rot{ ') = (oz)"t -*'tu-Il e qcns leql saoqs 8I'U eurtuarl;o ;oo.rd eq1 '61'6 uotls I=J{"f} eleq} 'U s slslxe ecuanbas I ) ttla.te .ro; leql qcns J ul -odor4 r(g'g 0z asoddns'f1asra,ruo3 e xld'd ='!9 = J l u r o d ) ! ,e WW ''! g - '! g susatu y?lq,ll 'J 3,L freaa .rog "!6 -'!g 'ecue11'u uo reqlo qt€e qll^{ }ueplf,ulo, arc (L)'!g pue (r)Yg suol}eru -roJsu€rl snlqgl tr'J ) L frarra ro; ueqtr 'U uo z! - V teqt asoddns '1srtg /oo.l2'

"!e='!6 frlstlos 'vuo z! =r! (7,'l= f) lS *A puo!? fi fi1uo

: I

'Z'9 BtuuraT s|utddoul lotn.totuousonb om1 'Tcodtuocs, A 7oq7 asoddng

:3ura,l,o11og eql a^€q ear 'f1en1cy 'e?eJrns uu€uelg 'are11 pasol, e q J I H - Ar ueq!\ (t) +J qll^{ luaPlculot sI (J)J leql a}ou altl '(,f)Z f" aceds pacnpar Jo PUI{ € sI '(t)*J1o (D+J 1eq1des uec e^a'acua11 lurod aur€s eql osle a,rrE(.i')13o 1ulod etues eql Surururrelep s3urddeur o,lrl l"ql r€elf, sr 1t'4reureg e^oqe eql tuoq ('fem eures eql ul J dnor8 uersqcng ,t.re.r1rqrslre roJ PaugeP eq uec (.7)7 aceds rallnurqclal eql 1eq1 elop) '1 1o acodsrelptaq?Nal eql (J)J II€c aA\

'{Q)cbe-l[rn]]=Q)t les 'rnJo

fq elouaq 'lI uo eot = rlt reqlo qcee ol Tuapatnba ate ss€lc acuel€Arnbaaq1 JI [nr.] 'sdno.r3 osle are r-rnJrn }eql uslsqlnJ Q)Cb ) zm'rm sluetualeoA{1}€tll ,tes a6 ar s3urddsu qcns les aqt eq (l)CCI l.l IIe Jo lecluou€? IeurroJuoctsenb C ;o 'sa\olloJ s€ J laPolu aql eugap e^{ rPu€rl raqlo aq} uO uersqf,ng aql Jo (.7)"1' aceds re[nuqcla;

'(6 ul ,ra1deq3 8't$ aq} uo Jo Z {rsrudg ur uartS s?^\ uolllugap esoq,r,r)J Jo (U :t) ("f)Z }as lIruIT "!6- V6 : f s3urddeur leur.ro; z! = rlgr,{po pue y. r(;sr1es(?,'l = D lS r-g -irocrs6nbo Ll l"q? 'rnolaq U'g eurual 3o;oo.rd eql q se 'noqs usc aM 'tlrDureg saoedg rafilrurlf,ra.1 Jo uollf,u]suo3

tzr

ctlfpuy

'1'9

124

5. Teichmiller Spaces

definition of T#(R), we set \ = gi(f). Then /is projected to a quasiconformal mapping.f of R = H/l onto'^9 - H/\, and hence determines a point [S, /] e ?(.R). Thus the original mapping is also surjective, and we have the first assertion. The second assertion follows by Lemma 5.2.

5.1.3. Teichmiiller Distnnce We shall now introduce a topology on "(.R). For this purpose,we define a distanceon ?(,R). point [,S,fj e T(R). Let py be the complexdilatation of the canonica] lake a lift / of / with respectto l-. Then we have

0 i Q ) o f= i o t ,

'rer.

Hence, for almost every z € Il, it follows that

10i0)'" f) . f" = (f, ot). t' and

(oi0)'"h.i,=(iz"i.V. Thus we obtain

pi = jti " t)|/l

a.e.on ff,

t ef .

( 5 .1 )

Conversely, if (5.1) holds for every 7 € f, then we can see that 0i0) = a holomorphic homeomorphism of 11, i.e., belongs to Aut(H). Hence, i"l"i-'is we conclude that d;(l-) is a Fuchsian group, which implies that / is projected to a quasiconformal mapping of .R onto a"notherRiemann surface H/0iQ). We call a bounded measurable function 1t on H satisfying (5.1) with p instead of py a BeVrami differentialon I/ with respect to f. We denote by B(H,l-) the set of all Beltrami differentials on 11 with respect to l-. F\rther, we set

B (H ,rh = {p e B (H ,r) | llpll-< t} . We call any element of B(H, f)r a Beltrami coefficienl on H with respect to l- . Simila"rly, we call a measurable (-1,l)-form p = p(z)d//dz on R such that llpll- = ess.supz€Rh(r)l < - a Beltrarni differcntialon r?. Denote by B(R) the set of all Beltrami differentials on ,R. Further, we set

B (R )'= {p e B (R )'I lpll- < t1. We call any element of .B(^R)1a Beltromi coefficient on R. Remark /. By the definition, B(.R) and B(H,f) are canonically identified togather with norms. Also, for every quasiconformal mapping f of Ronto another Riemann surface, the complex dilatation Fi e B(H,l-)1 of the canonical lift / of / determines

'0 < I f.ra,ra .rog 'P aruet$P Jellnurq?Ial eql 1eq1 sarldurr p Jo uorlgsep eql 'too.r'4 o1 lcadsa.rqty( (lI)J ul r1:{[V ''S] = ud] acuenbasfqcne3 krc a4ea 'ecualsrprallnuty?tal ay7 o7 Tcadsa.tql.tm sy (os1o (,t) *J acuaqpuo) (g)a acodsreIInu'UcNU eVJ '?'9 tueroaql aTayihuoc 'y

{sp }fun aq1 uo ecu"?srparccurod aq1 sr d araqar'(g')pO)

6't {rare ro;

(6d ' Iil)d df,' 't'"=

H -\ / (lttA- rl . ,\ ) I A tu,r i l l n - r l_ t) I VAlt)}3o1dns'ssa= t \l - {,t I

(,-ro'r)v3o1

1eq1saqdurr91'7 uorlrsodord'6 ,lJDu?A 'n raldeq3 '[gg-V] olqel elus]sul roJ aas '.1 dnor3 uetsqcng preuaE e 'uorlsrgrluepr qql repun (U),2 f" leql ,(q PeugeP sI Jo (J)J uo r(3o1odo1e roJ 'acuel ("f),2 oo f31odo1 e '(t'9 uorlrsodo.r4) (U),2 qt1,r.rPeUIluePIsl (J)J a?uIS -srp ra[ntuq]lal qqt ,tq (u,)-t uo f3o1odo1 e eu$aP errr 'lcedutoc sI Ur UaIIA\ 'zd = rd 1eq1 saqdut sq;

'J ) L 'r - (.t)'-!!"t e 13ql aPnlluoc

el\{

'NI u'J ) L'(L)'!e=(L)'-!:'{o

e?urs'pu€q raqlo eql uO '.& uo .,(lurro;runf1pco1 pr o1 saErarr,uotuQ lsql 98'7 uorlrsodor4 "0rl e?uIS'u,tra,ra ro; Q)'lg o1 lcedse.rqlrrlr 3 r, s" urorJ s1r^olloJ l!'oo 0 zt'r{1 u1 u0 1:r.-l'oo "f u se I ("6))I leql qcns Jo IJII Iecruouec eql aq t;:{"f} ecuanbes€ sr areql leql e}oN '0 = (zd'td)P tet{t asoddns',t11eutg

'(26)>t

'?xa1q .(4X ) (0" t6')y teqr l)eJ eql ,tq sno11o;flqenbaut a13uetl1aq1 'crrlaruurfs sr p eruelsrp Jellnuqlletr aql terl? I'g sururarJ;o;oord eqt ul '1srrg aq1 Sutsn fq aas u?c a,ri\'1e.raua8ul (t-t))I = (6)X ecurs se luaurn3re 'ecuelsrpJo suorx" eql sausrl€suorlcunJ sII{l }eq} {caqc o1 oaau aa.zd eqlJo etlol{f, al{lJo luapuadeput { (zd'Id)p pqt aas pue IdJo se^rleluesa.rdar o1 fsea s-r 'zd pu" Id uee!\laq (U),2 uo e?uDIsrPr?nntuq?try eq1 flrluenb stql tI 'arag ''a'l) dJo uolteteilP Isurxeru eql slr€atu (6)y IIe? a11\'(f .lo tltt e Jo l€q1 z1'tt r)6 '(f)y so1 - (d' ril)p ' las eA{ FV o zI o1 crdoloruoq ere qclqa zS' otuo rg ;o sEurdderu IsturoJuortsenb IIe Jo las eql eqz['tI1 ]al'(U)J ) lzt'zg1 = zd'lrt 'IS] = Id slulod o,r,r1fue ro3'lltop 't

'I(u)g 1o Tuatc$eoc,tuo4pg aq1 r/ slt{t IFc e^t

'{d fq lt e}ouaP PUB 3 r/ lueurala ue f11e.rn1eu

sacedg raflnruqf,Ial Jo uollrnrlsuo3

9Zr

ctp{puy

'1'9

126

5. Teichmiller Spaces

we can find a sufficiently large N. such that, for every n, rn ) N., there is a quasiconformal mapping, say fn,^, homotopic to f^o fi-l and satisfying that ( e, where pn,^ = pl..^.In particular, we can find a subsequence llp",-lland a sequence{fni,n;+, }p, of quasiconformal mappings such that {l"r}pr

l ? tn r'n ,*Jl<- z-i , i = r ,2,3,"" Next, let ps be the base point of T(R). Since {d(ps,p")}Lr is a bounded sequence,we may assume that K(/r) ( 1( for every n with a sufficiently large 1( (> 1). Since 1Lr-j

K(fn,,n,*)s#
for every j, we see that gi = fni-r,n;

o fni-z,ni-r

o "'o

fnt,n,

o fn,

is a quasiconformalmappingof R onto .9,.r,homotopicto /r'

and satisfies

i-r

K ( s i 3) K . l l ( 1 + 4 ' 2 - i ) . j=1

Hence, {f(gi)}r_4,

is a bounded sequence.We denote by Kr the supremum of

{rcki)}.

Now, let ii be the canonicallift of gi with respectto l- for every j. Then Fj = lii; belongsto B(H,f)r, and llpill- ( &1 = (1 - K')10a Kr) (< 1). Also, we have

1"" " rr -llPi*'-Pill (2-r ,llui- rr+rll-slll:F;r,,.rll_= llru,,"i+,llfor every j. In particular, {pi}p, is a Cauchy sequencein B(H,l.). Hence, pr = limj**ti exists in B(H,f), and satisfiesllpll- S et. Let / be the canonical p-qc mapping of 11. Then we can show that / belongs to QCQ). Let p = [S, /] be the point in "(.R) determined by d1. Since

,tann ^ ^ , (I d- ( p " , , p ) \ - l l p - p i l l

\

;:)sllT:wll-

I

,, ,r s Tl6yttt'ir'tt*'

we see that pn, convergesto p. Since the limit of a Cauchy sequenceis unique, p, also convergesto the same p. This implies the completeness. tr Now, fix a point lPa-1_,ftl € T(R) arbitrarily. By setting

[.f,].([S,/]) = [S,f " f{r],

[S,.f]e r(R),

we can define a mapping lf1l. : T(R) "(Rr) of "(E) onto the Teichmiiller space ?(.R1) with base point [R1,fd]. Moreover, we have the following proposition.

'lertuereglp € pell€r-os crqd.rouroloq alprpenb Jo e^Il€luesardat e se 'e.la11 '.le1e1ureldxa pereprsuo?aq uec urroJ stql ul a(r?) ,t1r1uenbaq1 II€qs a^\ se

l''

lJr 20q)

I

|

. .,

ltutz\'qlt, ' + )pl"l,,tl

zl

sauroceqzlzpq+ zpl w1t eesa^r '()),t = e fq z ralaure.red z Eurddetu l€?ol eql 3u€ueqc uodl'.leP{ l€r.uroJuo? ctrlau ueapqcng uroJ eql seq repun aueld-areql uo + + zpl / )/)) .lrnpl zklfi 'I< eqlJo {teq-1nd aq1'(t + >f)/(t - N) = { les erlruaqo,'acue11 >I auos roJ z J--frp+ex<---12 -z r - ) I ' a , l+>I uroJeql 'aue1d-rn .ro) EurqclerlsPue uollelor eql o1 (Eutsse.rduroc ul ?)t allr^t uec aarr. elq€lrnspue'aue1d-zeqt of uoll"lor alq€tlnse 8ur,t1ddy'9 3o Sutdderueuge ue aq (15/[< lol'C > d'o) 4d * zn = (r)l - n Io"l Sur,rrasa.rd-uort"lua-Iro '716 rBIIluIs ssn?$pfleqs eA\ as€c eql lo; s3urddeur snuaS;o ((lecluouse,, '3;o Surddeu aulgpEut,rrasa.rd-uolleluelro u€ Jo uollf,elordaq1rapun I; ur lurod raqlo rtue ol lues sr'1 snueS;o acedsreilnruqcletr, aqt'U ur lurod uu"tuerg pesolc'-t.ro1 fue lerll ueeseleq s,lr 'auo snua3go sa?€JJns ;o aseceql uI slerluara.DlqcllBrpen$ crqd.rouolog'I'Z'g srrroroaql

s6rallntuqclol

pue sturddel4l

rallnuqr.ral'Z'9

'1 raldeq3 ur paugep esoql qlla luePlruloc aJ€ uorlres sql ul PeuueP 6J Pun (g),, t*qt uollces lxeu aql uI ^^.oqsoA\ 'k?) 0 snual eq11-III€c Pu€'t;.'(q aceds lo acods;a11nuqz??J e qcns alouap feru a,s.'ecue11'1urod aseq aql Jo luepuadapur $ Ur qcns ro3 aceds rallnuqcrel aq1 '{laurep '(27) 0 snue3 atues eql Jo U sac€Jrnsuu€tuaw pesolt ge ro3 crqd.roruoeruoqfpenlnur are (g)g 1eq1saldu1 g'g uorlrsodotd'6 qroueq

aqTto uorTolsuvrt e (rA)t

'(g); aceds rallntuqtlal er71 1o Tutod asoq (U)Z : *[t/] srql se qcns Surddeur e ilec aiA

*

..,l.r1aurosr ue sr _[r/] ]erll Jeal, tr sr lr ocueH 'r-rto6'r-r{o/^/ qtl^ seplf,utoce?uelslp rellntuqclel eI{} Jo uol}IugeP e'41q 6'tl rtlT,o*Jeql '(U)J I [f ',,S] =b'll 'S] = d slurod orrtl fue .ro;']xaN 'uorlcafrq € sl -[V] fpealc (,U)-f : -[r-VJ ecurs'1srrg 'too"t4 '-[V]Jo Surddeu esreAuleq1 sar-r3(U),-f --

-euroq

'(a),f, u1 s? (rA)J'.topcr,7.tnd ot nr1ifu,ou,oatuott 'se?ullsrp rellnruy?t4 aq7o7 qTtmutstyd.t'ou.to Tcadsa"t (A),t : *lrll |utddDur styJ '9'9 uorrysodo.r4

lvrNrleuros, uo n (rg)a -

sueroeqJ s.ralFurq)ral puc s8urddel4lrallnurq)ratr'Z'9

LZI

I28

5. Teichmiiller Spaces

A family p = {pi} of holomorphic functions pj on zi(Ui) for all coordinate neighborhoods (t/i, zi) of a Riemann surface ,R is called a holomorphic quadratic differtntial on .R if it satisfies p*(rn) = pj o z1r(zx).(z1e'(21,))o 2n Ui fiU3,

(5.2)

where zi* = zi o zk-r. We express (5.2) simply as

e*Qp) = elQi)(lzi /dr*)'. We also write g = 9e)d,22. Denote by A2(R) the complex vector spaceof all holomorphicquadratic differentialson r?. A holomorphicquadratic differentialcorrespondsto a holomorphic automorphicform of weight -4 with respectto a Fuchsianmodel lof .R acting on the upper half-plane1/. Here,a holomorphicautomorphicform p(r) of weight -4 with respectto ,f is, by definition, a holomorphicfunction 9Q) on Il suchthat

p\eDt,e), = e(r), z € H, 7 e r. We denote by A2(H,l-) the complex vector spaceof all holomorphic automorphic functions of weight -4 with respect to f . Remark. From these definitions, Az(R) is canonically identified with ,42(I1,f). In fact, any element of A2(H,,l-) clearly determines an element of ,42(r?). Conversely, for every p = {pi} e A2(R), formula (5.2) implies that the family {piQi o r)((21 o r)')'}, as a whole, determines exactly one single-valued holomorphic function on I/, which belongsto A2(H,f). Here, r: H R= H/f is the projection.

5.2.2. Teichmiiller

Mappings

As a "locally affine" quasiconformal mapping of .R, we take a mapping / such that for some constant &(0 < & < 1), it satisfies

fz = kf" for a suitable local coordinate z around almost every point of -R. More precisely, we discuss a quasiconformal mapping / whose Beltrami coefficient 11 satisfies that

w = kl e&l with a suitable p e Az(R).(See Proposition 5.19 below.) Let a positive & (< 1) and I e Az(R) - {0} be given. Then we call a quasiconformal mapping f a formal Teichrnilller mapping of ,R for the pair (ft, p)

'oqy 'arrlcefu1 pue peugePlla^{ q 34i leql f1dutl 96'6 tueroeqtr,Pue t"qt ilecer 't=[{(llgD.l '(tfy])Y} = (S).1eraq^r 'I'9 "ruuarl 1"qt eloN 96'6 uorlrsodord

'(a),t>ll'sl '[(r)Y's]= ([/'s])3o

,(q ue,rr3

o,!J* (a)1, so (pueq .raqlo Surddeure a^"q aa,r'g,uo t=[{[t7]'llV)] = 3' Surrgeur€ Eqrxg a rlrefrnse sr oI ,- orlJ, oJ ?eql os 6 raldeq3 go g$ eqt uO 'ursrqd.rouroatuoq ur.o,!J uo f3o1odo1e ernporlul e { l€q} pus 't'I tueroer{I uI Palels n ,r"(A)J IIll^{ pegrtuapr.4 pl"J teql IIsctU 'g 3o sursrqdrouoaslPSur,uase.rd-uolleluelro '(e Sursn,(q 1 .ra1deq3 ?) f snuaE1osacsJrns Jo g$ ul paugep?sq1q pp(A),t Pue 'fleurep 'ra}}sl eql roJ aW sl uusruarH pe{rcu pesolc tes priJ lle Jo o,!J pu" 'uotlaasslllr uI '1 raldeqS ut PausaPesoqlqll^{ PeU pp(a),t uollelou eql e6nea,r lit,r"pl are .raldeqcslt{l w pel?nrlsuoc 'J put (U)Z r"qr ^{oqs11"qsem '1srrg '6 raldeq3 g$ ul peugaptg Jo eql esn aal 'asodrnd $q? roJ 'ursrqdroruoauroqarrtlaalrnse u (gt)J aeedsa:1cr1E + t(g)zy : 7 Eurddetusrrll leql ^toqsol $ uorlcesslqlJo esodrndulerueq; '0=6 to! p.r- / pue'0 I dt tolEutdderure[nuqclal € sl (U)/ =,9 tt(g)zy ) dt '$'Sl = Qt)-f

g' : 3f araqm fq paugap

(U)-r *- r(g)cy :1 Sutddeur € eA€q am ',uo1q 'aceds ('leuorsueunp auo sr (A)"V'g' snrol e Jo es€eeql w l"tll ilersg) rolf,el xalduroc leuorsueurp-(g-rg) e f11en1ce{ (U)zy teql s[el rueroaq] s.qoog -uueruerg 'laloeto141'tll . ulou slql qtl/'t acedsqeeueg xalduroc tssE ParaPlsuoc ll '(Z q (U)zy pue '(U)e!. ) dt f,ueroJ ellug q llldll 'esec stql uI <) f snuaE;o sace; -rns uueruerH pasolc Jo es"c aql ,(po rePrsuoc eal 'uolssncsrp Eurmollo; aql uI '1) 'dt railntuq?Iatr, .rred qql roJ Eurddeur e rqlnutlcyaa ro1 |utrltlout (dt 'r(A)zV 9 d luaurala ue rod I€nrroJe IIec e^ytlldll = { leqt arunssefetu am se flduns uelllr^{ ualJo sI 1e.rta1ursrql) W\uil

"[[z='llall 'npxpl(z)d>l 1nd eru

'"zp(z)dt = o5Eurltraa 'uo araq uro.rg

.{r > Illall | @)"v ) 6} = r(a)zv

tes

q ld',lldttrto;lffolt:;"Jtif;:"; e,,nueq,opaiueqcun qlrmdc dq o5aaeldar

asec aql o1 puodsarroc q?Iqa 'sEurddeur Jallnurqclel leuroJ se oqe s3utddeur 'Vllaq o1lenba sI /Jo /r/ ?uapgaoo rurerlleg aqlJr I€ruroJuospreSar an'ara11

6Zr

suaroaql

s.rallnruqtral

put sturddel4l ra11ltnq]ral

'Z'9

130

5. TeichmiillerSpaces

the identification betweenT(R)'td andTftd was given by the same @s. Hence, @e is clearly surjective. Thus we have the following lemma. L e r n m a 5 . 6 . T h e m a p p i n g s i D y : T ( R ) - - - - - - T ; ' oa n d F s o i D y : T ( R ) bijectiue.In particular, Fo = Fo oAy(T(R)).

Fg are

= fs o(Dy o 7. Then we obtain the

For the sake of simplicity, we set t following: Lemrna 5.7. The rnapping t

+

: Az(R)t -

Fc is continuous.

Proof. Let {p"}f;=r be an a.rbitrary convergent sequencein A2(R)1, and rps be its limit. For every n,let fn be a Teichmiiller mapping of .E for gn, and fn be the canonical lift of /. on fI with respect to f , where f is the normalized Fuchsian model for [,R,I]. Set

f^= i,f i;r,

n=0,I,2,...

a point of 1, representing /l. by definition. Then t, =i(g)_is = We set lrn in o /;1 and Fn = pi1^ for every n. Then we obtain

oi;r. @) ,,=(r'i^-'i, ' \'- l'r. Fi" (fo)"/ Sincelim,,*- llp"llt = llpollr ( 1, we can find a positiveI < 1 suchthat

llp"ll< - r (t =0,1,2,"'). When gs = 0, then lim'-llp"llr = 0. Hence,by Proposition 4.36, f,' .orru"rg". to id locally uniformly on I/. Even when gs f 0, we can show the same assertion. In fact, since lim'-llp"-pollt = 0, Qn(z) convergesfo tfisQ) locally uniformly on f/, where Q^Q) is the elementof A2(H ,l-) correspondingto rp,. Hence,letting H' = {z e H I ti'sQ) I 0}, we can show that pr, convergesto 0 locally uniformly on Ht, which is enough to show the locally uniform convergenceof {h"}[1 on .I1. However, since it needs a fairly long argument, we first finish the proof of Lemma 5.7. Since f,, converge-sto id locally uniformly on I/ in any case, i- " t " f;t convergesto iso"l" i;L for every 'l e l, which implies that t, convergesto ts. Thus we have proved the assertion. Now, we return to the proof of the locally uniform convergenceof {i"}p, to id on.F/ even when gs f 0. For every n, we set

z€H I u"Q),

v"(z)=\0,

Itla,

z€R

zeH*

tr

'(t)".t l?)"t = (r).ul

saop os a)uar{ pue'g uo r(pr.rogrunf11eco1pe o1 saS.ra,ruocuy roJ ",_{uor}nlos 'y 'acueg 'V 'e'e ol .reldeqC Ierurou eql Jo g'U$ ur 6 f.re11oro3fq ure3e C uo 0 sa8rerruocosle uy ?eql ees o1 fsea q ll 'C uo flur.rogrun,(11eco1 p.r o1 saS.re,ruoc "r/' e?uls 'oo 1- u s€ y uo 'e'e ol saEra,ruoc"y ux\oqs a^€q e.f$,snr1J l€ql 0 'y uo 'e'e g o1 seS.reauo" un 1eq+epnlcuo? e/d ,_(Irl) o '{,H ) Z to ) z 3 z} uo fpuroyrun f11eco1g o1 seS.reruocun acurs'os1y rH lV 'y uo sorez ou ser{ r(:"/) r(t"nn ecurs'y uo flurro;run,(1eco1 1o1 sa3raluoc

/ ,(t^l) \

vQ'l)"{-r

, \ ,\i,I) ) ,acuag.3 uo flurro;run f11eco1pr o1 saS.raluo, ,_(ynl) 1eq1luaurn3re prspu€ls e fq rrroqsuec arrr'os1y 'g uo fpure; e sr acuiq pue 'snonurluocrnbafgeool pue papunoq I"ruJou .{1uro;run ,{11eco1 I?{r-(1"/)} ,(tg"l eql eas uer eM'gi.,'V tuaroaqJ, Jo leq} { '7 .ra1deq3 segsr?esr.,!' drerra aours 'alog (VZ'V) ;oord eq1 uI pel€ls se,lrs€ Jo '3 uo {puroyun f1eco1 o1 pr sa3rearioc osp "r,f }eq} ,raoqs 'popunoq fpuroyun a.re uy 're1ncr1.redu1 'u d.ra,rero; Ileqs a A ;o sl.roddns aq1

'o

( v ) i ^ l- c ) '

(v)i^t>,

/ty"D \ ,r_(i^l)"\ffi-)

) )=(r)"Y

)

ueqJ 'irl o "r/ se "r3f esodtuocap'u fraaa rog 'lxaN 'p uo {pr.ro;run d11eco1 pr o1 seS.re,ruoc (z l1)ug lG)g = @)l^l 'oo ol spuel u se 1"q1 aes o1 fsea $ lr ueql C uo fpr.royui pl "i se3rer'uoc u4 tol uotlnlos leturou aql '7 ratd€r{C Jo g'6$ ur 6 ,(re1oro3 fq 'ecueg '3 11, uo'e'e oo u s€ 0 +- u/t pue u frara roJ { > -ll"4ll 'f1.rea13.@/1)"a{lI = (z):^t leql ^rou{ a/rrpue'y ur paureluo) sr urt,f.tare;o lroddns aq} uaqf

e Qr=G)'l 'v-J),

'u*;j

v)z

=Q)i^

lASA^r'U Are^e rOJ '1s.rl,{'09't 'esodrnd srql .rog uraroeqtr yo yoo.rd eql ur s? esoduroeap arrr .rjf 'p raldeq3;o 6$ ut pa1e1ssuorlnlos l€urrougo serlredo.rdesn dluo aM roJ'qceo.rdde ,(reluaurale raql"r 1nq 3uo1 fpre; .raqloue e{el aal '1ce3 slrl} Jo yoo.rd e ua,rr3 1ou eler{ e,!r ecurs 'rela,!ro11'flalerpeurrur uorlrasse eql urclqo eilr '7 .raldeqC g$ pue aql Jo Jo 'c u.l leql le {I€Iuau aq} ul 1"3J eql esn eai!,Jr'aeua11 uo'e'e oo <- u sp 0 + pue u drarra roJ { > -ll"rll leq} aou{ a.tr 'os1y '(gg'7 uorlrsodor4 ;o;oo.rd aq} 'Jc) I/ o1 ? Jo ,r/ Eurddeur cb-un lecruouef, eql Jo uorlcrrlser aql $ "? uaqJ srueroeqJ s.rallnruq)reJ pue s8urddeyq rallBurqf,ral'Z' g

III

5. Teichmriller Spaces

132

Simila,rly, (but more easily by applying Proposition 4.36) we can show the following lemma. Lemma 5.8. The mapping f s o@2 :f@)

5.2.3. Teich-iiller's

--

Fe is continuous.

Theorems

The injectivity of t foilows from the following Teichmtller's uniquenesstheorem. Theorem 5.9. Let f he a Teichmiller mopping for an element 9 e A2(R)1, and letT(p) =fS,fl. Then eaery quasiconformalmapping h of R Io S which homolopic to satisfes is f

llpr,ll-> llprll-. Moreoaer,the equalilyholds if and only if h = f . A proof of this theoremshall be given in $3, for it needssomepreliminary discussions.Returning to the proof of the fact that T is a surjectivehomeomorphism, we note the followingcorollaryto Theorem5.9. T and t are injectiue. Corollary. The rnappings Proof. By Lemma 5.6, it sufficesto show the injectivity of 7. Assume that f (pt) = T(pz) for somepr,pz e Az(R)r. Let /i be a Teichmiillermappingfor gi and.T(pi) = [Si,/i] for eachj. Then the assumptionimpliesthat there is a conformal mapping h of Sr onto ,92such that h o /1 is homotopic to /2. Thus Theorem5.9 gives

> llp,t, ll-. llp,r, ll- = llttnoy,ll-

Similarly, since h-l o.fz is homotopic to fi, we have

l l p r " l l2- l l p r , l l - . Hencewe concludethat llp;,oy,ll- = llprrll-, which impliesthat h o h = fz againby Theorem5.9. In particular,F!, = Fiz. Thus if pt = 0, then 92 = 0. If 9r * 0, then ll91ll1 - llrpzllr,and n/lprl = pzllprl a'e' on R. Hencewe concludethat 92/91 is positivea'e. on.R. Sincepz/pt is meromorphic,it should be a constant.Namely,there is a positiveconstantc with gr = cgz. Sincellrpllll = llpzli, we concludethat c = 1, tr i."., pt = 92, which showsthe injectivity of 7. Fo is an Lerrrma 5.to. The imaget(A2(R)) of A2(R)r undert : Az(R)r -------+ onto its irnage. open set, andt is a homeomorphism Proof. By Lemma 5.7 and the Corollary to Theorem 5.9, we see that t is a continuous injection. Since,,{2(,R)ris homeomorphicto R6c-6, Brouwer'stheorem D on invarianceof domains(Theorem3.11)givesthe a.ssertion.

'(t@)"v)t)r-(soo o.d)= ((a)zv)-t = a '0I'g pue 1aBaru 8'g seurueTurord '-L to1uorlress?eql ^loqsol seclgns 'g'g eurural 8ur1ou[.9'too.t4 1r ,(A),r .69 = (r(g)zV) [. ruo = (I(U)zy) 1, 'Qauto71'aatTcaftns ?ro -L puo -L sfutddpru ?ttJ .gT'g BrrnuaT tr

'1xag 'pe1)euuocosle $ '.{ t*.{l ,tldtur g'g ptr€ g'g seurural

'p91?euuocasl/rrJI3 sl (U)Z snr{I'[.f ',S] prr [p!'lf] fuloa es€q aql uea^r?eqstceuuor qctq,rl (g)g ut a , r . r n cs n o n u r t u o c eu r " ] q o s a a u e q t r ' @ ) r l = r g 1 a sa / v y r / l {t I l;0l[t/'rg]] e aq T lel'I > I t 0 q rur€Jllag esoq!\ g;o Eurddeu ?uel?Ueoc l"uroJuorrsenb 'too.r4 qll,r{ 3 fra,ra rog 'd - Iil tas pus'(A),f,>'[/',S] futoa f.rergqre u" xld 'p?l??uuo?ero 6l puo (Ah :3urao,o11o; eq? ilecal am '1srrg '7;o

secoilseUJ 'ZT'g BtutuaT

,(1v'tlcekns eql r'roqs ileqs aal 'fgeutg

'snonurluoc q -t ?3r{} aPnpuos ein'{rerlrqre s o1acurS 'urErro eqt ts snonurluo? st t; snq;

'(oo - u) o--

((4)ra'6)'t')p +l4ll+?o1'

elsrl a/$'u fra,raroJ (Illuf ll - I)/(Illutlll + I) o1 pnba s1 u4l ro; Surddeurrallnuqrral e Jo uorlelepp Ieturxeur aql aculs 'oo 'nog r?{ ",/l} eauenbas r(ry)zv <- rll"/1ll fue ar1e1 ttr s€ q?ns ul t"ql 0 'ur8rro eq? (sEurddeurrel1ntuqcrelSursn,(q o1 ,{pelrurrspauyap) ts 7 t(rg)zy : 11 to1enrl osle sl slql yr dluo pue JI a51e snonurluocsr (-U)-f + 'o1 'acua11'aEeurr oluo ursrqdrouroatuorl € s-r pue slr 1, Jo pooqJoqqEtauatuos r ( U ) z VI L o * l f l " r - ( l r ) = ur peugap{lam s r t ( t g r ) a v+ t*Ut Io,-(!) saqdur 0I'g surureT'(O)? = (a1)/.acurg'trJo esecaql ur se fe,n etrrcseql uI t,tr-

r(rg)zy :r1o Keo ul = ,!lU)o V

Surddeur" eugapeA\ 'f.rlaurosrarrlcelrnse q '[I/] g'g uorlrsodor4fq ,raouqe^{ ueqJ '(U),1 f"ql '1urodes?qeql Jo (IU)J .- (g)Z Io Wg'tgr] lurod es€qeqt o1 d spuesqerq,r : *[t/] uorlelsrrerle reprsuoC'[V'IU] = (6)L - d 1aspue'fprerlrqre r(U)zy 3 d lurod e xg 'asodrnd stql rog 'snonulluo? q 7, lsql ^roqs ol sulsrueJ?[ '3r uo snonurluocsl r--r. ryqI ,(1dung1'g pue '(r(U)zV)I = '(Keo ul) o g'g spurrrurarl A uo paugapjla^rsr ,_L = r-L a?urs 'loo.t4 r--L leq} s^{olloJ1-t'6'9ureroaq; o1 drelloroCaq} {q e,rtlcefutsry a?uIS 'afputt, s?, oluo rustrldlnutoauoq

o q (U)J

1-

r(A)zV

: l,0utililout

eVJ '1.1-'g BurrrraT

suaroeqJ s.ranlruqrlal pue sturdduq ralllurqf,reJ 'Z'g

ttI

134

5. Teichmiiller Spaces

which is an open set in ?(,R). F\uther, Lemma 5.12 implies that 7(r?) is connected. Hence the assertion follows if we show that the relative boundary 0E of E in T(R) is empty. Now, suppose that AE * 6. Take any [S,/] e d,E. Then there is a sequence {p"}L[r in,42(R)s such that T(p") .* [,S,/], and llp"llr ---- 1as n .--oo. Let /, be a Teichmiiller mapping for gn. We set T(p") = [S,,f,].By the assumption, there is a quasiconformal mapping h, of S, onto ,S which is .-* 0 as n I homotopic to / o fil for every n such that llpl"lloo. In particular, for a suitable & < 1, we have

l l P r " l l * < i ,n = I , 2 , " ' , where g, = h;L o f. On the other hand, since g,, is homotopicto fn, Theorem 5.9 implies that

l l p r" l l *2 l l trt" l l -= l l p l l --

1 (n-

oo) .

This is a contradiction. Thus we conclude that 0E is empty.

tr

As a corollary to Lemma 5.13, we obtain the following Teichmiiller's eristence lheorem. Theorem 5.14. For euerg quasiconformalmapping f : R -

S, there edsts a

Teichmiillermappinghomotopicto f Lemma 5.13 finishesa proof of Teichmiiller's lheorem. Namelv. we have provedthe followingtheorem. Theorern 5.15. The mapping T: A2(R)1* phism.

"(Ii)

is a surjectiuehomeomor-

In parl.icular,T(R) is homeomorphicto Az(R)r, and,hencelo Roc-o In the course of this proof, we have also shown that all representations we have considered as the Teichmiiller space of a closed Riemann surface of genus S P_2) are mutually homeomorphic. Corollary. The spacesf@), T(R)otd, Ts,T;td, homeomorphicto each olher.

Fs, and R6c-a are mutually

,tlluarcgns" qly{'{t > > > z?rc > a z} ureurope sdeur) srql 0'!?, O C lrl I eq }ouue? leql Jeplsuoc{eru ear'{errrfuy'0dJopoor{roqqftaufue ur panlerr-a13urs

= zlrd"onf = rD er(z+*)z ry

?"q1 ees aA\ 'acua11'od lo l) pooqroqq3rau etuos uo z eleutproot lecol elq€llns € roJ 6 'zP*z ruJoJ eql ur uellrru. sr d 1eq1pa,ro.rdq 1l'(I {) u^llepro p 61o '0d fes'oraz e 1y ('61'g uorlrsodo.r4oqe aag)

" #'=$'= 1eq1 uorldurnsse aql urorJ sl',rolloJsnll

. m l l r r=/ l1l , ) t t + ) = O)s ,{q uaar3 g Eurddeur euSP u€ se ,{11eco1 paluese.rda.rsr a1.ro; / Surddew railntuqcretr e 'seleurprooc-ol Sursn fg ('sapurprooc IerolJo luapuedapur ele a1;o soJezleql lecag) '(4 ur 'ro) 0d punore aTourp.tooc-6e Surddetu qqt IF? a7yyC o?q 72 ;o Surdderu leruroJuoc e sarrr8 2 >d

'zlt6

"oof = "'

fq peugep uorlcunJ eq1 pue(od Jo , pooqroqq3tau auros ur qcuerq crqdrouroloq panl€^ e13urse seq zp"1rQ)dt - t(A)"V 3 d = ueql'dt oraz € sr ue xlJ ) 0d lou luaurele 1o A JI'{0} "1rdt 'I'g'g IBI+uara.SrC crlerpun$

crqd.rouroloH B ,(q pacnpul

r(rlouroag

'fgatrq 'uorlcesqns ((crtrleru,e qf,ns ureldxa aru. lxeu eql uI 'rtllncgrp ou qlr^r pernporlur aq u€c tl13ua1pu€ €are s" q?ns suorlou eql 'reae,no11'dt go orcz.r(.ra,re sale.reue3ap 'Surddetu Jallntuqcrel 1e (f,rtr?eur,,srqS uarrt3 aq1 o1 Surpuodsarroc I(U)uy Jo luauala eq1 x "zp(z)d - o5 a.raqr* '9. uo "lzpll(z)61 = "sp ((crrleur,, eql raprsuor ain 'crrleur e qrns sy 'Surddeur Iellnluqcletr uartrS aql qll^{ pel?I?osse cularu eruos 01 lcedser q1r,n scrsepoa3 raprsuoc ol lernleu aq feur l.r 's3urddeur rellnuqcrel Jo eseo eql ul uelg 'rrJ -1eu ueeprl)ng aql o1 lcadser qll/'a scrsepoa3e.re C uo seuq teql ileceg 'Eurdderu rellnurqclel e o1 lcadsar qll^{ (seull, ;o turueaur eql ssnrsrp aill '1srrg 'seur1 ol saurl puas ,{aq1 leq} sl s3urddeu euge Jo sellradord elrsrcep eq} Jo auo 's3urddetu re[nuqoretr pa11ecsSurdderu,,aug:e d1pco1,, 3ur,t1dde fq uear3 are ,,fcuercge,, lsaq aql q?!la seJnlcnrls xalduroc aq1 Jo suorl"ruroJep 1eq1 slrasse (6'9 ue.roeql) uraroaql ssauenbrun s.rellnuqcrel

uraroaql

ssauanbrun s6rallntuqrlal

Jo Joord

uaroaqJ ssauanbrull s.rall$uqrral

98I

'g'g

Jo Joord

'g'g

5. TeichmillerSPaces

f36

2)t, 0 < small r, conformallyonto a "domain" {( e C | 0 < ""g( < (-* (-plane. we also call ( a may Hence, over the spread lClcZrt^+')lz/(rn*2)) p6. g- coordinalearound Now, we considerthe "metric" ds2 = lgQ)lldzl2, which is nothing but the pull-back of the Euclidean metric on the (-plane by an arbitrary grcoordinate

c.

To make discussions clearer, we consider the lift Q € Az(H,f) upper half-plane ff with respect to a Fuchsian model I of R. For every piecewise smooth curve C on I/, we put

of I on the

9 b = [ l,vQ)lu2ldzl. JC

We call this lCl,p the Q-lengthof C. For any two points 21,22€ I/, denoteby L"r,"" the set of all piecewisesmooth curvesconnecting21 and z2in H. We set

do(rt,rr)= cr9!,,,"1c1a. We call itlhe Q-distcnce between 21 and 22. An element Cs of L2r,7, is called a t/-geodesicbetween 21 a\d z2 if lt satisfies lColo = d,v(21,z2)' Now, we describe how a r/-geodesic looks. Assume that there exists a t/geodesic C6 between 21 and z2 in H. For every P € Co which is not a zero of ',i, tn" lenglh-mlnimaliiy implies that G should be a segment nea,r ((p) on the (-pla,ne, where ( is a rlcoordinate, i.e., the composed mapping of a g-coordinate C6 may and the projection of I/ onto.R. At a zerop € Co of rf of order n)0, be broken. However, the angle at p should not be less than 2tl(m* 2). (See Fig. 5.1.)

yi

(lctlo < lCol,pfor 0 < 2r /(rn + 2)) Fig.5.1.

We call a closed arc -t on H a tf-segmenl if, for every interior point p of.L, L is mapped by a r/'coordinate at p to a segment. By the definition of a tf-segment, it

'syutoil pua sp 6u4cauuoc crcapoaf-dtanbtun ?Ul s? ,I 'z(.re11orog Tuau0as-dty (zz .ro 'UOllf,IP€rlUOC e SarttSqCrqar D rz ol Eurpuodserroc 1ou f, fre,la roJ (Z + !u)/t7 { fB ,ra,ra,nog .s,f aerqt ls?el 1e rog (6 + !u)/vZ ueql ra1ea.r3lou aq pFoqs !0 wql s,rrolloJq'0 < N eours t=!

'k+ til"Z=(e(Z+!ut)- "dZ

r€rrt apnrcuo,"^1"r.t"" 'luaur3as-de uo 0=Qp?w)p7+Q)QEwp leql

1lrou)I ellr 'raqllnJ

I=f

'tty= (t6- ")T + Qparqpoef er\eq ern 'pueq raqto eql ug '^r(1rcqdr11nur Surpnlcul O q dt 1o sorcz Jo reqr.unu aql $ l.r ereqru t - f

o e f . t t = 't0[ut'3+ - (z)/t?wp rLNT, I

'aldrcutrd 1€IIt ^{oqs us? e^\ luatun3re eql ,(g'f,.,tre,ra roJ O u-r t+!7 pue f7 uee,lrleqa13ueaq1 eq (0 <)fd tat 'os1y 'oraz aqol !s, irro1€a^{ pu" I? - t*-I araq,u'f, frarr.ero; r+!7g ll +n 4 t--,j{|il fq uraql alouap pue 'O p Oe Jo sorezJo rapro aql eq lur p"l'Q 7 u") f.repunoq eql Jo uo.rleluerro elrlrsod eq1 o1 lcadsar qlr^r Japro us ureql e,rt3 e7y1 'sluaur3as-dJo raqunu elrug sJo zC prre rC(eroJeq uees ueeq seq sy ls.rsuo? 'H ul CI ureruop ueprof e spunoq e, n I, uaqJ '{zz'tz} zCl) Ig teqt r(lqereua3Jo ssol tnoqlr^{ erunss€{ew au'.,(ressaceu;r slutod3o red e1qe1-tns e qlytr zz pue Iz Sutcelder ,fg 'alduns arc zC pu€ rC ',t1.rea1c ueql'cz pue rz Eurlceuuoc zC'rC scrsepoeS-al o,rr1ere eraql tsql asoddng ;l'oo.l2' 'anbtun s! zz puo rz futlcau -uo? crsepoe6-0"rtt ' H ) zr'rz slutod l?urlstp omy fiuo rof, .gl.g uorlrsodo.r4 'crsapoaS-de;o ssauanbrun3uuao11o;aq1 aleq elr 'os1y 'uraq1 turlcauuoa crsapoaS-q!s slsrxa araql 'Il ;o slurod oa,r1f.ra,re ro;'relncrlred u1 'acuelsrp4 sgt o1 lcadse.rq1u,relelduroc q I/ teql noqs r(lsea u€f, e^\ uerlJ 'u eceJrns uutsuaru pasolc € Jo es?? aq1 ,(1uo Jeprsuor aiu 'are11 '0 ur repro < Jo otez e 1e (Z + *) l"Z ueq? ssel 1ou a13ue ue e{€ru slueurEas-d qens o^rl uaes elsq elr.r'ra,roaro141 l"ql '61o sorc2 to'zz lo (Iz raqlla are slurod pue esoq$ sluauEas-ol Jeqr.unualrug Jo s Jo slsrsuoc .g ;o slurod olnl Surleeuuoc crsapoaS-d fra,ra 'Suro8aro; eql uroJ.{ '(t - z i e sr (uo Suop 116olnpour > I 5 O) (l)z luaurEas-ol,tue luslsuoc I eraq uorJ zzp@)QEre su uatlrr^r fldurrs q qclq,lr) "(l),r((l)r)fEre 1eq1 r"ep sr uaroaqJ

1,8I

ssauanbrull s.rallgruqf,ral Jo Joord 't'g

138

5. Teichmiller Spaces

Now, to prove Theorem 5.9, the following lemma due to Teichmiiller plays a crucial role. To state it, we prefer to returning to ft and rp. In particular, the g-Iength lll, of a curve L on R is defined by

l L l v =J[Lv f , ' . The projection of a rf-segment to -R is called a g-segment. Lel h : R + R be a quasiconformal selfLemma 5.17. (Teichmiiller) mapping of R homotopic to id. Then lhere is a positiue constant M depending only on R, h, and g such that

l h (L )l ,2 l Ll*- M for euery g-segment L. (Here, h(L) may not be reclif,able,i.e., it may happen

that lh(L)1,- x.)

Proof. Let h b" th" canonical lift of h with respect to the canonical Fuchsian model f of .R. Then it suf;Hcesto find a constant M such that

li'G)lq>lLl,e- u for every rlsegment L r" n. fo, every7 € f. Hence, First, by Lemma5.1,it followsthat 7'rol :1oh connectingz and h(z) for everyz € H , we have letting C, be the rp-geodesic lC"lo

-_

lCrpllo

for every1 e f . Since-Ris compact,we seethat M=2sup{lC"l,ilzeH} is finite. (Note that lC"lq _is continuouswith respectto z.) Now, let a rf-segmenti be given.Let z1 and z2be the end points of i. Then the curve C"r.ir1L7. Cr"-'aiso connects21 and 22. Sincei i. . rf-geodesic connecting21 and.22,we obtain

* lc""lqs li6)la+ M. lLlos lc",lq* li,G)lo Thus M is a desired constant.

5.3.2. Preliminary

D

Considerations

A prototype of Teichmiiller's uniqueness theorem is the following Griitzsch's lheorem,which treats the casewhere .R is a rectangle {z = x*iy eC | 0 < c < r,0(y(1).

'l = rl 'sl 'p! - 6 D leqJ WLll saoqs l'? uotlrsodor6 ;o ;oo.ld ar{}Jo JIeq puoceseql ul se luaurn3re arueseql 'g 'S' ecueqpue 'cb-1 sr ;o secrlre^[e saxg d ecurg Jo rorrelur eql uo l€r.uroJuo? - d lsql saqdurrqqtr 'U uo 'e'€ = 'trt l€r{l aese^\ snrll 'U uo 'e'€ I ,-t o rt

l"(V)lt= l"(V)l Pue

' l ' ( V ) l +l " ( V )=l l ' ( V )+ " ( V ) l to'y

ul€tqo e^1, ueql 'sp1oq,{1r1enba aq1JI '}xaN ('g'7 eunual;o;oord eql qll,r.ra.reduro3)'tl < rtl lerll ',(lluap^Fba = .rls < r)1 1eql epnpuor arlrarueH '(tq - t)/$t + I) = rX areq,$ ,s. (.rrrr)l

-,y$0"[ . np,pel,(!)t [( uo,offi

[I t

ar f \

l'('/)l)JI JI )lt "' |np,p(l'(t)l+

z (

sa,rr3dlrpnbeur(zre^rqcs'z(tl) *'(!)

='(t/)

acurg

'nprylasy11"U t,

JJ 1aBa'r'r '[1 'g] .re.,rof o1 '[I '0] 3 n fra,ra 1sou1e ro; lcadsal q]l^\ seprs qloq Surler3e]ul

opl@t+ ')"(V)l

of

il

- @t+r)r/l = s t Koul''t '1sug'too.t4 leqt a?ou

'l = V Il fi1uopuolp snoq fiTtpnbaayy'taaoato1tg 'q (I + X)/ft- X)={'I

<'t/s- X

pql peutnsso st 7t'a.tep'g lo.touaTu,eql uo cb-y st,ycryn

'tu+sx=#=?)! fiq pau{ap 'O] x = g a16uo7ca.r pesop re1?ouoo7 f1'0] x [r'0] = U a16uo7 [t ["'0] 4N, pesop o {o 0utddotuTotu.toluoctsonb au$n uo aq t pI 'g1.'g uorlrsodo.r4 ueroaqJ ssauanbrull s.ralpruqrral Jo Joord 't'9

6tI

5. Teichmiller Spaces

140

Thus an affine mapping as in Proposition 5.18 is extremal. Returning to a closed Riemann surface R, we find that any Teichmiiller mapping looks very simila,r to an affine mapping. Actually, we have the following proposition. a r b i t r v r i l y .L e t f : R + S beaTeichmiller Proposition5.19. Firg€Az(R\ = a holomorphic (< Then therc etists unique g, and set k l). mapping for lltpllt quadmtic differenlial tlt on S satisfging the following conditions: (i) tf p is a zero of p of order m, then f (p) is a zew of r! of lhe same order m. (ii) Let p be an arbitrary point of R which is not a zero of g, and ( be a 9coordinatearound p. Then therc erists a $-coordinateu at f(p) such lhat

'

u o f=

C+n( L- n

(5.3)

We call cp and ry' in Proposition 5.19 the initial differential of / and the Ieryninal differcntial of f, respectively. Proof. For every p which is not a zero of g, we define a mapping u = up i\ a neighborhood of /(p) by (5.3). Then in some neighborhood U of p, we have

,dc , p

F r o ! = t ca 7 = E ; i ' @\

lYl

Since p.o1 = kpllpl for every local coordinate ur on /(t/), we see that u ottt-r is l.qc, and hence conformal on f(U). Thusar is also alocal coordinate around

f(p).

Next, for a zero p of g of order rn, we have seen that g = z^dz2 with a suitable local coordinate z. Define a, as a continuous branch determined by

,"f=(

z ( m + 2 ) 1I2 1 , 2 @ + z ) l z

1-&

)2t(n+2)

Then we can see similarly that c..ris a local coordinate in a neighborhood of /(p). in a neighborhood of every point /(p) such that p Finally, consider (fu)' is not a zero of rp, where @p = u is as above. Then we can show that these a single holomorphic quadratic differential on ^S,which we denote (fu)'give by /. From the construction, ry'clearly satisfies (i) a,nd (ii). The uniquenessof ry' follows at once from (i) and (ii). 5.3.3. Proof of Theorem 5.9 Assume that the assumptions of Theorem 5.9 a,resatisfied. Let r! be the terminal differential of / obtained in Proposition 5.19. For every p e R which is not a zero of rp, take a g-coordinate ( a,round p, and a ry'-coordinate r.r around q = f (p) "horizontal dilatation" as in Proposition 5.19. Consider the

(z.q)

sff '"prps f f l4opr(b'6)y lJ lJ

u-relqoa/',r'(g'g) o1 flrpnbaul (zreaqcsEutfldde '1xep

.a uo.e.e @)('!)r'x> + (o)lr('/)l)) ,@,r!)u ,(tolt:tv)l - r(o)l)('/)l = @)(rilt a,nuaqa'r(6)lJ('/)l 1aB

p u e ' 1 - )6 r t o o = V ' ( I { - i / $ t +

I ) = t ) r ' ' - l l ' I r l l l = r 1 1 a sa m ' t s r l . {

aues aqr,(q "^.,0p",8Id. o.suaroaqr ;iltff"Hi"i:i:lii ;iT'":.t:iT# (g's)

'"r"0 t4op(b'qy fl [[ o1 lualerrtnbaq (t'9) flrlenbaur aq1 'aeua11 sf f 'tpop(b'6)y IJ

L= upy1a,'nu 4op((b)r-!,'nutfl ll aleq e^r 'bplpy - tpop acurs 'U J d ,{.ralalsourt€ roJ

(q.s)

(d,tt)u= (@)1.6)yN segsrlsspue'elqernseaur sf lI'S'uo'a'€ pausapu f;o oo

..1 tollffil

I

=(r'r)v

(uorlel"lrP .!.t+o = r'l pu" .frpxpl(z)dtl= bp?pareq/{ leluozrroq,,eql ueql ,_I ort = f 1a5

'tptpN"fl,hp''p@'.ny || :U Jo eJnlcnrls xalduroc eql Jo uorleurroJap ro; ,,fcuarrge,, lseq eql seq / Eurdderu rellmuqcle;1 e l€ql lceJ eql luese.rdar o1 ,(e.n euo s? ,(lqenbaur 3urmo11o3aq1 preEar feur e,n '91'g uorlrsodor4 ;o ;oord aq1 Suqlecar 'no61 '@ 'U 'uorleas slq? f d fre,ra lsourp roJ t)/Q + I) = l9r las air Jo 1se.ratll uI

= ffi =(o) |1ojjy)sltor |#DI

=(o'r).

urclqo e,n ,o pue ) o1 lcedsa.rqll,rrr/ lo (d,/)V .(uorlelelrp '1eq1 selldurr 'uorlcunJ alq"rns Ieluozrrorl,, erlt roJ 6I'9 uorlrsodo.r4 leql II€ceU -sarue s.rprrp'g uo'a'e peugap q (d'V)V slqJ'ra pue )ot leadsar qfl,r VJo

'trt * ?=)'tol115f;;aal=(o'v), I?I

uraroarll ssauanbrull s.ralpruqf,ral

Jo Joord

'g'g

5. Teichmiller

t42

Spaces

Hence, (5.5) and (5.7) give

ff

, - f f (^$r,d\'

s JJ"(tf-, JJ,ooo'

Kd(drt

='# | I,o,o, d(dr1 t+ il "t (r,)(p) Thus, K1 2 K, and hence &1 ) /c. Finally, if &r - t, then both equalities in (5.7) should hold. Namely,

= l(/r)e l(0)+ l(/,).1(0) l(/r)e+ (n)el(o) and

l(o)= &l(/r).1(0) l(/r)e

a.e. on -rR.This implies that py, = kQ/lpl.Hence, 9 is l-qc, i.e., it is conformal on R. Since g is homotopic to fd, the canonical lift of g on H is coincident with fd by Lemma 5.2. Thus we have h = f . Now, to prove Theorem 5.9, it remains to show the following lemma. Lemma 5.2O. The inequality (5.6) holds. Proof. lt is in this proof where we need Lemma 5.17. For the sake of simplicity, we assume that rltlz has a single-valued global branch on S, which is a holomorphic Abelian diflerential on S. If not, take any local branch of rrLl2, and continue it analytically as far^as possible. Then we can construct a two-sheeted branched covering surface S of S, with a branch point at every zero of t! of odd order,such that ,ltrlz becomes a single-valued (holomorphic Abelian) differential on s. Applying the argument below to this differential on S, we have the assertion for the general case. When rl.'Llz has a single-valued global branch, say d, which is an Abelian differential, we define the geodesic flow {.F1 | t € R} on S with respect to the "metric" ldl. To explain the construction, we always take

u = irrr(q)=

t loo

as a ry'-coordinatearound p which is not a zero of ,!.We continue the inverse mapping tr;l along segments on R in both directions as far as possible' Then we get a locally biholomorphic mapping, which is denoted by the same notation V;l , ol a domain containing an open interval 1o = (r1,ur)--of.R into .9, where -'oo ( u1 1u2 ( oo. we set flo =vor(I). This flo is called the |-horizontal lize passing through p. It is also called a horizonlal trajectory of r/. (See Fig. 5.2.) Note that, when we trace along flo in one direction, either flo ends at azero of r!, or lc.rltends to oo. In particular, restricting ty'-distanceon Hp, we can identify.[/o either with a circle or with a subinterval, say /o of R, preserving orientation and length. Let

'lfil "l 1eBarrr lradsa.rWIm oraz €ere spq u - s - or ecurs'u x et uo uorlDunJ alq€rnseeue s (?'D'd)y ueqa 'u I ? ' U ) b ' ( ( b ) , u l ' 6 ) y= ( t ' b , 6 ) y '1xag 1esa.n 'S'uo seJ€,, eqt s€ tpop srql esn e.rrr,uoa.raquolg .ltil ((crJleru,, aql .(lueruala o1 lradsar ql1,rnSura.raserd-eare q t4, ''a.l ,U Jo X ?esqnselqernseeufraaa ro; (x)urr xrr rpop ll -"pop ll JJ

JJ

'.re1nar1redu1 'saleurproo, leql aes o1 .,(seasr 1r -@ o1 lcadser qlr.&rI i{q eq} sluesa.rdarqcrqrrr ,,uor1e1sueJ} I"luozrroq 1a11e.red,, '6 ;o Surdd€ur-Jleselq"rns€aru arrrlcaftq e sl ?d fraaa 1eq1 ees uec elrr uer{I 'U)d 'U)?'(t)dV=@)tg 3ur11as,tq rg augep eAyA - S - t;i las pue 'arogeqse eq gr p1 .lfil (clr?aru), 'a,ro51 aql o1 lradsal qll^{ S uo {U ) ll tdl ^rog crsapoe3aq1 augap yleqs e,r,r, 'S uo ldl ((crrleru,, aql pue U uo )rrlatu u€eprlcng aqt of lcadsar qlr^\ rrrler.uosr f11eco1sr qrlq&l 'd = (0)ol

'dH *E,dl

Eurddeu € ur€lqo a,u 'ul ,!n = d7 3ur11as'fl - S 3 d f.rala roJ snqJ .lr7tl,,crr1eur,,eq1 o1 lcadser ql-r.{ oraz "ere s€rl 3' }eqt ees uB) e1r{,sotrezJo Jaqrunu elrug e fluo seq qt aeurg 'U Jo Ie raturqns radord e sl dI leql qcns S ) d Jo tes eql eq A (g p ol:,z aldurrs p rpau saurl pluozrroq-p)

'z'9'tlJ

\\ \

l - ,l, \\ t| t/r,r /

r \ ll l tlI

uaroeqJ ssauanbrun s(rellnruqf,ral Jo Joord 'g.g

5. Teichmiller

t44

,=

dt I"(ll,^r,t,t)dodr)

Spaces

(5.8)

=l: 1,,,',)(g' at c)a"a') "(l =

l:"(l l,^r, d

q)dodr dodr) dt= 2L I l,^ro,

for every positive .t. and henceis ACL, we seethat On the other hand, sinceg is quasiconformal, Fubini's theorem gives

, =I I,(l _"ur,t,iat)aoar =

(5.e)

ilrls(Lr)ledodr.

Here,weset.Lo- lr([-t,.L]), andhencelLolq=2L. from (5.8)and (5.9)that Finally,applyingLemma5.17,we conclude 2L [ [ \(s,q)dodr>(21- tt1 [ [ a"a,JJS JJS

Divide both sides by 2L, and let ,L tend to oo' Then we obtain the desired tr inequality (5.6).

Notes For the reduced Teichmiiller spaces,see for instance Earle [57]. When f = {id}, we denote by "(1) the corresponding Teichmiiller space ?(f), and call it the aniuersal Teichrniiller spoce.see Lehto [A-68], Chapter-III. The fact that "(1) is contractible was firstly shown in Earle and Eells [62]. In Douady and Earle [53], it is proved that "(f) is also contractible for every f. For investigations on the Teichmiiller metric from the differential-geometric viewpoint, see Kravetz [128] and o'Byrne [169]. The Teichmiiller metric on ?, is not smooth. See Earle and Kra [65], Royden [184], and Gardiner [A-34]' "curvature" with respect to the $9.4. Moreover, ?o does not have non-positive Teichmiiller metric, as is proved in Ma.sur [142]. See also Theorem 6.21 in Chapter 6. Teichmiiller's theorem gives another compactification of the Teichmiiller space ?e, which is called Teichmiiller's compactification of 4. This is different from Thurston's one defined in chapter 3. see Kerckhoff [111] and Masur

- .itr" lr46l.

original *proof' of Teichmiiller's theorem is found in Teichmiiller [A106].The proof in this chapterfollows that in Bers [23]. We also refer to Abikoff

'[Orz] '[Oea]qcea1 pue ,[666]rqcn8ru€J ,[67I] ,[gtt] ,[rtr] ,[qrr] ,[Wt] rns"trAtr'[gtt] "I[*S pue msstr l 'goqqo.ray ,[lg] pr"qq.,H pue ,(penoq ol reJar osp a \'[ZO1-V] Ieqarts pue'[Z]-V] sur{uaf ,[lS-V] raurpreC ol reJer a^{,g$ ur se sl€rlueJagrp )rlerpenb crqdrouroloq Jo setpnls lecr.rlauroa3 .rog ,flpurg seloN slr pu" y xpuaddy ur pelels are sef,eJrnsuueruorg praua3 yo suorleuroJep FuJoJuotrsenb uo scrdol aurog '[291] ue4eg pue'[08] .raurp.reg'[62] ue{es pue uueurlqeJ '[Og-V] (p{qsnry alr) e,tr,s3urddeurleruroJuocrs€nb1eure.r1 -xe uo suorle3rlsa,rur raqlo sy'[016] Iaqerls pu€'[0/I] a{"tqo,[gg1] ue11nrycry aes 'l"uerlxa dgressacau lou $ e?eJrns Eurrarloc e o1 Surddeu l"urroJuocrsenb 'alourrarlllng 's3urdderu Isuerlxe u€ Jo lJrl e IsruroJuocrsenb leurerlxa Jo sseu -anbrun uo s?lnser ureluoc ,I"l"U pue treurfell ,os1y [961] sareqles pue,[6] '[gg1] 'l€tuer]xa ,(lanbrun aq ol 1aqa.r1spue qcleg eldurexe roJ aas lou paeu lr 'leurarlxa sr 'letuerlxa Surddeur ra[nuqrral e ua^g fpressacau ]ou arts FurroJ Jr 'sSurddeur s8urdderu rellnurqrreJ aIq.,rr l"ruroJ leur.ro;uocrsenb leuerlxa flanbun 'ece;.rns uueuarg Ilrls a.rc s3urddeur rellnuqcrel lelauaS € Jo es?r aql uI 'qrstep e.rour roy [gg-y] '([I ';r) otqerl pus .reurpreC ees auo e oEe sr uozlrpuo? tuercgns IA] [pg-y] leqarlg ,la,roero141 .uaroeq? sseuenbrun spollnang slql pa$oqs pue qrrsg 1eq? leqarts s(rellnur{f,ral alord o1 ruaroeq} s(uollrru"H slql asn uec a11t .Fuerlxe aq ol Surddeur .roy e uorlrpuo? fressaceu e srsfleue IsuorlounJ Sursn ,tq leruroJuof,rs?nb '(asuas eruos ur purrunu) pe,lord [69] uollueg Itsuerlxa sr Surddeur JellnuqcraJ e Jo luel?lgaor nueJllag eql leq] sa]€ls rueroeql ssauanbruns(rell3ruqtlal .[I-y] ItI

seloN

Chapter 6 Complex Analytic Theory of Teichmiiller Spaces

We introduce a natural complex manifold structure of the Teichmiiller space "(R) of a closed Riemann surface R of genus C(> 2), which is realized as a bounded domain in C3s-e. Furthermore, we prove that the Teichmiiller modular group Mod(R) acts properly discontinuously as a group of biholomorphic automorphisms of ?(,R). In this chapter, unless otherwise stated, we assume that l- is a Fuchsian model of a closed Riemann surface of genus S (Z 2) and that each of 0, 1, and oo is fixed by an element in l'- {i,d} (cf. $1.2 of Chapter 5). In Section L, following the idea due to Bers, by using Schwarzian derivatives, we prove that the Teichmiiller space?(.1-) is realized as a bounded domain Tn!) in the space A2(H. lf) of holomorphic quadratic differentials on the Riemann surface H*/f , where 11* is the lower half-plane. The Riemann-Roch theorem shows that A2(H. lf) is a complex (39 - 3)-dimensional vector space. Hence TaQ) is regarded as a bounded domain in Csg-s. Identifying "(l-) with Tn(f), we see that T(f ) has a complex manifold structure of dimension 3s - 3. In Section 2, we show that this complex structure of "(f ) is independent of ,| , that is, ?(f) is biholomorphically equivalent to T(ft) for another Fuchsian model f' ol a closed Riemann surface of genus g. It is verified in Section 3 that the Teichmiiller modulat group Mod(f) of f acts properly discontinuously as a group of biholomorphic automorphisms of has a nor"(f). Thus we conclude that the moduli spaceMn =T(l)lMod(f) mal complex analytic space structure of dimension 3g - 3. In Section 4, we shall explain Royden's theorem which assertsthat every biholomorphic automorphism of "(f) is induced by an element of Mod(f). Finally, in Section 5, we give a brief exposition of the Thurston-Bers theory on the classification of Teichmiiller modular transformations.

,tq uarrrSe * ? :/ ursrqdrouroeuoqe el€q elrruaqt '11 uo ,rn - ,1m11 'too.r6 ' nn = nn (rr) *H uo 'U uo nn = tn (\) -oamba a.r,o6utmo11ot eW'r(J'H)g

:luel omy fi.uoJotr 'T'g BrrruraT > n'rl s7uau,a1a

'fle,rtlcedser 'n.t nJ ld H o1 crqd.rouroloqlqarc l!,U pue 'uotlezrurroJlun snoeuellnruls *U Jo U a3eurr .ror.rrureql pue S ?erll epnlf,uoc errr 'S o1 'trl - r/ '/ tas *2f ;o / Surddeur Jo luelcgeoc rurerlleg aql ,s.reg ,tq ueqtr 'U;o *gt e3eurt Jorrnu eql Jo J lepour u€IsqcnJ e rlctd IeuroJuocls?nb e ar1e1Pu€ '1ceg u1 ',S pue g flsnoeuellnurls sazlurroJlun qclq^r /.7 dno.r3 uetsqcng-rsenb e u1 due ro; 're1ncr1.red pug e r 'f snuaS;o S pue U sec€Jrnsuuetuelg pesolco,ra.1 '([76] q slr{J, snoaunlputs.sreBr srag aes) uotToztrulolrun Pelle? ttg 'd.7 dnor3 u€rsq?ng-Isenb e13urse ,tq ,(lsnoeuellnuls pezluroJlun er€ +Urpu€ saceJrnsuuerualu orrrl'f1e.,lr1cedset'd1flg pue nJ/nH,{q peluesarderer€ *Ur = g;o e3eun rorrflu et{t sl *2f eraqlrr pue ttg saceJrnsuueualg o,lrl eculs'lltt 'Ulln = dA o+ JIH oI Jl *H = *U Jo Surddeur crqdrouroloqlq e pu€ h/nn = sacnpur trn, g' e Surddetu Surddeur leuJoJuoctsenb eq; Jo leuJoJuoclsenb 'in put rtg qloQ uo slurod paxgou seq {pl} -nJ Jo }uatuala f.rarra1eq1eloN'paxg C uI elrnf, pasol) eldurts 'uorlrugap fq'st dno.t0 palcerrp € sa^eel q)lqrtr (c'z)lsa 3o dno.rqnsalarcslP e uDrsqcnl-rsnnDe 'a.re11'sdnor3 uelsqcnJ-Isenbgo eldurexa lecrdfl e sr d.7 dnort slce e q?ns'GH)n^ - ltl pt* (U)'* = "-t1 qtoq uo {lsnonurluo)slp.{1.redo.Id q c r q , n ' ( 3 ) l n v J o { J ) L l ( L ) ' X } = n J d n o r 8 q n s€ e l e q a ^ \ s n q l ' ( q ) t " V 1 " ''e'r'uorleurro;su€J1snrqotr^tr e sr (1")/X leql easeu'g raldeqCJo t'I$ luerueleue ul ?eql o1 SuruosearrBIIruIsfq 'r-(/rn)o Lodm = (1,)/X 3ur11nd'J f ,Lfue rog "_' m .{q 1r alouep a,u '(gg't uorlrsodo.r4) H lo ,tn Eurddeur ob-r/ lecruouec eq} uorJ srql qsrn3utlstp ol repro u1 'flerrtlcadse.r'pexg oo pue '1 Surddeu IsurroJuotlsenb '6 saaeal pue '/ uorlel€lrp xelduroc eql seq qrlq.tr go Surddeur leruro;uocrsenb ? e ''a'l '? 3o Surddeur cb-r/ lecruouec e dlanbrun slsrxa areql '08'? ruaroeql tuord

'H-c.)z H>z

=?)d ,(,)2\

''e'l'r(l'n)S 'J roJ Ff uo r/ luercgaor rruerlleg e

les e1( 3 r/ lueuele uarr,rEe rog '*H

aueld-;1eq re^\ol eql uo leuroJuoc er€ q?rq^\ C a.raqdsuuetuelU eql;o s3urdderu 's.reg Surrrrollog IeruroJuocrs€nbfq (.7)g aceds rallnurqclel eql luasardar lleqs elri uol+BzrruJolrrlf snoauBllntllrs'T'I'9

Jo arnlrnrls

xalduoC

arBds rallntuqrraJ, aql pue Eurppaqtug (srag 'I'g Surppaqurg,srag'I'9

LVI

148

6. Complex Analytic Theory of Teichmiiller Spaces

f (") =

z €H

zeIl*UA.

{(")-t"''(")'

Since (tot')-rotul is quasiconformal on C, we see fhat f is ACL on C. Thus, by the analytic definition A of quasiconformal mappings ([1.1 of Chapter 4), / is quasiconformal.Hence,g = wpof o(wr)-1 is a L-qc mapping on C, i.e., a Mobius transformation. Since g leaveseach of 0, 1, and oo fixed, g must be the identity. Therefore, we have up = u, on I1*. Conversely, if wu - IDv on I/*, then utt = It)y on Iy'* U R. Thus we obtain a quasiconformalmapping h - wpo(wp)-Low,o(w')-L: H - H.By the same argument as before, it follows that h must be the identity, which means that wF = w' onR.. Now, for two elementsp,v € B(H,f)1, wu and'wv are said to be equiaalent if wu- w, on H*. Denote by [ror] the equivalenceclass of wrfor every element p e B(H ,.1-)r. Let fB€) be the set of these equivalenceclasses[tor]. Lemma 6.1 shows that the correspondencel*,) * [tor] is a bijection of "(l- ) to "p (f ) . The topology of TBQ) is induced from that of "(f) under this correspondence.In other words, this correspondencegives a homeomorphism of 7(f) onto fBQ). In this way, we can identify fpQ) with "(f) as topological spaces.We also call fp!) the Teichmiiller spaceof l. onto TBQ) given by 00t) = [or]. Then by Let B be a mapping of B(H,f)r the definition of topology olTB(f), we immediately obtain the following. Proposition tion.

6.2. The mapping B: B(H,f)1

*

fBQ)

is a continuous surjec-

One merit of the Teichmiiller space TB(f) introduced by Bers is the applicability of the theory of univalent functions, i.e., conformal mappings of H* .

6.L.2. Schwarzian

Derivative

A quasiconformal mappin E u p 6 defined in $1.1 is conformal on the lower halfplane 11*. Now, assume that wu is a Mcibius transformation. Since uru leaveseach of 0, 1, may and oo fixed, tuu must be the identity. Thus we have [tou] = [rd] in fBQ).It be considered that the diflerence between [tou] and liQ in fpQ) is indicated by the difference of the conformal mapping wu on H* from Mobius transformations. To measure the difference of a conformal mapping on I/* from a Mobius transformation, we shall find a differential equation which all Mcibius transformations satisfy. Let 7Q) - (az +b)/(cz * d) be a Mijbius transformation, where c, b,c,d e C and ad - bc = 1. Take derivatives of 7 to eliminate a, b, c, and d. Since j'Q) = @z + d)-2 and l'(z) - -2c(cz + d)-", we obtain = -z12 - dl2c. Thus we have (lf (7" 1t'))' = -1/2. Consequently, l'G)/t"Q) we get

uo ndt - ndt leql arunsse'f1as.ra,ruo3',I/ uo pruroJuor e Surugap uer{J'*Il ndt = f,dt 1eq1 sa11dur1 uo not - drn ueql '(,1)d,t ul ["m] = [dt]lI {rpla'.II '(6'9) elnur.roJe eq a.trsmlJ'*I/ uo = "(z),L{(z)L'dm } 1aBa,u 8'9 {r'n^} ut st "urrrutrrerl Aq(rt6ortL - LodmJo e^rle^rrap uerzrs^{qcs eq1 3ur:p;'t(;(g)g 'loo.t4 r/ asnecaq 'uotleurro3.suer?sntqoni e sr ,-(/rn)oLodm = dl uaql 'J > L 11 '*H uo nd - 'td) fi1uopuo fi ) , 7 ' t t e t u ? u e p o m yf , u o . t o t ' t a a o a . t o 1 4 1 fi Q)gl,q [,n] = [n.]'t(J'H)g 'J / ,H acottns ',1 o7 uuoulery D uo lorlueta[ry c4o.tponb ctyd.toruopy o co papto|a.t st 7t puo q??n *H uo V- Tq|nn lo ut.tot ctyrltou.toTtoatyiLtoruopy o sN ndt 'fipu.to7J Tcadsat, (z.g)

.*H)z

,(t)nd="(z),L((z)L)d6 ueql 'J ) L lI 'V'g BturuaT '3urmo11o;aql e^er{ e.&rueqtr,

'*H)z

'{z'dm}=(z)ddt las e,/rr'r(J'H)g

ra[nurqcral

Jo arnl"nrlg

xeldurog

3 r/ r(re.rltqre rog

aql PrrB Eurppaqrug

6srag

acudg 'g'I'9

'uolleturoJ tr -suerl snrqontr" q / feqt apnlcuoc aan'uorlenba FlluaraJlp slql 3ul^los 'O uo

- {''!) -,,((r),!sor) o= r{,((r),!sq)}f l e q ?l n o s u r n ll r ' O u o 0 - { r ' I } '/} sagsles uotpurlo;suerl snlqotr{€ 13ql uees 'O uo = 0 / ;r'flasreluoC {z itpearle e erl e1yuorlress? tsrg aql sn sarrrSuolleln?Fc preruro;lq3ler1s'rg'loo.r'4 'O u o 0= {t't} o 'teaoa.r,o141 sn qory o s? CI{o furddout'Tortt"t'otuoc 19fi1uopuv fi uotyout.rotsuorl

(r'g)

'Q) z

'(O)l 'fi1aat1eailsa.t

'{r'l}+

'f "(z),t.{Q)t } = { z'to6}

u?tll pu, q to s|utildotu lotu.totuoc atp 6 puo I It 't'g BtutuaT

(94\s-!',)'Ii={,'!} (z),,,!

,\(r),J ) t

,(q / lo {t'l } aaqoarrePuotzrDnqrs aqt auuep e^{ 'C uI ururuop e uo 3[ Surddeu l€trrroJuoc frerltqre ue rod (r),L ( (z),L

, - . \ I ; ' Z )\ ET,- G l J ,

Suppaqurg .srag 'I'9

6'I

150

6. Complex Analytic Theory of Teichmtller

Spaces

mapping F: wu(H*) - u,(H*) by ,F = w,o(uu)-L, again by Lemma 6.3 we see that g,(z) = {Fowptr} = {F,wuQ)}'.'uQ)' + pp/) on 11*. By the assumption that g u - g v on H *, we have { F, z } - 0 on uu(I{' ). Thus .t' must be a Miibius tra.nsformation. Since f leaves each of 0. 1. and oo fixed, we see that F is the identity. Consequently, uu = '.i)v on f1*, that is,

tr

l.ul= [u'"]in TBQ).

Let A2(H*, f) be the complex vector space of holomorphic automorphic forms of weight -4 on -I1* with respect to f . Since it is identified with the vector spaceA2(H- / t) of holomorphic quadratic differentials on H* f | , the RiemannRoch theorem shows that A2(H*, f) is a (3S - 3)-dimensional complex vector space. Now, define a mapping B of :tBQ) into A2(H*,f) by B(lwrl) = pu, where g p = { u p , z } , t h e S c h w a r z i a nd e r i v a t i v e o fw u o n f I * . T h e n , b y L e m m a 6 . 4 t h i s 6 is well-definedand injective, and that is called Bers'embedding.The mapping iD: B(H ,I)t ---*Az(H* , f ) given bV @(p) = 8"0(p) is called Bers' projection. In $2.2 of the previous chapter, Az(H* , f) was considered as a complex Banach space with trr-norm. In this chapter, in connection with the next subsection $1.4, we introduce on A2(H* ,f) the hyperbolic -L--norm by using the Poincar6 metric dss.z = ldzl2/(Imz)z on f1* as follows. By formula (6.2) and the inva,riance of the Poincar6 metric under P.9.t(2,R), every element peA2(H.,f)satisfies

(Im1Q\2le(.r("))l= (Imz)zleQ)1, z e H*, 1 e r. Thus, (Im r)2lp(r)l is regarded as a function on .R* = H*ll. L* -norm of 9 in Az(H* ,l-) is defined by

The hyperbolic

z)'leQ)| llpll* =,s.S.(Im Here, note that the supremum suffices to be taken over, not the whole If*, but only a fundamental domain in 11* for .l-. In our case, .R* being compact, we can pick a relatively compact subset in f1* as such a domain (see Example 5 in $4.2 of Chapter 2). Therefor", llpll- is finite for any p € A2(H*, l-), and hence Az(H*,f) becomesa complex Banach space with this norm. Throughout this chapter, we assume that A2(H*,1-) is equipped with this norm. P r o p o s i t i o n 6 . 5 . B o t h B e r s ' p r o j e c t i o nQ : B ( H , f ) 1 embeddingB: TBQ) - A2(H* , f ) are conlinuous.

*

Az(H*,f)

a n d ,B e r s '

Prool. Note that { p"}T=t convergesto
I=l

7r - u7_r7l"ql" 3

N

1eq1 seqdur qcrqrrl

t = 'o- / - "v -\ l\ , r - ' " 1o" s 1 " 3"/ t l o

al"q eA\ ' gz?J= m rc1 mf ".t - qr,3ur1ou 'snq; "3f

'@),rp@)t"l *I = " , - J fq ua.,rt3sl iC ,tq pepunorrns uretuop pepunoq eql Jo 'y eere aql 1eq1 sarldurrelnuroJ s(ueerD ueqJ 'd repun { .r = lnl I C > .l = "C elcr.nreqt yo a3eun aql aq ',C 1el 'I < .r ,{.rerlrqrero;'1ce; uI'I t |tql ,r"qt

( e' g) eql 'y -

?

-

'

+t*t*oq*m=(rn)g=)

fq ua,rr8q rorre?xe aes e 'y aq} 1er{} lrun rf JI A Jo lleqs {slp '*V uo '1srrg Jr uorlcunJ luale^run € Jeprsuof,aiu *y ereqa 3foo"r4, 'H)z z ' { } 1 " ( z wd1n) s- * l l { " ' / } l l tr; l{

'!

sa{sr,7ns*H uo uotTaunt lualDarun fitaag (snutx

puB r.reqag)

fr,Tqonbaue ayl 'Z'g BrrruraT

'sneJy pus rJ€qeN ol enp sr rIJrrIA\'*Il uo suollounJ luale^Iun ro; ,(lqenbaul ue Jo ecuanbesuoeel€rpeuurr ue sr rueJoeql slt{I

u? nvq uado aql u, peurDlu@ s? (ilal '(J'*H)zV

' 1 f g s n t p o " pr u D 0 r ? l u n q ? g n( J ' * H ) G V acoils .tapnu,?pl4 eUJ 'g'g uraroaqJ

ur ureurop pepunoq e sl (J)sJ

l€rl1 slrroqsuraroaql 3urno11o;eq; Q)slJo

ssaupapunog

.t.I.g

('9'6$ aas 'oqy 'asec l€uorsueurp-auo eql JoJ leql ol relrturs sr ploJrrr€ur xelduroc Ieuorsueurp raq3rq s Jo uorlrugep eqJ) 'J/H = Ar eraq/!\ 'sp1o;rueu xelduroc Ieuorsuaurp-(g - 0g) se peraprsuoc osle are (g),, p"n '(l)gJ'(I)t sacedsrellnurqcral aql '(J)aJ qlr^r uollecgrluepr rapun '1 1o acodsrelpu.tqcteJ aql palleo osle sr (l)sJ slql '(J'*H)zV Jo ernlcnrls ploJrueur xalduoc aq1 slrraqur (l)al,'aceds rol?el xaldtuoc leuorsueurp-(g - 0g) e sl (J ',p.)zy ecurg 'uruqdrouroeuoq e q (J)sJ -ur ureruop e sr prre'(J'*H)zV Q)d,l:g (J'*H)zV * (t)d,f,:g uorlcafursnonurluoceqlJo (J)sJ a3erureql1eq1se11dur1 sureuopJo ecuerJsAuruo rueroeql s(rea{norgsnql'(.i,)ag sr os pue'g-ogtl ol crqd.rouroauroqsl (J)J areds rallnurqclel aql '91'g ue.roeql ur pelels sV 'I'9 Surppaqurg,sra6l

I9I

152

6. Complex Analytic Theory of Teichmriller Spaces

for any positive integer N. Letting r * the inequality

1 and then letting N'--+ oo, we obtain

i"'ul, s 1.

n=L

This is the content of the so-called Bieberbach's arta lheorem. In particular, we have lDll S 1. Differentiating the series in (6.3) term by term, we obtain

{ 4 , }=- # 0 , * i # ,

weA*-{*}.

Hence, we get

. 1 4 $ l . ,r' ,{ . } l = 6 l a r l 6<. Now, let / be an a,rbitraryunivalentfunction on I/*. For a givenpoint zo = ro * iao € H* , first supposethat f (r.) * oo. Taking a Mcibiustransformation T: H* ---+^4* definedby "(z) = (z -Z;)lQ - zo), we put

F(u)=r@w*,

w€A*.

Then F is a univalent function on A*, and has an expansionas (6.3). Flom formula(6.1),we have{ f ,r} = {F,T(z)l.T'(t)" on I1*. ThusnotingT(t") - %)n, we concludethat oo and T'(r)'= -4v|fQ)alQ

l { f , r , } l= , l l l }l"{ r , r p 1 } . r , Q ) r l

=.rgg l*n{r,, }l.,!g" &

S#"

Next, supposethat f(r") = oo. Then by the relation{f,tol - {Ilf,r"} and the aboveargument,we see again that l{ f,zo}l t Sl!yl). Sincezo is o arbitrary, we completethe proof of Lemma 6.7.

6.2. Invariance of Complex Structure of Teichmiiller Space which is introduced in the Let us prove that the complex structure of "(f) preceding section is independent of the choice of the Fbchsian model ,l- of a closed Riemann surface of genus c(>2).

(g'g)

1-\bru-zultt eleqeir\'*I1uo g= ,(lltrlt,- z&l&)eculs

(q'g)

't = (t-)zu = (l-) I& \ O= Q-)It" = (p-)rh ) uorlrpuor uorl"zrlerurou aq1 f;st1es zb,pue It t"q1 etunss€ a,t. 'ata11

(r'g)

'g=bdt9q,,tt

uorlenba lertueragrp freurpro repro 'd -puo)es aq1 ;o z& pue tlr suorlnlos arqd.rouroloqluepuadapur fl.reauq a4e1 e,rr1 '{oo.r,4 -e^rrap uerzr€^rqrs qll,lr *I/ uo uorlcunJ luele^run 3 Irnrlsuoc o+ repro u1 'dt = (l"dnl)g sa{n1os d>uto.{ p?prul.suo? 6111o4uau$rp Nravrqeg ?Nuoruroy?Ul '2,/I > -lldll qlr,n (J'*H)zV 3 dt Tuauale fiuo.r,og (tU"rf4, pue sroJIqV) 'O'S tuaroaqtr, ttt'",ttt 1"" sroJlr{y or anp ruaroaq} Bur,u.o11oy aqr Jo acuanbesuoc "r"ro"**,'|lJl 'p! = o ,utddout ctyd.tou-to1or1 ttrog \?pm ''a'!'A<'(l)dt ut T u t o da s o qa y 1 / o 2 : g l o e s r e a u f, 1 6 u ro s ! / ) < - A i 4 p u D pooty,oqq|nuuedo uo s! (A)4 - 2 'uoqon7ts|utpaca.tdell repun '8'g uraroaql '6'9 uorlrsodoJd ruo+ snonur?uocsr qtlq^r

'[^n*) - (d),n tq Q)d,t +- A i 4 Surddeur€ euuape,tlsnql ' A ) o\ itra,raro3.1,o1 laadsarqll^{ }uenlgaor nuerlleg e ''e'l 'r(t'n)A o1 sEuolaqdrl ueqa '(l'*n)zv ur urSr.ro eql Jo pooq.roqq3reu 'crrteru -lldll (t'.n)"V eq} fq pecnpul n'{Z/t 6} = > ) A1e.I ?r€f,urod | qlr^aurroJ(t 't-) cruorureq€ sr lI ef,uls roleradoruerlleg-er"1de1aq1o1 lcedsa.r 'p4uataSrp ,urDrlleg?ruouuDqe pell"c osle st 'o1uror; pelsnrlsuo, 11 lo4uetal -lrp tutotTlag.sreg eq! pellsc sr "/.HI'(*t'n)A 3 drl luarueleue ulslqo aa{ 'H ) z

'(z)dt"(zwl6- =Q)drt

snql 'J/.F/ uo rurerllage s zp/zp(z)dtr(zul = Itp/"r!Q)fi_ 3ur11es lerluareJrp p.I'H 3 z fue .roy = ueqJ'II uo rrrleru er"f,uroderll eq Itp ,(zu4)/"lzpl '(J ' *H)zV 3 al luaurela Q)a = Q)fr fq paugap(l'tt)"V ) ql luatualaue 1aB"^ fre.rlrq.reu€ roJ 's,raolloJ * (,1 ';J)g 3 drl luarualaue qlra (tr' .11)cy ) dt '(;)ag e ur (,t)s,l ur urSr.ro eqt luauela qcsa elercossee,ra Jo pooq.roqqSreu * (l)d,l:g Eurppaqure (sregJo asralur crqdrouroloqe f1t1c11dxa lcnrlsuo? oJ tmppaqurg t9I

6srag Jo asralul

IBcoT

'l'Z'g

aredg rallnurql-ral Jo ern?f,nrls xaldurog Jo af,u"rr"^ul 'Z'g

6. Complex Analytic Theory of Teichmiiller Spaces

t54

on ff*. Set /(z) = ,nQ)lrtz(z) for any t e H^*.From (6.6), we see that / is a locally biholomorphic mapping of 11* into C. A straightforward calculation gives{f,z}=9onH*. Now, we put

F(z)=ffi,

z€H'

( o . {)

Then ,F' is a real-analytic mapping of /1 into e , b"""n." its numerator and denominator do not vanish simultaneously from (6.6). Bv a simple computation, we see that Fsf F, - Hv on I1. Since llprll- ( 1, the Jacobian of P is positive on 11, and hence F is locally diffeomorphic on 11. Next, we set :. \ | F(r)' z€H

I\z)=l/(r),

z€H*.

We need to prove that f exte^ndsto a quasiconformal mapping of e onto itself in such a way that upn = So/ for some Mijbius transformation S, which implies that{up,,z}=p. For this purpose, first suppose that g is holomorphic in a neighborhood of ,F1.U fr. in 0 and lp(t)l = O(lrl-n) as z ---+oo. Then ?r and qz are defined on a neighborhood of the real axis \, and so are / and F. Since f = F on R, we obtain a continuous mapping f of C into C by putting f = f on R. This extended mapping f is locally homeomorphic on a neighborhood of R. In fact, for any point z on R, choosing a small disk D with center z, we see that both f : D--- /(D) and F: D---+ F(D) are homeomorphic,and f = F on DnR. Since both / and F are orientation-preserving,f(Dn I1*) and F(DnH) do not intersect, which implies that / is injective on D. It is easy to see that f is an open mapping, and hence f^: D --- /(D) is a homeomorphism. Moreover, it is proved that f also extends to a local homeomorphism on e as follows. Since l9(z)l = O(lzl-a) as z'--+ oo, r/1 and \z ate expanded near oo in the form

qte) = a1z* bt + O(lzl-r), nz(z)= azz* bz+ O(lzl-L),

where c162- a2b1= 1. Hence,f = rylnz is a univalent function in a neighborhood of oo, and .f(*) = at/az. On the other hand, we have

F(z) =

atz*bt+O(lzl-1) (z * oo). a2z!b2+O(lzl-t)

Thus F is an orientation-preserving diffeomorphism on a neighborhood of oo with ,F(oo) - ar/az. Therefore, putting l(o") = at/az, we see that / is a Iocal homeomorphism of 0 into itself. Then Lemma 4.28 implies that i: e --* e it u homeomorphism. Applying Painlev6'stheorem t?.r,o(.f)-1, we seethat there exists a Mobius transformation S with wp, = Sof . To remove the hypothesis that g is holomorphic on R with zero of order at least 4 at oo, we pick up a Mcibius tra.nsformation fl. given by

uorlenba rtuerlleg aq1 3o (1'9) ut.to; eql ul uoltnlos e pug ol poqlau cll$rnaq e ureldxa ol e{ll plno^^ eM'6 qrvueq 'lcedtuoc sl JIH leqt srsaqloddq eql asn lou plp a,r,r;oordaql q ecurs'.-;'dnor3 uetsqcng.,(ueroJ sploq 6'9 tueloaqJ,'[ slrDuev !

'6'9 ueroeqJ ;o Soord eq1 salelduro? slqJ

' {r'los} = {z'6rim) = (l^'^l)g *H uo 6 'drnor-5 '3 uo Surddeu urelqo eal 'ra,roe.roy41 leturo;uoctsenb ^amor-S = u1 € o l s p u e l x a / t e q t s e q d t u rq c l q , \ { ' l I - C u o ;l'pue'(q)nV "S ler{l eesarr,r'relncryed uI 'U - C Jo slesqns lcedruoc uo (orrleur lect.raqds S '?l.l '.{1a,rr1cedsa.r 'uorlcnrlsuoc fq snqa eql o1 lradsar qp.,u) fpro;run 3l <- $ ud t'o1(7'9) uotlenbe pu€ Il, oI *H Jo slesqns lcedruoc uo ,{pu.ro;tun a3ra,ruoc u'zb pu€ u'rl, suollnlos pezllerurou eq? 'pueq raqlo eql uO I€rluereJ-rp"ql p ^dot *- "dotr 'C Jo slasqns lceduroc uo flurro;tun ud teq} ees am'2'9 '.{lluanbesuoC'C Jo slasqns lcedtuoc uo fpruo;run P! * 'ecua11'U eurrreT ;o goo.rd eql ul se luarunSre atutss eql fq 3 Jo slesqns "6rl '.t"ql Wql sarldtur gl't uoltlsodor4 ur (8't) lcedtuoo uo ,{prroSrun 0 *

+ I)/-lldllt; -ll"u'lll r > (?llalF urctqo a { snql 'z{(-lldllA - I)/(*lldllZ + I)} J uotlelepp l€ulrxeur 'r-(^ilm)o"dm = u6 3ur11e1ug e seq 'f leql ees e/tr 0I'7 ueroeqJ uror.;: u{ous = "drn 1eq1 qcns 'u5 snlqgl uotleu.roJsuerl tr etuos roJ tol 'uant3 uollrnr]suoc "! Surddeur ler[roJuocrsenb e sernpord Jleslr oluo e J" t"6rl = 'rl 3ur11nd'rrlo1q a q t , { Q ' e c u e 1 1' t > - l l a l l Z S - l l " d l l Z = - l l " t / l l 1 e Be r ' t

' 9 t - l l r l l; T L

z;2 ur1;f,riJ; l(@"t)al.11 l(r)"otlr1rqlf,,i."= -ll'dll e^eq e.!\ snql

*H)'

. ,.\,

-fFtt-'

,z((t)";*I) =eQutl)

r

'1'g 1eEa,r.r, uotltsodor4 'oo de'*11 ) Gn)"t Eur1otr1 le ? ls€el 1€ reProJo otaze seq Pue'(p'),l,Z.lo "6 aprslnopaugep uollcunJ crqdrouoloq e sr ud>teql aes el.l.'"(z)ia' (Q)"a)d = ' "J pue 3ur11ag'oo F u s€ pl o1 C Jo slesqns lceduroc uo ,tpu.ro;run sa3re,ruoc H ro; ueql'u.re3alut arr.tlrsod,'{ue Jo ur€uropqnslcedruoc flarrtlelar e sr (g)1lj u7 -t zl

= (z)".L ?-zuz 991

'Z'9 acedg rallnurq)-reJ ]o arnlf,nrls xalduroS Jo af,u"rr"^ul

156

6. Complex Analytic Theory of Teichmriller Spaces

w, =

I,

)(z

_ z)2ee)w,

on fI for any p € Az(H*,1-) with llpll- < 1./2.Thisis due to ShigeruFuruya, and the authors learned it from K6ta^roOikawa. setting z - y and z = sin the aboveBeltrami equation,we have apartial differentialequation L w,=i@-x)ze(o)wu. Denoteby w(x,U) = C with an arbitrary constantC a generalsolution of Riccati's differential equation I

u'=-|@-r)2p(r). Then this u gives a solution of the above partial differential equation. Putting u=Uo, weget

u'+r

t

- - rIv @ ) '

Thus, settingu= -af a/, we obtain the second-order differentialequation I

u , ,= _ ) v @ ) a . Take linearly independent solutions [1 and rp of this equation. Then we see that

--. -. - T t + ( Y - ' ) n ' r

ffi|;--;rt'

and hence we obtain F' in (6.7). Corollary. For eaery g e V, therv ezists an element p e B(H, f)1 such that w, is real-analytic on H and B-r(p) = lup). Mortoaer, eaery point [S,f] of the Teichmiller spoce f@) "f R - H/f is r e p r e s e n t e d b y a r e a l - a n a l y t i c q u a s i c o n f o r m a l m a p p i n gosf R l o S , i . e . , [ S , . f ] =

lS,slin r(R).

Proof. In the proof of Theorem 6.9, we saw that tnp, = S"i, i is real-analytic on ff , and B-t(p) - lwp,), which shows the first assertion. Let us prove the second statement. Denote by D the set of all points [S, /] g ?(R) such that [^9,/] is represented by a real-analytic quasiconformal mapping of .R to ^9. Let [So,f,] be an arbitrary point in D. From the first assertion, we find a neighborhood LI of the base point in "(Sr) so that every point of [/ is given by [S, /] with some real-analytic quasiconformal mapping /. Then 1[S,f"f,] | [S,/] € U] is a neighborhoodof lSo,f,l in "(n) and contained in D. Thus D is an open subset of "(,R). Next, let { [S",.f"] ]Lr b" a sequencein D which convergesto a point [S, /] e "(,R). We may assume that each /, is real-analytic. Since [^9,,f^of-tJ converges

taE tdor.acurg aar'seueselqnopuo rueJoeq? (ss?rlsrale1turor;'*g uo crqdrotuoloqsr

(o'g)

uoy?-ffo

,^"[[ !r- = e)v]q

araq.n 'g (- ?'se c Jo slesqns lceduroc uo dluro;tun

(7)oa (z)lnlp?* z - (r)'n* eler1 a^\ '19'y ura.roaqa 'tg'loo.t4

(e'g)

* H) z , t f i p % ? . [

uaal6sr pu, ,7.rrr, lrtfo1aatTonu?peql'(.t'tt)g

;-

=e)ta]06

rol ) /7fr,r?a?

fiq 'Ol'g uraroaq&

'0 = es?t aql JaPlsuocairn'1srtg ereqr$ tt 'II'9 ulaloaq; ur ue,rt3 sr 'uol]?as Eurpacerdeql uI turou [n]nO lo uorleluasarder ler3alur pu" ef,ua]slxaaq;, -oo? crloqredfq eq1 o1 lcadsar Wlu ecuaS.raluocrurou st acuaS.re.tuoceql areq^r

'((,t)o-P|d: ?-fi'=[nlne t ueqJ '0 {- I se g..- -ll(l):ll P* '("1'g)g ,tq paugep q [n]dO qcns ur8rro eql Jo pooqroqq3rau r1t d' - trl o1 sSuolaq n araq^r '(l)al + + leql uI lueruala ue aq trl 1a1 e ul I reqtunu xalduroc fue ro; paugeP t(t'n)S 'Q'n)g I r/ fre.rlrq.re :l z pue r(J'n)g 'lro11 [n]dO e^IlsAIreP eq] eugoP of qsIA{ arrr ro; r/ 1e z uorl?arrp eqt ul O p

uollcaford

'z'7,'g 6srag Jo uor+Bl+uara:gTCI

'dd'[Og-V] 3eplo '6VT,-ZV1, o1 {ooq eq} ol reJarear's1te1ep.rog'uap,tcrg enp pu€ elr€g ol enp ere qf,rqrh ueJoeql qqt o1 saqceordde reqlo ere eraql 'gtI-IgI 'dd'[e-V] sroJIr{Y aas) (;)a; '([6] prr€ srolgy uoll?as pcol crqdrouroloq " ltnJlsuoc pooq.roqqSrau ? uo fue ur lurod O Jo ;o (uorlceger 'uorpes o1 elqrssod sr 1t 11a1y-s.tol1qyeq1 leruroJuocrs"nb Sursn ,tg ur lurod as€q aqt Jo Poollroqq8rau e uo 6rt <-+a1 uorlcas pallsc fl qc1q,r,r'(;)a; 'E pcol crqd.rouroloqe seq O uorlealord (srag 1€ql selsls 6'9 rueroeqJ, ,lroue[ tr ecurg'(g)gJo ur [u/orJf 'S] .ro3["f',u5] = -leer s slswe L9l

'(U)Z of lenba aq lsmu O'palaauuoc q (U),2 'snql'(U)J tasqns pesolce q O ecuaq Pue'6'ol s3uolaq [/'S] = cl$pue-par sr "/o,lf snqtr'u e3.re1f11uategnsdue Pot [/',S] ud Eurddeu "s o1 l"uroJuoarsenb c1ld1eue s;o fr_to"t,".g] q]l^ eJeql 1eq1 saqdurr uollrasse lsrg eql '(,S); lo lutod eseq eql ol acedg .ralpurqrleJ Jo ernlf,uls

xalduro3 Jo ef,u"rn

uI 'Z'9

158

6. Complex Analytic Theory of Teichmiiller Spaces

*'rr=I*tiltfu]'+o(r), - ' 1 , , = t r b f r ]+" o ( t ) , t'1,',=trbfv)"'+o(t) uniformly on compact subsets of .F/* as I ---+0. Thus we see that i D ( p r )= { u p , , z } - t t t f v ) " ' + o ( t ) uniformly on compact subsets of f1* as t - 0. Since f/./f is compact, it follows that iDs[v] exists and is equal to rblv]"' . Further, formula (6.9) provides (6.8). tr T h e o r e m 6 . ' L L . F o r e a e r yp e B ( H , f ) t erists and is giuen bg

a n d v e B ( H , f ) , t h e d e r i a a t i u eQ r [ u ]

ouv'11'v=f-* lLmaea,t]-u1"1' ,

zeH* (610)

Proof. Set f = *r, !11= wprowrl , and \t = pcr.Then we have

A , (' -c )(=L f 'r -- ,F p 4)) "" 'r - , ( ( ) ,c € f ( H ) . \E

Thus, putting

)\ / (l \ o =/\ ;f "L d'"' r -\ ' { c ) ' we get )t = l) +t6(t)

on /(f1),

wherel16(l)ll-- 0 as I + 0. On the other hand, from the relation O ( p , ) ( z= ) {s*f ,r} = {gt,f(z)}.f'(r)'+AQt)(z), we obtain

ib,1,11,7= f,k)r, z € H*. hi"r 1{n,,tel}1 Lr+u, I Then by the same argument as in the proof of Theorem 6.10, we see that @r[u] exists and is represented in the form

f t ibu1,11,1= z e H*. ' ' Jy'e')2, "' ' l-9" l [l 1[ q 1 -fg-'.,n ( C - f Q ) ) ' alartl ,

Therefore, by substituti"g /(O formula (6.10).

for ( in this integral, we obtain the integral tr

sr C acurs'(.t)al> ('*DA lurod fue Jo pooqroqq3reue ur crqdrotuoloqlqsl d 1eq1arro.rdo? peeu am'crqdrouroloqlq8I r-go*(rr))org d }€ql aesoa'too.r'4 .(r,t)al, - (J)sJ : r_Ao*(o)or1 ,(Il)dt * (1)0a :*(o) ,(tt)t* (1)a:.la'l ,(tU)Z * (U)Z ,.[V] eqJ 'ZT'g uaroaqJ :ctrld.totuolorlrq nD eto sfutddput.6urmo11ol .(J,H)zV p (J)sJ ureuroppapunoq€ se pazrleerq (r"f)dZueql'(IJ'H)"V o1u1(y)dglo Surppaqura (sregeq Ig ta1 '7utodasoqay1 uotyolsuDrle *(o) .ro *[o] lec osle elA to 'ln*7 = ([nt]).(t),--, ^] [n rq (I.r)d,Z+ (1)da:.@l 'r_mod(nora usrqdrouroa{uoq e euuepelr'ralroaro141 nn Ir-qI qtns pu€'oo pue 'I '0 l€q} IlsnsuollsruroJsuerlsnlqotr{IeaI € sI D arel{l!\ Jo qc€asexu r_oott(nolo 'l.n^l = ([n.])'[r] lanl :(lt)t - (.7)g :.[o] uotyo1suo"r7 ursrqdrouoauoqs secnpur(tU),-U-- (U).f :*ltl) Turod?saqeql to '(t '(g)g - rJ ,(gluapr e^\ uer{11y eq1'flerrrloadsar 7)"6'(,1),2qtt^ (tU)-f r-nJn 'IU *-g:tf eJo l;ll € eq t't taT SurdderuleruroJuocrsenb 1€r[]eunss€,teure11 '{pp} - r.7ur 'g luaurelaalqe}rnsefq pexgsr oo pue'1 slurodJoqceet€q} qcns pesolcreqloue;o t7 Iepouru€rsqrnde e{€I 3 snue3Jo rU. a?€Jrnsuueruar11 .J

JO

acroqf,eqlJo luepuadapursl'(J)sZ * Q)dl,:g Surppaqure(sreg Sursn,tq g'1$ ur peusep se^r qf,rq^\ '(;); lo ernlcnrls xaldruoc eql l€tl1 a,rord 11eqsa,t. '.nog '1 reldeq3 'ltt-Vl srrcpox pue rrorrol{ pue 19 raldeq3 '[Og-V] srrr€H pue sq]lgrJ9 o] reJeraa,r's1e1eprod 'uo os pu€ 'sp1o;rueur xalduroo uee.&\1eq s3urddeur crqd.rouroloqlqpue ctqdrouroloq'p1o;tueur xelduoc " uo suorlcun; crqdrouroloq'sp1o;rueurxalduoc leuorsueurlp-u eugep uec arrt'1 raldeq3 Jo I'I$ ur sploJrusu l"uorsuaurp-euo Jo es"f, eql ur sy '(6'd 'hf -V] srag aas) dlrnurluoc s1r sarldurr ,f go ,tlrcrfleue aleredas leql slresse uaroaql (raqlrn{ 'f1a1e.redes alq"rr"^ qf,ea ut crqdrouroloq pu€ O uI snonul}uo) ,sEo1re11 sl 1€ql pepr,ro.rd ur erqd.rouoloq sl lerl? uaas fltsea sr 1t 'elnurro; 1e.l3e1ur 6. / / = z IIe JoJ saS.raruocqctq,u. s,dqcnep dg 'o yo pooqroqqEraus ur (ur'"''rz) t { ' '

.r(uo - "r)'

'tq:, ,r(ro - rz)"q

'r{

T

= Q)l

roJ JI O uo uorsuedxasarrasrau,ods s"q tl'O ) (up' "''tD) - n .r(.ra,ra ctyrl.totuoloy pell€? sr uC Io CI ur"ruop e uo peugep 3l uotlcun; panlerr-xeldtuoc y 'uorsuermp raq3rq Jo splo;tueur xaldruoc fgarlq /rarleJ ero, 'q1t.tr ut3aq o5 (,f)^Z f" arnlcnrls 69I

xelduro3

Jo acuerrBlul

't'Z'g

acedgralnurqrlal Jo ernl)nrls xalduro3 Jo af,u"rr"^ul 'Z'g

6. Complex Analytic Theory of Teichmiller Spaces

160

homeomorphic.Let f p = wpl(dr)-l two mappings

and Bu be Bers'embedding of TB(f ts).If.

-Bo((wP)-t).oB;t : TB(IP) -Ts(f), Fz =B;.o(uo(.t')-r)*oBpL: Ts(f P) * "s(|1) fi

are biholomorphic in a neighborhood of the base point of Ta(fP), then .F = FloFlL is biholomorphic in a neighborhood of B([trlr]) eTBQ). Thus, it is sufficient to give a proof that F is biholomorphic in a neighborhood of the base point of TaQ). Take a neighborhood, say V = {9 e A2(H* ,f) | llell- < 1/2}, of the base point of Tn(l). For arbitrary p,t €V,we set D = {t e C Itb+t9 € I/}. Put p(t) = t+t9 and p(t) = Its(t). Flom Theorem 6.9, we have B-'(p(t)) - [urr1tyJ. Let )(t) be the Beltrami coefficient of wr(t)ou-t, which is given by

(t)-u, \^,.,_, ^. \\o ( rt-) _( r " r-F -*{ a ) "t \q on f/. By the construction, we get f(e(t)) = {ror(r),2}. Since,\(t) is holomorphic with respect to t, Theorem 6.11 implies that F(g(t)) is holomorphic on D. Since g and ry' a.re arbitrary, .F is holomorphic on V. Since .F' is injective on ?s(i-), from the following lemma (Lemma 6.13) and the inverse mapping theorem, we see that .t' is biholomorphic on I/. O By the definitions, the rest of this theorem is trivial. Lemma 6.L3. The Jacobian "Ip = det(0Fi/0zp)4i,1'=n of an injecliae holo' o f a d o m a i nD i n C " i n t o C n a a n i s h e sa t m o r p h i c m a p p i n gF = ( F r , . . . , f ' " ) no points on D. Proof.We prove this a.ssertion by induction for dimension n. First of all, clearly it holds for n = 1. '/.,, we assume that the assertion holds for any positive Given an integer n ) integer S n - 1. Let Do be the set of all points in D where .Ip vanishes. We want to prove that Do is empty. Suppose that Do is non-empty. F\rrther, assume that the Jacobi matrix of F is of rank r with 1 S r S n- 1 at some point a € Do. Then we may a^ssumethat det(d.Q /|rx)1.5i,x9, does not vanish at c. The inverse mapping has the inverse theorem implies that G(z) = (F1(z),...,Fr(z),zr*rt...,zn) mapping H = (Ht,. . . , Hn) in a neighborhood of a. Then we have F"+r(() = G + t , . . . , F . F / " ( O ( ' , m d F o H G ) = ( ( r , . ' . , G , F " + r o I / ( ( ) , . . . , F n o H ( () ) i n a of neighborhood G(c). We set W = { ( = ( ( r , . . . , ( , ) € C ' | ( r = F r ( o ) , . . . , G = . F " ( o}), W t = { w = ( w r , . . . , w n ) € C ' I w r = F r ( a ) , . . ' , u r = t r ' ' ( o )} . Then the restriction FoHllat is an injective holomorphic mapping of a neighborhood of G(a) in I,7 into Wr.By the hypothesis of induction, it follows that

'8'I ul $ pa?npurauo aql o1 luap,rrnbes1(.i,)dg uo ernlcnrls xaldurocstql 'ecue11'Q)aJ, ol (J)dJ;o Surdderucrqd.rouroloqlq e q g Suppaqrue(srag 1eql reelt q ?I '(t)d,1, uo arnlcnrls ploJlrreu xelduroce se,rt3{(.t)d,t > ln^) | ("nlnl'ntl)} 'erogaraq; 'p nn U nn q1r,u rp pue n2 f.rcra ro; rtqdrouroloqlq sI + (nnUnn)nt *_ (nnudn)df : r_(d,{)on,{ 'zI'9 tuaroeql ,tg '[nr] 1l l€q} s^irolloJ Punorepooq.roqqtraue]eulProoc e se ("n1,tg'nn) n{rt uec et$'acua11'd4 oluo /4go usrqdroruoeruoqe secnpur d4 pqt s?ress"g'g uraroaql ueqJ nd por '(,1)d.t ul [/.]Jo pooqroqq3raue s1 ' ( n t n ) r - ( n d=) ' n

pu" {zlt > -lldll | (rJ',H)zv ) d} = nA

+es 'g '(rJ'*H)zV o1q (n.i,)dggoSurppaqura al111\ (sregsr dg araqn = ([dnt])dgqlrm *(nmlod - ag g G,t)st * (.t)dt : Eurddeur :snolloJn (l)g,t > ln*] lutod fue e,rrlcafurue a^€qaar'r-(no-l).ir{n = tJ 3ur11as 'f1aure11'lelluesseur,tlen1ce fl ploJllrelu Jo pooqroqq3raualeurp.rooce e)p+ ea.l, sI pcrSolodolleuorsueurp-(g- 0g) I"er € sr (l)gl, leql lc€J aq1'ctqd.rouroloqlq g qcns ernlonrls plo;tueruxalduroce seq (.7)dg eeso1 'ralarrroll l€ql leql 'crqdrouroauroq -sl Surppequre (J)sJ (l)dl,:g (sreg l€q1 aasol sureuopJo parEolodolleuolsueurP acu€rJe^ur uo rualoeqls(JeA{norg Pasna/$acurs'p1o;rueur -(g - 0g) I"er e sr (l)dt leql /$oul ol pepaeuaarr'eraq; '(J' *H)"V acedsrolcaa xalduroceql ur ur€ruoppepunoq" $ qcrq^{'(t)s,t uo arn}rnrls xaldurocaq1 r.uorJpernpul seo,(;)dg uo ern?f,nrlsxalduroceql'g'I$ ur pelsls sy'tlrautey

tr

'u ror sProq s."."er eqr acuaq oT"i,il}i,T ffifi;"_;:':iil:*",,

qcrqar'1ue1suo?" aq lsntu .!,{rarra'acua11 'In uo saqsrrrelttg/lge fra,ra'06, uo qsrusA tzg/lgg 1e acurf 'Ln uo uorlcunJ crqd.rouroloqe s€ pereprsuo) sI q)lq^{ 'n U oO ol .{4';o uorlcr.r?saraql aq fd fe1 'rC u-r ul€ruop elq€?rns€ sI '{tn = n U o O 1 6 q 1q r . , r 1 - , C 4 I p u r e u o p e u o > " | ((t)f'r)} n araq^\ 4l uorlcun; crqd.rouroloqe slsrxe araql leql arunssedeur er,.r'(07'd'[ft-y] srag) ueroaql uorlerederd (sserlsJaralt urory 'g uo if uorlcunJ crqdrouroloq eql Jo soraz Jo 1as a{l sr 06r acurg 'O Jo lesqns radord e 4 oO 1eq1 aurnsse '1xe11 'O uo e^rlcelq q A 1eql slcrp"rtuoc slqtr'O uo tl$uel^ tzg/lgg 'lrolq '1ue1suoce aq 'O = oO lsnru fg qcee ueqJ l€q? arunssr IIe esneceq 'oO uo qsruel plnoqs tzg/lgg i(ra,rasnqa'I -u t {u€rJo sl J leq} slrrpsrluoc qcrqin 'o le u {u€I Jo aq lsnur dr Jo xrr}€ur rqo?ef aql 'p Jo poor{Ioqq3rau e ur crqdrouoloqlq q 5t ecurs 'pu"q reqlo aql uO '(r)g tr u {uer Jo s! IIoJr Jo xrrl"ru rqocsf aq1 'acua11'@)C t" r - u {uer yo sr zltlgog Jo xrr}etu rqocef aq} acudgralpurqrlel Jo ernlf,uls xalduro3 Jo a)u"u"^ul 'Z'g

I9I

162

6.3. Teichmiiller

6. Complex Analytic Theory of Teichmiller Spaces

Modular

Groups

We shall prove that the Teichmiiller modular group Mod(,R) of a closed Riemann surface of genus C(]=2) acts properly discontinuously on the Teichmiiller space "?(r?) as a subgroup of the biholomorphic automorphism group Aut(f@\ of

r@).

6.3.1. Definition of Teichmiiller Modular Groups Let.E be a closedRiemannsurfaceof genusC(22). We define the Teichm[iller rnodulargroupMod(R) ofR asthe factor groupofthe groupofall quasiconformal self-mappingsof ,Rover the normal subgroupof thosehomotopicto the identity (cf. $3 of Chapter 1). The elementoI Mod(R) definedby a quasiconformalselfmappingf " of R is denotedUV[/,].The action [/,], of an elementlf "l e Uoa@) on ?(r?) is given by

[/,].([S,/])= [S,f"f"'l for every [S,/] e T(.R) (see $1.3 of Chapter 5). We call such an [/o]. a Teichmilller moilular transformation oI T(R). Let f be a Fuchsian model of ft. By lifting, a quasiconformal selfmapping "f, of rR corresponds to a quasiconformal self-mapping ar of the upper half-plane I/ with c..,fc.r-l = f . Let or; be a lift of a quasiconformal self-mapping f; of r? with u;f (u;)-r = l- for i = I,2. By the same axgumentas in the proof of Lemma 5.1, we see that [fi] = [/2] in Mod(R) if and only if u2 = ca1o7,holds on the real axis R for some jo € |. With this in mind, two quasiconformal self-mappings {rr1and = f U = 1,2) are said tobe equiaalenlif there exists u2 of H satisfyingu;fulr an element 7o ol I such that e2 = uroTo on R. Denote by [c.r]the equivalence class of ar. The Teichm'iiller modular group M odQ) of f is the group of all these equivalence classes[a.'].The action [c.r]-of an element [w]e Mod(f) on T(l-) is given by tf [r]- ([ru]) = laowq owfor every [ru] e 7(f), where a is an element in Aut(H) such that eorlrou-r fixes each of 0, 1, and oo. (cf. $2.3). Theorem 6.12 asserts that [cr]- is a biholomorphic automorphism of ?(f ). Furthermore, this [c.r]*induces a biholomorphic automorphism (c,r),of TaQ) defined by

(r).([ru]) = lw,) for any lrul e TB(f), where z is the Beltrami coefficient of oowrou-l . We use the same notation (cl). for the biholomorphic automorphism ol Tp(f) instead of Bo(w),oB-1, where B is Bers'embedding of Tp(l). We also call [ar]- or (c.,)* a Teichmiiller modular transformation. By the construction, it is obvious that ModQ) is isomorphic to Mod(R). By the identification of ?(f) and "(rR) (Proposition 5.3), the Teichmiiller distance on 7(-R) induces the Teichmiiller distance on ?(f). Then Proposition 5.5 implies the following.

',t1duns arou pagrre^ q tnq 'ZI'g rueroaql u€ql ra{pellr sr q)rq^r 'l1nsa.rqcns raqlou" ^roqs lleqs a,r'are11 '(61'gure.roeql'Jc) UJo J Iepour u€rsqcndpezrl"urrou€ sa)npulU uo scrsepoe3pesolcJo sqfuel crloq.redz(q;oles e '61'g ureroaqtr uI u^roqs s€^r sV 'L to aco.t7aqyto a.tonbs e ? i ls e l . o u e p s ! { J ) L l ( t ) z r t } ? ? se q J ' f r e 1 1 o r o 3 (L)"r1e;eqn'1gur a1atcslp ',t.re1oroc3urmo11o;eql ol peal g't pu€ gI'g suorlrsodor4 8)'urut 'gl'g eururarl slcrperluoc slq,L'u .{ue ro3 o W > (il1 = (Q)uL'z)d u,0} uaql'u,[3o leql qrns JJo sluauele lcurls-rp,tlen1nur;o ecuenbase q r?{ "ty s l x " ee q l s l e r a q , u ' p * " t V U d t t s q l P u e ' ( 8 r e l d e q 3 J o 8 ' I $ e a s )" 7 s r e l o s 'g ut ",1 u,L sI qclqrr J roJ .4 leeduroc ,(1errt1e1a.r leql os .1, ) luetuale u€ e{sJ ur€urop Ftueu"punJ e esooqC '"I Io t{fua1 cqoqradfq eq} sl ("7)/ araqrrr',tr4l raqurnu aarlrsod aruos roJ W j (l)t seg$les u7 ,trerle leql qcns g' uo scrsepoe3 pasolc lf,urlsrp fgenlnur Jo r=.:{ u7 } ecuanbese slsrxa eraq} ?eq} aurnssy /oo.l2, '7 y76ua7ctloqtadfiy IlWn A uo sctsapoa0 paso1cfruorufipyru{ Four ID Isrre N,eq?'7aa4tsodfiuo .tol 'taaoano147 'E u, ?pns?p s! 6 snua6to g acottns uuDureNypesol? Q,7) eqoq.radfiqto 1as eqJ 'gT'g uorlrsodor4 D uo s?r,sepoafpasolt 1r'oto st176ua1 'uorlslPeJl tr -uoc e'(n)lnv;o dno.rEqnselerf,srp€ lou $ J'gI'7, euurerl ,(q snqJ,'Q1)nV o1 s3uolaq / feqt srrrorls8I'Z €rrrueT'17 ul sl orn acurg'oot = (oz){ ateqell^ uo peugep / uotlcuny ctqdrouroloq uer{J 'Il uo fprue; I€Lurou e fl lI asn€caq '.,ir1, etunss€ osle .{eur arrr e ol sles uo fpulogrun sa3ra,ruoc I*{ "f teqt lcedruoc } '.raq1rr1g'oo + u w oz +- "z om <- ("r)"L prc leq+ eurnsse fetu )1 ) H ) e,rl 'r(resseceu;r acuenbesqnse Eut1e1 'alalduroc sr d pue lceduroc $ I4 etuls 'X u,L.r(ra,r.e ) "z auos roJ W j (("r)"L'"2)d sagsrles leql qcns J uI slualu -ele l)urlsrp ,t1en1nurJo I1{ u,L acuenbas3 slsrxa araqt }€tl} asoddng /oo"l4' } 'ytr 'H uo e?uolsrp ?rmurod aqy st d a.taqm Y)'uutt j ere1l '14Jtaqu,nu aatT q??n L fiuout fr.1a7tu{ ((r)t'z)d ) lsoluu ID lswe J -tsod puo 11 auoyd-l1ot1 fitaaa "tog 'gl'g Btrrwarl .r,addnayl u, >J psqns Tcodu.toc 'suorlrsodord auros e.reda.rdem '(.7)"6 uo rllsnonutluo?slp fgedord s1re-(,1)poq

1eq1 a,rord o;

slas qnpotr I 'z'8'9

'ecuDlsNp qTtmfi.tyau,os,uD sN (l),f, lo *lnl tustrld rellnuqral eyq oy Tcailsa"t -routolnD ctyd.r,otuo1otllg eW '(J)poW > l'r,l lueuep fi"raaatog '?I'g uraroaq5 t9l

sdnorg r"lnpol4l ralllurqf,raJ't'9

6. Complex Analytic Theory of TeichmiillerSpaces

164

Proposition 6.17. Let f be a Fuchsian model of a closedRiemann surface of g e n u sg ( 2 2 ) . L e t { t i } T = , b e a s y s t e mo f g e n e r a t o r sf o r I s u c h t h a t T h a s t h e repelling fired point 0 and the attractiae fixed point oo, and such that 72 has the repelling fired point r with r < 0 and the attractiue f,red point I. Then each 7i is iletermined bE lhe absolute ualues of traces of elements in the fi'nite sel g = { 1 p'l z, 7j, lft ol x, ^tt' oj r, (lro7r)*' o1,r}, w h e r cj - 1 , . . . , m , a n d l c= 3 , . . . , f f i . Prool. First of all, we may a^ssumethat 71 has a matrix representation

' ,= Lf o) ' lol 'I l ' A > 1 ' Thus 71 is determinedby the absolutevalueof tr(lr). Next, by the normalization condition we may assumethat y has a matrix representation a , b , c , d . ) 0a, d - b c = 1 ,a * b = c { d . a =lo',.|, LC dJ' Thus we get

a * d,= lt.(rz)1, a.\* d)-r - ltr(71"72)1. by the absolutevaluesof tracesof 7r,72, Henceboth o and d are determined and 7p72. Since the quadratic equation l2(z) = z has a solution 1, we have

2 c = a - a + 1 f t 1 a 1 2- 4 , b = c * d - a . Consequently, both D and c are determined by c and d. Therefore, 72 is determined by the absolute values of traces of 7r,'Yz, and 71o72. Now, for every [ - 3,...,rn, the Mobius transformation 7; has a matrix representation

" =ll

p s- q r= r ' : ] , P , e , r , ER€,

Here, by Theorem 2.22 we may assume that

P+s>2' Then, using the relation

tr(.a)tr(C)= tr(AC)* tr(.4-rC),

(6.11)

we see that tr(,AC) is determined by the absolute values of traces of lt, 7* and 7fo73. In fact, since the left hand side of (6'11) is positive, we have

tr@Cj = ltr(,aQl - ltr(71o7r)lprovidedthat ltr(,Ac)l ] lt.(e-'q)|. tt It;(/i)l < ;tt(e-rc)1, then*" oLtuinft(A-tC) = 1tr(A-rc)l = ltr(7!r"7i)1, andhencetr(AC)= ltr(7r)l. ltt(z*)l- ltr(7r-1o7j)1.

ulstqo ar$snql ',Lo1 seEra,ruorI?{ 1-(rloo"o)oLo(rlnouo) } acuanbas aql 'J =l Lt.ra,re roJ ?"ql aes a^{ '(,t)z ol [pl] ot se3re,ruocI;i{ [rloo"o1 1 eturs 'oo pu" 'I 'g go qcea saxg ,loouo leqt qcns (g)lny 3 uo ereq^{ '[1loo"n1 = ([pp]).["]

-JIesIeurloJuo cIse"o "-:::;t,"f t ":i j"; ;o uoEutdd€ru

;; \7"'lft:

l:T

"

'[prJ ot sa3ra,ruocr-*{Wl]"rl } feUt erunssr feu aar 'od olur (,f )Zfo lurod aseq aq1 Eurlelsuerl 'acue11'od o1 sa3ral -uoc Il{ ('d)"rt} ttql 'od - (ob)o6 = ('d)'rt 1aEaal '("d)"Sor+"6 - ("d)"U fg'od - (n)'0 ulelqo en'ud - ("a)uSou6uro14 'rtlaarleadsar'oq pr* od s$urd -deu crqdrouroloq ol (J)J ur slas lceduroc uo fpr.ro;run aS.rarruocr*{ 'rl } prt 'ob = (od)ol eleq e,a{'relncrlred u1 'of Eurddeur crqdrour t}{"}',t1rc1yu19 -oloq e ot (J)J ur slas lceduroa uo ,(puoyun sa3raluoa r-/{"t } te{t erunsss feu aaa'fressacau y acuanbasqns e 3uu1e1 'snq; 'f1rure; l€rurou B sl r?{ T } '(g'g uraroaq,f,) (.f)s,J ursruop papunoq e o1 crqd.rouoloqrq sr (;)g ecurg 'utorlufi - uq

'rlI

= "6

les e^\'u qcse IoJ 'Q),f,> oD '(J)J lurod e o1 sa3ra,ruocr?{ ("d)"{ } acuanbaseql e o1 sa3raluoe q?rq/r ("f)'f q ,-*{"a acuanbesur€lree e ro3 'leqt 3 od lurod } I=g{ qans (.2.)po7g ur sluetuale flenlnur 1?urlsrp Jo $ } acuanbas s s}srxe ereql ueql '(J)J uo flsnonurluo?$p fl.redord lce lou seop (1)po1,tg1eq1asoddng '(6 reldeq3 Jo g$ ul {r€urag aas) 11'g uorlrsodor6 ur srsaqlodfq aq1 Surf;slles srolereua3 ;o urelsrts e seq J leql IFcaU 'too.t4

'(Q)Dt"v dno.tf rustrlil"rotuolnonyd.toutolotl?q?W Io dno.tfiqnso so (1)a uo filsnonutTuoc -stp fiy"tado.rd sTco (.1)po141ilno"rf l,o1npou reIInurUoNeJeUJ '8I'g uraroaql

sdno.rg Jslnpor4l rallnruqcral,

tr (zL 'rL

yo z(lrnurluocsrq'

g' g'g

(qL (eLorL ' q L o + ( z L o r L )p u e ' l L o f L ' t t o l L Jo sarcrl Jo 6anl"A etnlosq" aql ,tq paunuretap st {,L }e{l epnlcuoc e^r

'r_ysp* r_yDa+ yrg + ydo= (CaV)rt 'sp+bc+rq+do-(gg)r1 'r_ys*yd=(gy)r1 's

1d - (g)r1

suorlenba reaurl Jo uralsfs eq1 3ut,r1og ''tLo+(?'LorL)pue '{1, (zLorL saf,erl 6enlp^ e}nlosqe eql ,tq peuruJelep Jo Jo sl (CSy).rt 'f1.regu4g 'cto*L pus '{1, 'zf Jo sare.rl Jo senle^ elnlosq" eqt fq paurrurelep sl (CA).rt 1eql aasam'y;o pealsur g Sur.reprsuoc'feaa euresaq1 u1 't'9

sdnorg r"InpoIAI reillurqrlal

99I

166

6. Complex Analytic Theory of Teichmiiller Spaces

since { t,2(z) r,,. ;;;,i;":l;"

"i,];J,'.;..,*ition 616), and

since every wno7ou|r belongs to f, we have tr2(uf,ro1ou,) = tr2(7),

I eg

for every sufficiently large n, where g is the finite subset of l- given in Proposition 6.17. Hence, Proposition 6.17 implies that for every sufficiently large n, there exists an element 0" e AutlH ) such that unro1oun -

|ito',to\n,

I e l.

This shows that Bn belongs to the normalizer N(f) of f in Aut(H), and [c.r,,]*[f"]-. tfrus every such [c.,,]* fixes the base point [fd] of T(l-). By the definition, it is easy to see that the isotropy subgroup of Mod(f) at . On the other hand, it is well known that N(l-)/f [id] is isomorphic to N([)lf is isomorphic to the biholomorphic automorphism group Aut@ lf) of the closed Riemann surface H f f , and that Aut(H/f) is a finite group (see the following Remark 1). Therefore, { [ar,,]*][1 should be a finite set. This contradicts that D { f" }T=t consists of infinite elements. Remark -1.Every element o € N(f) induces a biholomorphic automorphism [o] ot Hll defined by [a]([z]) = [o(z)] for any [z]e H/f .It is easy to see that the mapping a r* [a] is a homomorphism of N(l-) onto Aut(H/f) whose kernel is l'. Thus N (f) I f is isomorphi c to Aut(H / f). H. A. Schwarz proved that Aut(H/i') is a finite group. F\tther, A. Hurwitz showed that the number of Aut(Hlf) is not greater than 84(9 - 1). For these facts, we refer to Farkas and Kra [A-28], p.242; Siegel [A-98], Vol.2, p. 91; Tsuji [A-108], p. 496; and Imayoshi [104]. Remark 2. In the proof of Theorem 6.18, we have used the fact that "(.1-) is biholomorphic to a bounded domain. However, we can also verify Theorem 6.18 from Theorem 6.14, i.e., the fact that [r.r]. is an isometry with respect to the Teichmiiller distance on "(l-) for all [w] e Mod(l-) (see Gardiner [A-34], $8.5). Moreover, by using a theoremof Nielsen on topology of surfaces, we can show directly that M od,(R) induces a discrete subgroup of the biholomorphic automorphism group of 7(,t). The proof of this kind is in Nag [A-80], $7.1 of Chapter 2. Now, we have the following fundamental theorem on the moduli spaceMn. Theorem 6.L9. The moduli spaceMo of closedRiemann surfaces of genus g (2 2) has a norrnal compler analytic spacestrtclure of dimension 3S - 3. This theorem is an immediate consequenceof a theorem due to H. Cartan [48]. Namely, for a given discrete subgroup G of the biholomorphic automorphism group of a bounded domain D in Cn, the quotient space DIG has a

'd W ) b

' (b 'd)n{,p (b 'a) } ,l{,p

pued - odr{ly*w > ,d'"''od,slurod IIera^o*orrt;tffifdtJ,i;';# I=!

r 1t4,r1d)\pT

r", = (b,Q n{p

'u 1nd arrr .ra3alur arlrlrsod ,tue rog 'b = (ilt pue d - (r)/ qtp W - V :rf Surddeur crqdrourolotl e slsrxa eJaq? t"ql qcns y ) g'o slurod IF ra^o ue{el fl rumuuul eq? alaq/{ '(q'o)d

lur

- (b'd)\p

'W 'd ) b slurod ollr1ue^l9 'ploJlueur xalduroc e eq W pj les e^r 'ploJrueu xalduroc preua3 e oI V ,srp lrun eql uo d acuelsrp eJe?urod eql Jo uorlezrleraua3 e sr qcrqan'acuelsp rqser(eqox aql ecnporlur a/rr'sureJoeq}s,uap,tog eqrmsep oJ 'uapfog ol anp osl€ eJ€ qcrq,rl (66'9 pue *? eql f,q pe,rord sl ,tlurlcafrns eqa IZ'g $ualoeq;) suraroaql Eur,rlo11o; Jo '(tZt-gZt'dd '[OS-V] 8ep aes) - 6 e7 peprno.rd ol cr.rqdtoruosr sr +r'q1 Z 'lce; srql uord 'C ralo +r;o eceJrns 3ur.re.,roc paqcu€rq ll 1aura1aql l€ql s^r\^.olloJ palaaqs-orlrle ,{q paluaserda.rsr U 'sl leqt 'cr1-dr11a.raddq sl ZI acurs ,orll1 .rap.ro 3o rusrqdrouolne rrqd.rouroloqlq e seq oar? snua3 ;o Ur er€Jrns uueruarg pasols ,{ue'pueq raq}o eq? uO'e t }eql papuo.rd earlcalursr *z ecuaq puts ,{ [pp]] of < '(1 uorlrsodor6 o1 frelo.roo ,[6lI] q?n€U pue ,[0gI] *? lenba sr Jo leura{ aq} snql sapr/{arl :gZ,Z'd'[Og-V] srrreg pue sqtgrr5 aas) {pz } = (A)?nV qll,!\ gr ereJrns uueruerg pesop e slstxe eraql l€rll u^{oDI sl ll'e 4 f .rog '(,DpoW ) [o] ,(.re,re roJ +[rn]= ([r]).1 fq ((.7)g)nv ot (J)poW;o +r tusrqd.roruotuoq€ auuep eM

'k
( t ) p o lw. e z/Q)p o wJ = \\J) J) ? n v (uapfo11)'0e'9

uraroaql

'[gzt] '[gg] '[?t-V] ntx pue €rx pue a1.reg .raurpreg'pg1] uep,(og aas 'sgelap 'llnsar 'drqsuorlela.rSurr'ro11o; rod srq ureldxa fger.rq eql e^eq II€qs eM (;)g eceds rallnurqcrel er{} Jo ((l),Dl"V dnor3 ursrqdrouro}n€ crqd.rotuoloqrq aqt pu€ (,ilpoW dnorE relnpou rallntuqclel eql leql parord uapfog .1 .11

stuoroaql

s(uap^oll

?'g

'7 tueroeql'[6lt] pw '[911] qcneg eag 'slurod .reln3urs setl d11en1ce (Z 7 0) 674Jacedsrlnpou drarra1eq1uldou{ $ lI 'g raldeq3 '[gI-V] dgng ol reJar am 'tueroaql s(uelJeC .crqd.rouoloq sr Jo U$ ;o;oord e rog CIO - O :1, uorlceto.rdeql teql qcns arnl)nrls aceds cr1{1euexalduror leurrou sruaroaqJ s,uapfog'p'g

L9l

168

6. Complex Analytic Theory of Teichmiller Spaces

for all positive integers n. The Kobagashi pseudo-distanced7,,1on M is defined by

du(p,il = "1!g{u@,q). It is an ea.sy matter to show lhat dva: M x M -* R is continuous and sat- d1,a(q,p), and isfies the axioms for pseudo'distance: dru(p,C) > 0, ilu(p,d dM(p,C) * du(q,r) 2 d,a(p,r) for all P,Q,r € M. It is said that d74 is non' - g. Note that du is not always degenerateif d|,a(p,C) = 0 is equivalent to p = du = o.If dru is nondethen obviously M C, nondegenerate. For example, if and M is called a hyperbolic on M distance generate, dy is called the Kobayashi , said to be complete if. it M is manifold compler manifolil. A hyperbolic complex is complete with respect to d1'a. The most important property of dru is the distance decreasingproperty,the proof of which is trivial by the definition: let M and N be two complex manifolds and let f : M - N be a holomorphic mapping. Then it follows that

du(p,c)Z dv(f(p),fkD, p,qe M. In particular, every biholomorphic mapping of a hyperbolic complex manifold M is an isometry with respect to dya' Theorem 6.2L. (Royden) Let rQ) be the Teichmaller space of a Fuchsian the Teichmiiller rnodel I of a closedRiemann surface of genus c (22).Then dr1). dislanceiI on T(f) is equalto the Kobayashi dislance Proof (an outline). We give a sketch of its proof' In order to prove dre) = d, it is sufficient to show that df,.y = d. In fact, if dl(r) = d, then d|1"; satisfies the triangle inequality' Thus, by the definition d|,r, for any positive integer n, and hence d71;; = d. we have 4
di(i4,[r^]) : dt (lidl,lw^l), [r^] e T(f') for any [trl,] e "(l-). In fact, if this equality holds,then the relations 1]), it(lrul,[r']) = dr(1i4,[to'o(up)-I di (1.'1,[to"])for all [tue],lw')eTQ). To simplify the notation,lp,dp, and u) in the aboveequality are rewitten as f,d, and roP,respectively.Then, we shall show that

dl
.p =p .(,f teq? u^,\oqssr tr uaql )Z ut b pue d Eururolp sqled qloous esrmacard1e ier.o ua)Fl sl runtugur eql ereq^{

,@)tfri = (b,d)p 1nd aal

'(,t)l>

D'd slurod oir,r1fue .rog

of '?p((t),c'G)c),t = O)t 1J

,C qled qlootus asrarecardf.rerlrqre ue .rog (rr) las e^\'("1)Z *[t'O] 'ooC sstspJo lou sI d teql uarou{ oEs sr tt ('[Og] alr"f, osle aeg) 'uorlras oraz eql 3o slurod 1e ldacxa (.f); fo epunq luaEuel eql uo 1C sselt Jo sl ,t teql paUIreAsl lI 'I = Illolll q1,r ("J'H)zV 3 ol ge .rarro ue{el sr urmuardns eql pu" 'H u\ nJ roJ ureuop leluaurcpunJ " sl n,{ erar{^{

,loo*o (z),-(^q"lrlnl| *1"" e)d>

= (v,[,.]),4 [ [ "rldns6

" ' | " ( n m t 1 r r V | I 'p ?rJlalu Jallnuq?ratr aql uorJ pacnpur 1eq1 pa,rord q lI (.f)Z f" elpunq 1ue3ue1eql uo f,Irlatu leursalrugul aql se pareprsuoc sl 3' slql

3

G;WW

o
tas e,$'(,1'tt)g ) y pue (.f),f f [,ar] lurod due rog (r) '1 raldeq3 '[ru-V] reurpre9 pue 'F8I] uap{og aas 's1re1eprog 'flqenbaul qq};o;oord aq? urc1dxa i(gerrq e11 'uapfog ol anp ;oold aqt go 1rcd l"rcn.rc lsoru eqt q sHI

' (0,r,> [an]' (lr*l'Wpl)p ? ([n']'W'4)Q)1p a,rord o1 paau aa,r'fleurg

'Q),t)

'(lr4'Wtl)p j ([n'] 'WtDQ)*p

la^l

a^eq era 'ecueH

ffi

sor- (r'g)d ; (lr*J'Wp))op

= QD^!'W!l=(o),/ qq^ Burddeur crqd.rouroloqe sr[,nm]=*[j[it;]* 1-V:d/

iY;

rrl Eurllas'pueq reqlo aql uO

Eurddeureql l"ql aes e{'ldtl/$t

'p.Pl)P = ([nrn] 3o1

ffi

spled (6'9 ureroaq;) ueroeql ssauanbruns.railnuqtrel uerlJ, 'I > { ; 0 '{ euos roJ Vl/{q - orl qt.r^ [,rm] = [",rrn] ]"qr qans (.7'H)zV I d luaurale ue sl$xe aleql '(91'g ura.roaql) ruaroaql ecualsxe s(rallnuqcral ruoq 's^rolloJ se suaroaql s.ueP^o'u't'9

69I

170

6. Complex Analytic Theory of TeichmiillerSpaces

(iii) It is verified that every holomorphic mapping f : A-

F(f (r),f'(,)) S

:W

?(l-) satisfies

r e a.

To prove this inequality is essential, though we shall omit the details. with f(") = (iv) Take an arbitrary holomorphic mapping f : A - "(f) points (ii)_ and (iii) imply that a,b A. Then (O) for some and e f [fd] - [rop] d(lidl,lwpfi = d(lid),[ru]) S p@,b). By the definition of d|r,rr, we get

d(lidl,fwp]5 dlr.l([id], [ru]), fwp)eTQ). This completes the proof of Theorem 6.21.

tr

Now, Theorem 6.21 asserts that every element f e Aut(TQ)) is an isometry with respect to the Teichmiiller distance on ?(f). we set q = Take an element f e ,+".t1f1f)). For every p = l.ul € "(f), = p of linear isometry "f at is a complex (p) derivative The f f f Q(7(f )) [tr']. and with respect to the infinitesimal metric F, where Q(?(f)) to fo("(f)) Tq(I:(f)) denote the holomorphic tangent spacesof "(,l-) at p and q, respectively. is canonically isomorphic to Here we use the fact that the dual spaceof fpgg)) the spaceAz(H,l.p) of holomorphic automorphic forms on H lor lP , a fact which is proven in the next chapter (Theorem 7.5 and Proposition.T.8).Similarly, the is identified with ,42(11,f'). Hence,/ inducesa complex dual spaceof foQQ\ with respect to the infinitesimal linear isometry o of Az(H,1"t) to Az(H,f') cometric induced by the Teichmiiller distance d. Here we know the following fact. Theorem 6.22. (Royd,en) Let a be a complexlinear isomelrg of A2(H,lp) lo Az(H, f') with respeclto the infinitesimal comelric inducedby lhe Teichm'iiller - HllP and distanced" Then lhere edsts a biholomorphicmapping htFll' ' = (h')2 caoh that a(9) 1 such a compler number_c with lcl for all 9 € of h to H. Az(H,lP), where h is a li,ft For a proof of this theorem, we refer to Royden [184], and Gardiner [A-34], Theorem 5 in Chapter 9. Proof of the surjectiaity o/i*. Now, we return to a proof of the surjectivity of f.. Frorn the previous observation and Theorem 6.22, for every / e ,Aut("(f)) and there exists an element ["0] e Mod(f) with [c..'o]-(p)= every point p e fQ) /(p). W" need to show that [c.,o]can be chosen independently of p. Fix a point q e TQ) arbitrarily. Recall that "(l-) is biholomorphic to a bounded domain (Theorem 6.6), that the Teichmiiller distance d is complete (Theorem 5.4), and that Mod(f) acts properly discontinuously on "(l-) (Theorem 6.18). Then we can find a positive constant 6 so that d(p,[r].(p)) > 26 for any p € "(i-) with d(q,p) ( 6, and for any lule Mod(f) with ["]-(p) lp. Thus we have

'([69] uralsdg 3c) crdolosr are feql;r fluo pue;r crdolouroq ers eceJJnspesop e yo s3urddetu o,lr1 eql ll€car osl€ aA '*u fq l-relouep pu€ 'ar sr a?eJ }eql l?eJ u^rou{-lla^{ ,{11ecrsse1c -rns Surr(1.repunasoqAraceJrns uuetuerg e xg e^\ '.raq1.rng'Surarasard-uolleluauo pue crqd.rouroauoq sr Eurddeur fra,ra 1eq1 pu€ '(Z ?) 6 snueEJo er€Jrns ("lq*lf -ua.ragrp) pesolc peluerro rre q U ?eql etunsse sfeule aa,r 'uorlcas sH? uI '([tSZ] uolsrnql pu€ gg'g ruaroeqtr aes) ace;rns pesop e;o s3urddeur -JIes rueroeql uorlecurssel, s(uolsJnr.{tr-uaslarNel€q er* 'acuenbesuoc e sV Jo '[rzt]"rx puts'[28-V] uaslerN'[Ot-V] ralralg pue uoss'eCot raJerosle e11\'[OZ-V] nr€ueod pue qoequapnel 'yq1egaas 'sluatuleerl IInJ rog 'adf1 elrug {leerldpue Jo eceJrns(pasolc fl.ressaceu 1ou) e;o ese) eql ur paraprsuocoqe sr uorlersrsselc e qcns '6 snua3 Jo e?"Jrns pasoll e go dnor3 sselc Surdde- "qt Jo sluetuele Jo uollc€ aq1 'f11uap,rrnbaro '(Z < 5) t; ;o suorleruroJsueJ?relnporu railnuqcretr '[gg] srag Surr',ro11o;'uorlcas qq] uI Jo uorlecgrssel?Jelnurs " ssnc$p II€qs eAr 'rqoqered pue 'crloqradfq 'cr1dq1apelle) ere qrrq^,r 'sad,t1eerql olur raldeqC ut pegrsselcueeq elerl suorleuroJsuerl snrqotr{ 6 IeeU

rBInPotr I rallnuqr.ral

'p1otruoruuNals D s? (ilJ

suorlBrrrJoJsue.I,I, '9'9 Jo uorl€rgrss€lc

acods.ralputp?eJ eqJ 'e7'g rrraroaql

'ueJoaql eql e^Bq e,.l\aJueg Surr'ro11o; ',,tqdrouroloq '(trtt 'A '67 uraroeq; Jo ursuop e s1 (.7)ag leql luep^rnba sr srql '[rt-V] srag) ura.roeq]s,e{O rg '(1, .ra1deq3ut ?'t tuaroaql'[tq-Y] rqsefeqoy eas) xa,ruocopnasd sl (.f)s't snql 'ecu€lsrp rqse{eqoy eq} o1 lcadsa.r q1r,rl elelduroc sl (J)sJ lerll epnlcuoc airn'16'9 pue l'g $uaroeqtr uroq 'fleurg '02'g ureroeq; pa,rord el€q e,lr snqJ'(J)J uo rf - *['o] 'pelleuuoc s-r 1eq1 sarldrur suorlrunJ crqdrouroloq roJ ruaroeql ssauanbrun aql (.i,)g acurg 'g > (d'b)p qlt,r (J)J f d 1p .rog(d)/ = (d)'fdnJ - (d).[to] ''e'1 'il = (d),ldnlor'[to] 'ecua11'g > (d'b)p qll,r (J)J ) d 1e .roy ]€qt s.trolloJ1r

9z > @'b)pz= (@)l'(l)t)p * (b'a)p((d)- [o"4'(D).[or])p + ((r) -[,r'l]'(d),lor))p ] ( (d). [dr]' (d).lorl)p = ((d). [do]o,lfb ol' a\p TLT

suorl"urroJsuPrJ r"lnPoIAI re[nuqf,reJ Jo uorl?f,ursselc'9'9

6. Complex Analytic Theory of Teichmfiller Spaces

172 6.5.1. f\rndarnental

Extremal

Problems

We can deal with the classification of real Mobius tra.nsformations relating to an extremal problem on hyperbolic translation length. More precisely, for every element 1 e PSL(2,R), set

a(t) = i\f_nQ,tk)), where .E[ is the upper half-plane, and p is the Poincard dista.nce on ly'. Then real

Mobius transformations7 are classifiedas follows: (i) 7 is elliptic if o(7) = 0 and there exists a point zt € I/ with a(t) = PQt,l@t)), i'e', z, is a fixed Point of 7, (ii) 7 is parabolic if o(7) = 0 but there exist no points z,l € I/ with a(t) = p(zr,yQr)), ar.d (iii) 7 is hyperbolic if a(7) > 0 (and then, there always exists a point z, € H with o(7) = p(zr,yQt))). Now, for Teichmiiller modular transformations, we consider the following similar extremal problem. modular transformations. Bers' extremal problem for Teichmiiller For every Teichmiiller modular transformation X of ?(R-)' we set

a(x)=

d(p,x(d)' o.#it".,

where d is the Teichmiiller distance on ?(.R-). Then find a point px € T{n.) such that a(x) = d(p*,x@)). If there exists a solution p, e T(R*), then we call p* a y'minimal point- We classify Teichmiiller modular transformations 1 into four types: (i) 1 is elliptic if o(x) = 0 and there exists a x-minimal point (which should be a fixed point ofX), (ii) 1 is parabolicif a(x) = 0 but there exist no x-minimal points, (iii) X is hyperbolic if o(1) ) 0 and there exists ax-minimal point, and (iu) X is pseudo-hyperbolicif o(1) > 0 but there exist no 1-minimal points. Note that this classification is independent of the choice of the complex structure on .R which is used to define the Teichmiiller space T(R-)' Recall that every point [,S,/] € "(n-) is representedby [Ro, fd] (see $1.4.1of chapter 1), where Ro is a Riemann surface equipped with a complex structure a on fi, and rd is the identity mapping of .R* onto fto. Hereafter, lRo, f dl is simply written * ["].Every Teichmiiller modular transformation is representedby [f]* for a self-mapping "f of .R (see $3.1 of Chapter 6). Now, to investigate x-minimal points, we may consider the following version of this extremal problem. Berst extremal problem for complex structures. For every complex structure o on .R and every self-mapping f of R, considering .f us a self-mapping

ursrqdroruolne atqdrouroloqlq e sl / ?Bql qcns X Surcnpur A Jo { Surddeur-g1as e pue Ar uo , ernl?nr1s xalduroe e sr ereql uaql '1urod pexg e seq x JI 'loo.t4 'ctpoutail s? t! l? fr.1uopuo 11 cpTdglast X uorTouttotsuo.tT rnlnpout reIInurUcNU v 'g7'g tuarooq;, ' u^,i,oul-lla^,rf gecrs -s"lt sr uaroaql Sura'ro11o; aq1 'suorleruroJsuerl relnpour rallnulqereJ cr1d11erog suorleruroJsrrBr,l cnoqradfll

tr

pue arldwlg'Z'g'g

'uorlresse aqt ePnleuol ein 'fFe[urrs ua{oqs st asra^uoc eqJ Pue '1eu1u1tu-*[rf]"l oa'acue11

.((td).[/]'rd)p] ((od).vl,od)p a^"q aal'acuelsrp rellnurqrral aql ol q1/rr (-U)Jgo frleuosr ue sr *[3f] ecurg laadser '(( rd), j[/]' od)p td)p ((od), :[/]' > o1 luap,rmba sl (UI'g) flqenbaur'I'e$ u! (,A)l uo rf go uortre erllJo uoltrugep eqt,{q ueqA 'flaarlcedsar'Io pue o o1 Surpuodsarroc (.U),2 ur slurod eql aq Id pue od p.I

(zrg)

'(l)'"x ) Pl)"x

eler{ ein 'g uo Io arnlcnrls xelduroc i(re,re pue / o1 ordolouroq U Jo V Surddeur-g1as.{rana.ro;'uorlrugep eql ,ig '(6'9 ruaroeqJ lc) ? aceJrnsuueuerg aqt uo / o1 ordolouroq (Surddeur lerueJlxa anbrun aql ''a'r) Surddeur reilnuqrral aql eq oI p1 'leununu-;| sru arnlf,nrls xeldruoc e l€rlt asoddng 'loo.t4 'lorututra-'fl) st o o7 |utpuodseuo? ("A)J ) [o] Tutod aq1lt fi1uopuo tt louttutul -l s! o ernl?nrls xaldutoc o'A D rof, '?Z.g uolllsodo.r4 lo t 6utdtlou.t-11?s 'uol?resse Surr*o11o;aql a^"q e,rl ueqJ :(6'9 ureroaqJ Jc) ooy * o"A , o1 ardolouroq eJe qorq^,rooy * ooq :rt s3urddetu t leur.ro;uocrsenb ge ;o flurey aql ur Surddeur leueJlxe anbrun aq1 ''a'r 'Surddeur rellnuq?rel e q o,U * ooU:o/ terl? palou $ ?I'o"U a?eJrnsuueruarll arll Jo |ut,dilpru -t1aspua.tTra fi1a7n1osqo ue org * ooy:o/ pue ernllnrls talihuoc lou.uutut-t ue 0p IIec e,rl uaql(paqr.rcsap s" (0/'0o) r-red e 'uor1n1os" slstxa areql JI '{ ot ctdolouroq A p rl Surddeu-gas f.rarla pue gr uo r, ernlcn.rls xalduroc {.rerr.a.ro;

(l)'"x > (t)'"x l€ql r{cns / o1 crdolotuorl U Jo o/ Eurddeur-;lase pue U uo 0, ernlcnrls xaldtuoc € pug uaqJ 'leuJoJuocrssnb sr * = aal 'ara11 lou 1nd / lt U)"X '/ go uotlelellp ',U ec€Jrns uu€urenl aq} I€rulxeur eW $)'>I {q elouap ar\{ Jo t/I

suorl" ruroJsu"rJ r"ln PoI,\l rallnurqf,ral Jo uorl"f, ursselc' 9' 9

6. Complex Analytic Theory of Teichmiller Spaces

t74

of Ro. Since ft is compact, it is well-known that / should be of finite order (see Rema.rk 2 in $6.3), and hence is periodic. Conversely, suppose that X is periodic. Nielsen showed that x has a fixed point in ?(E-), whose proof we shall omit here. (Actually, it is shown that the action of every finite subgroup of Mod(R*) has a fixed point in "(,R-). This is the affirmative solution for Nielsen's realization problem (cf. Notes of this chapter). D See for instance Kerckhoff [112] or Wolpert [256].) Remark. A weaker version of Theorem 6.25 is easily shown. Namely, it is easy to prove that a self-mapping f of R is homotopic to a periodic self-mapping of -R if and only if there exists a complex structure o on R, and a self-mapping /s homotopic to / such that /e is holomorphic on .Ro. Note that Proposition 6.24 implies the following theorem. Theorem 6.26. Let f be a self-mapping of R. Then lhere is an f-rninimal compler stracture if and only if the Teichmtiller modular transformation lfl. corresponilingto f is either elliplic or hyperbolic. Now, a finite non-empty set {Ct, " ' ,Cn} of mutually disjoint simple closed curves on .R is called admissible if every Ci is freely homotopic to none of {Cx,(Co)-t}*1i, and is not homotopic to a point. We say that a self-mapping ,Cn} if this set is admissibleand f of Ris reducedbv {G,"'

f ( C t u . . . uC " )= C rU . - . u C n . A selfmapping / of r? is called red,ucibleif it is homotopic to a reduced mapping, and.irceduciDleif not. Then we have the following theorem. Theorem 6.27. If f is an irteducible self-mappingof R, thenthe Teichmiiller moilular transformation [f]. induced bg f is eilher elliptic or hyperbolic. To prove this theorem, we prepare several lemmas. First, we start with the following fundamental one. Let f be a quasiconformal mapping of a Riemann Lernma 6.28. (Wolpeft) surface51 onto anolher 52, antl C be a simple closed,geodesicon 51 with hyperbolic length \. Then f (c) is freelg homotopic to a closedgeodesicuith lenglh lc such thal

t2< K(f)h, where.K(f)

(6.13)

is the maximal dilalation of f .

For the proof, see that of Lemma 3.1 in Wolpett [2aG)' Next, by the collar lemma (cf. Matelski [150]), we can easily show the following:

'acua11'elqrcnparrr ,elqrcnpe.r.rrsr {t1 osle $ d.rarra 1eq1 satldtut 0g'g etutuerl 3f ,.re1ncr1redu1 acurg 'f i{ra,re .ro; V > (0>t }€ql q)ns y luelsuoc € sr areq?

'(.[/]),= (t,ixao1ffif

eleq eal '(gt'g) dg ';l' o1 erdolouroq Eurddetu rellnuqcral aql eq ooq - t"A , !r! lel pue ,(!d)-Vl pue ld qloq ot turpuodsarroc ern?f,nrls xalduroc eql aq lo pI , [ ,{re.l,a.rog

(qrg)

'(.[/]), = ((td).[/] 'ta)pTir1! leqt qtns (.U),2 q r;i{fd}

acuanbese e{eJ 'La'g ueroeqJ {o !oo.r4

'(.a)l ut' sa6.t'aauoc r-;f {("!d)ux} acuanbas ?T?tDrg qcns (.9)alo suotTont.totsuorl rvlnpou rellnuy?reJ lo t--f.{"X} acuanb -es D puD 't=j{fd} 'g r-*{"!d} acuanbasqns o slsrce ereql ueqJ uDrll .ra7oa.ro lo sx fg qcoa uo crcepoa| pasolc alduts fiuo r176ua1 cqoqtadfr,tl aUl ?Dql lo Vxns g aatpsod D s, ?raql {fl asoddng 't fi.taaa .r,ol o7 |utpuodsa.u,oc aco!.tns 7oq7 .Ig.9 EurrrraT uuDurety aq7aq lg puD '(,A)J ur ecuanbas, ,q t{{!d} ?aI rog 'yc) ueroaql

'([69] srag e?u€tsur sseulcedruoc s.proJirunq 3uralo11o;eq? ilecer arrr ,f11eurg

'uorldurnsse eq? sl?rperluoc q?rqlr 'alqranpe.r tr e q p l n o q s / s n q J , ' ( s 1 a ss l u r o d s ) 0 , = ( " C ) , 1 p u e ( I - r ' . . . t 0 - 9 ) r + ! g = (C),1 qcns Surddeur e o1 flsnonurluoc urroJep u€r a..r{(ra,rau,o11 teql ,/ 3f '(s1asslurod se) 0, - t|'C pue lurolsrp e.re 'C' . . .'0, leql qrns r re3alur alrle8eu-uou e sr eraql 1"ql 6U.g€rrruarl ,{q aurnsse .{eur a,u '9'g uorlrsodor4 {q er€ tueql erurs .0g ueq} ssal sr lugofqp ?ou Jo lle , ( g I . g )f g . ( g - 6 t , . . . , I = e-rEC' ...'oCJo euo.r(ueyo { f u a 1 c r 1 o q . r e d .a { qq l [) "l crdolouoq f1ea.gcrsepoe8pasolc alduns eql eq fg 1"1 p,rn , = 0, tas (C) {

' o g> i. '-oe?))t ''n'l 'plo.I

lou seop(p1'g)l(lqenbeur'3[eurosJoJ'1eq1esoddng'i oo.t4 '64'9 DurueI ux ?uD?sunay7 st 09 a.taqm

(rrg)

? (/):r

,r-rrrrr(u?)

sa{sz7osS lo I 6uzddout-t7as elqr?nperr,fi.taaauaqA'k7) 0 snuaf lo g acnl.tnsuulurexq pesop n uo V r170ua1 ctloq.tadfiqy.tn nsapoa| paso1caldtuts D ?q C leT .Og.g eururaT :turrr,r,o11og eqt a^eq ar'r snqtr '09 uotg ssel ero uaqT ctloq.tadfiyIDW pepraord Tutotstp a"tp 6 snua| lo st176ua1 [o acot.tns uuDur?ry p?soli D uo sctsapoaf paso1calduts l?ut?srp omy fiuo 7ot17 q c n s ' 6 s n u a 6u o f r . 1 usop u a d a pW l q n ' 0 < 0 g l u D l s u n D s , a t ? q J . 6 Z . 9 e u r u r a r l suorl"urrolsusrJ

9LI

r"lnpow

raflnulrlf,ral Jo uorl")yrss"lc

.g' g

6. ComplexAnalytic Theory of TeichmiillerSpaces

L76

the hyperbolic length of any simple closed geodesicon each Eo, is greater than 6sA3-3t. By Lemma 6.31, we may a^ssume,taking a subsequenceif necessary, that there is a sequenc" {Xi}Er of Teichmiiller modular transformations such converges to a point q € ?(R-). We set q1 = that the sequence {Xr(pi)}Et j. eaih Since Xi is an isometry with respect to the Teichmiiller Xi@) for every gives (6.15) metric,

ilg

d(qi,Xi olfl* o (xi)-'(ci)) = o([f].)'

(6.16)

Again taking a subsequenceif necessary,we may a^ssumethat {xi o [/]- ? to a point q' eT(R,), for ?(r?.) is finite-dimensional (Xi)-t(qr))Fl converges and is completewith respectto d. since each1r.o [.f]- o (xi)-' is an isometry,it is easyto seethat

" (x.i)-'(q)= c'' o itg xi [.f].

Hence, by Theorem 6.18 we may assume,taking a subsequenceif necessary,that large j, say Xi "lf).o (Xr.)-1 has the same action on "(ft.) for every sufficiently j> jo. Then (6.16) implies that d(q,xioo [/]- o (xr")-t(q)) = d((xi")-t(c),[/].((xi")-'(q))) Thus there is an [/].-minimal point, which shows the assertion. 6.5.3. Absolutely

Extremal

= o([/].)' n

Mappings

Next, we cha.racterizeabsolutely extremal mappings. We have seen that, when a self-mapping / is homotopic to a periodic mapping, then the corresponding absolutely extremal mapping is conformal (Theorem 6.25). Hence, we discuss the case that [/]- is of infinite order. Here, note that the Teichmiiller space ?(fi-) is a straight line spacein the senseof Busema,nn (cf. Kravetz [128], and also see Masur [142]). In pa,rticular, any two distinct points p|,pz e "(ft-) lie on a unique stroight line, say tr' which is an isometric image of R into "(E') equipped with the Teichmiiller metric, and contains all points p such that d(pr, p) * d(P, Pz) = d(Pr,m) We also note the following elementary fact: Theorem 6.32 If a Teichmiiller moilular transformation 7 is of infinite oxler, lhen a point p € "(R,) is y-minirnal if and only if 7 leaaesa straight line thrvugh p inaoriant. Proof. Assume that p is x-minimal. since x is of infinite order, three points "segp, X(p), and X2(p) are distinct. Let p1 and,p2 be the midpoints of the -"tttr'; [p,x@\ and [1(p), xz(p)], respectively. Then it is ea'syto see that

1eq1 saqdurl (gt'g) €lnuroJ 'GI)X ! (V)y a?urs pue ,(U) Ot.l rueroaql fq zU)X ) ("t)X e?urs 'leurer1xef1a1n1osq€ sr / feqt etunsse,alolg

(srs)

'./ o1 ctdolouroq Surddeur Jallnuqcretr eql $ 11ereq^r

,(u)>t = "U)y se uellrrr,r,er q (ft'g) 'Surddeur rallnurqrreJ e sr 3l aaurg

(rrg)

'((["]).(.[/])' [r])p= ((["]).[/] '[o|)pz=

((["])'(.Ul)' (["])-[/])p+ (([o]).[/]'[o])p o1 luele,rrnbqfl s-rrll,69.9 ureroeq;, ,tg .leurut.tu--[/] sl (.U),2 ) [o] lutod Surpuodsa.rroeeq] JI {po pue 3r Isuer}xe flalnlosqe q ! 'V?,.g uotlrsodor4 fg 'o ernlcnrls leuroJuoc elqqlrns e rlty( oA = S 1n4 .Eurddew rallnuqcratr e sr erueq pue 'leuro;uoc lou q / 1tsq1erunss? Aern a71y.loo.t4 'z$)x = (z|)x qTtm|ut'ddotu rennuruz'eJ D oslD cr futrldou otp qcns |utildour rqpuq)r4 o ro |utddotu to1? 7t Totu^tot -uol D r?qpe s, 7t fryuopuv s! ,g g : I 6ut,ddou.t, o lt fi lourar?se fr1ayn1osqn ueqJ '(?, -) 6 snua| to acottns uuouery p?sop o aq g pT .gg.g uraroatll 'rualoeq? Eur,raollo;eql ur"lqo eiu 'alo11 '[167] ealie?ru€trosp 'auq ees l{Ere.r1s ?uelr?Aur auo ?soru +e s€q uorl"uroJ -su3r1relnpolu reilnuq?ral /tue l"ql ,(roaq1s(uolsrnqr ruog s^rolloJlr'I clrDuea 'luougau, autl D setuDel fr1uopuo tt u1oqtadfiy sg *flf uo4out^totsuo.tT 7t16to.r.7s 7g tg rDlnpout rellnuq?pJ eW'(A)poW > fl)Wauala atpouad-uou D ro4 .r(.re11o.rog O

'1eururur-X s1d pqt sarTdurrqclqa'(-U)J) ((d)x'd)p

d f.ra,raro;

2 ((d)x'd)p 'f.rerlrqJe sr u a?urs leqt apnl?uoc eal

' (( d)X' d)p . u ( d' d)pz * > ((d) "x, d)p = ((d)x, d)p . u teqt'flllenbeut a1Euet.r1 aq1 Eursn fq 'aas uec a^\ 'u reSalur a,rrlrsod fue pue (?),2 3 d lurod fue .ro; 'ueqA 'X rapun eull lq3rerls € lsql asoddns ,1xap luerr€Aur sr d qEnorql Z 'X .rapun luerrslur sr 'eU ((d).X (ed 'rd osp eruaq pue) (d)X ptre d qarq,u uo ,aur1lq8rerls aql leqt sarldurrqcrq^a([zd(rd]?ueur3asaq1go lurodplu eq] sr (d)X snqa'(X)oJo uorlru '(a)X ,pueq .raq1oeq} uO -gep zd aqt fq (X)o eAeq (za'41, a^{ aculs 7 '(x)' j (zd'rd)P 'acua11 leql epnlcuoc aiu'

.(x)"1= (d,(d)x)p= ((d)x,td)p LLI

suorl"ruroJsu"rJ r"lnpon railnuqtral Jo uorl"f,urss"Ic .g.g

6. Complex Analytic Theory of TeichmiillerSpaces

178

I{(h) = K(f,).Hence, the mapping .f2 is also a Teichmiiller mapping with I{(-ff . (Note that, by Theorem 5.9, we concludethat h = f2.) xift)= Conversely, if the mapping /2 is also a Teichmiiller mapping with K(/2) = E K(f)2, then clearly (6.17) holds. Therefore, / is absolutely extremal. Remark 2. we can show further that the condition that the mapping f2 is also a Teichmiiller mapping with 1{(/2) = K(f)2 is equivalent to the condition that the initial and the terminal differential of / (cf. Proposition 5.19) coincide with each other up to a positive constant factor.

6.5.4. Reducible

Mappings

and Nielsen-Thurston's

Theorem

For reducible mappings, we can show the following theorem. Theorem 6.34. Let f be a rcducibleself-rnappingof R. If f is nol homolopic to a periorlicmapping, then lfl. is either parabolicor pseudo-hyperbolic. Thus, by Theorems 6.25,6.26,6.27, and 6.34, we concludethe following: Corollary. Let f be a self-mapping of R. An f -rninimal complex structure exists if and only if f is either homotopic to a periodic mapping or irteducible. We shall omit the proof of Theorem 6.34. Instead, we shall explain the structure of a reducible mapping. (For this purpose, recall that the foregoing arguments still work even for the case of a surface of finite type.) Let / be a reducible self-mapping f of R. Then we can deform / continuously to a completely reduced mapping, or more precisely, to a self-mapping fs of ft which sa[isfiesthe following condition: there is an admissibleset {C1,... ,Cn} of disjoint simple closed curves on R such that, for every comp-onent H of R a.nd for the smallest positive integer N with /Jv('?') = R', the CtU-...UCn \ mapping /t l,*, it irreducible. L e t { . R i , . . . , R ; } b e t h e c o m p o n e n t so f R - C r U " ' U C , . A s i n t h e c a s eo f a complex structure oi on Rli and an closed surface, there is "r, (/", Ini )-minimrl (R!)"1for every i (i = 1,"' ,m), absolutely extremal mapping Fit (H)", -- Rli (We can where Ni is the smallest positive integer such that |Ni(Rii) further slow that (Hi),, is a Riemann surface of analytically finite type.) We can show that o([/]-) - max{Iio,(f'1),''', Ko^(F^)}, but there exists no [/]--minimal point. Hence, if all K"r(S) are equal to 1, or equivalently, if all .Q are conformal, then [/]- is paraboiic. If not, then [f]. is pseudohyperbolic.

'([t t t] 'auos(uotsrnt{J Uoq{rrey pue [6S]srag 3r) [911]uolsrnqJpueJoq{rrex ees ruorJ lueJeJrp sr Surppaqure(srag Sursn t 6 Jo uorlergrlceduroc aql leql alou osle e \ '[qql] uefln4ctr{ pue '[8gI] tl{sel tr'[96] srag '[t]goUqV o] raJeralll'1ca[qns q (.f)sz Jo (J)sJg srqt rod 'sdnor3 u€ruraly ;o ,froaql aql o1 palelar .{1aso1c frepunoq eql'0
saloN

'(yloq aosouy-opnasd \ou 1nq) alqtcnpa.tst .r'ouL,sttld.r,ouoa[r,p pu s, q?tqn'(27) 6 snua| to o o 7 c t , d o T o u t osyt ' a u o c t p o u a d o o 7 c t d o T o t u o U acottns pesop o to futildoru-l1as y (uolsrnql pue uaslarp) 'gg'g tuaroaq;, 'uolsrnql pue ueslerN rueroaql Surrrrollogeq1 fldtur ?g'g pu€ /6'9 suraroeql Jo snqJ 'rrloqradfq sr -[/] yr fluo pue yr usrqdrouoeJrp ^osouv-opnasd e o1 crdol 'tt'g '96'9 'e.royeraq; -ouoq sl sure.roaqa / leql fldurr g'g$ ur A {reru€rg pu€ 'rolceJ lue?suo, allltsod e o1 dn raqlo q)se ql-ri apIDuIo?slelluareJlp leurturel pue I€IlIuI esol1\^ Sutddeur rallnuq)retr € seurooeq/ tn,It qcns U uo arnlf,nrls xalduroc € ulelqo u€? a^! 'A tusrqdrotuoegrp ^osouy-opnasd ua,rrE due ro; '.{1esre,tuo3 Jo { 'u.tsrtld,totuoaficp 'ernlcnrls xalduoc eq1 aosouv ?a3.ro3a r uarlirr -opnesd pell€c-os e se,rr3rolceJ luelsuoc errrlrsode o1 dn raqlo qc"a qllrrr eplc '.{1eutg -uroc slsrluaraJrp l€unuJal pue Ierlrur esoq^\ Surddeur rellnurqcrel e 'cqoqered sl -U] 1"ql ees uec aal 'acua11'C - A ;o luauodtuoc qcee uo Surddeur .r(1r1uepr eql ol crdolouroq sr / pue 'asro.luauodtuoclu€Irelur C - A ser'ee1/ ueqtr, 'C a rnc pesolc aldturs e o1 lcedsar qlr^r lsrlrl uqeq eql ag U - A : I p1 'aldutotg 6LI

saloN

180

6. Complex Analytic Theory of TeichmrillerSpaces

Let "s(1) be the image of Bers' embedding of the universal Teichmiiller space ?(1), where I denotes the trivial group. Gehring [83] proved that "3(1) coincides with the interior of the set S(1) consisting of Schwarzian derivatives of holomorphic univalent functions on 11. It is also known that S(1) S "s@ (see Astala [19], Gehring [84], and Thurston [232]). Shiga [200] showed that if l- is a finitely generated Fbchsia.ngroup of the first kind, then fBQ) coincides with the interior S(l-), where S(l-) = S(1) n Az(H*,l-)' When f is of the second in the case kind, Sugawa l2,l2l has shown recently that S(l-) I fn!).However, whether it is unknown first kind, generated and of the where l- is infinitely or not. S(f) c ra!) For an arbitrary Fuchsian group i-, the Nehari-Kraus lemma (Lemma 6.7) implies that "s(l-) is contained in the open ball with center 0 and radius 3l2in Az(H., l-). The infimum of radii of open balls with center 0 in A2(H* ,i-) which includes fBQ) is studied by Nakanishi [166], Sekigawa [192], and Sekigawa and Yamamoto [193]. For connections with projective structures on Riemann surfaces, there are papers Gunning [88], Kra [120], and Shiga [202]. It was known by Fricke that the Teichmiiller modular group Mod(l-) induces (see, for example, Fricke and Klein [A-33]). a discrete subgroup of Aut(T(f)) The proof in this chapter is due to Bers [31]. The modular group Mod(l-) is studied in Bers [39], Hejhal [100],Ivanov [108], Kerckhoff [111],McMullen [154], Mumford [161], and wolpert [249]. For classification theory of Teichmiiller modular transformations, we refer to Casson and Bleiler [A-19]' Fathi, Laudenbach and Podnaru [A-29], Bers [38], Kra [121],Shiga [201],and Thurston [231].Kerckhotr [112] and Wolpert [256] solved the Nielsen realization problem which asserts that the action of every finite subgroup of Mod(l-) has a fixed point in ?(f). See also Kerckhoff [113]. For Bers' fiber space over a Teichmiiller space and the Teichmiiller curve, we refer to Bers [31] and Earle [61]. The relation between Teichmiiller spaces and holomorphic families of Riemann surfaces is found in Nag [A-80]' Chapter 5, Earle [58], Ea"rle and Fowler [64], and Imayoshi [102]. Their applications are treated in Griffiths [87], Imayoshi [103], [105], [106], Imayoshi and Shiga [107], and Riera [183]. For holomorphic sections over a holomorphic family of Reimann surfaces, see Hubbard lA-441, and Earle and Kra [66]. Many more details of the Kobayashi distance a.re found in Kobayashi [A54], Lang [A-64], and Noguchi and Ochiai [A-88]. For a distance on ?(,l-) invariant under the action of. Mod(f) other than the Teichmiiller distance, we have the Carath6odory distance. Earle [59] proved that "(l') is complete with respect to the Carath6odory distance. The connection between the Teichmiiller and Carath6odory distances is studied in Kra [122]. There are also invariant distances which are induced by the Bergman metric and the invariant metric on the Siegel upper half-space. For this subject, see Royden [185]. Even in the case where "(l-) is of infinite dimension, it is shown by Gardiner [81] that the Teichmiiller distance also coincides with the Kobayashi distance. See also the book by Gardiner [A-34].

'rrrell Jo lurod srql urorJ palPnts st se?eJrnsuuetuelg ;o t(pue; e ;o '1 uorprrel Isrursalruuw aql eraqa '[996] lradlol\ ol reJarosle eA\ re1deq3lo 7'6$ ur fgarrq pessncslp eq ll€tls sploJlueur xelduroc ;o f.roeql uollsturoJeP .racued5 -eiltspoy aql pu" saceds relpruqcle;, ,(.roaq1 eql uaa^r}eq uolleler eq& ;o .[996] r.radlolt pue [66I] e3rqg ur s;oord e^Ilsurelle errl areql 'ploJlueur ulals € st aceds railnuqoleJ I"uorsueurp alruS e t"ql h?] srarduarqg Pue srag fq pe,rord 1srg se^{ 1I I8I

Chapter 7 Weil-Petersson Metric

Unless otherwise stated, a Fuchsian group l-, considered in this chapter, is a Fuchsian model of a closed Riemann surface of genus g (Z 2).We also assume that each of 0, 1, and oo is fixed by a"nelement in f - lid\. As stated in $1.3 of Chapter 5, the Teichmiiller distance on the Teichmiiller space 7(.l-) measures a kind of magnitude of deformation of complex structures of Riemann surfaces, and ?(f) is complete with respect to the Teichmiiller distance. We also saw in $3.4 of Chapter 6 that the Teichmiiller distance is equal to the Kobayashi distance, which is defined complex analytically. However, the Finsler metric induced by the Teichmiiller distance is not of class C-. The purpose of this chapter is to introduce another natural metric on ?(l-), which is called the Weil-Peterssonmetric, and to prove that it is a Kd,hler metric whose Ricci curvatures, scalar curvature, and holomorphic sectional curvatures are negative. The first section is preliminary and devoted to studying the Petersson scalar product on the space ,42(.I/, f) of holomorphic automorphic forms of weight -4 with respect to l- on .Il, and related topics such as the reproducing formula for holomorphic automorphic forms, Poincar6 series, and the Bergman projection. In Section 2, we see thaL A2(H,,l-) is the dual space of the holomorphic tangent space Zs(?(f)) of TQ) at the base point. We also represent elements by harmonic Beltrami differentials. Then the Weil-Petersson metof "r(Z(f)) is given by the dual metric of the Petersson scalar product ric on "0("(f)) on A2(H,l-). In Section 3, we define the Weil-Peterssonmetric on 7(f) and verify that it is Kiihlerian and that its Ricci curvatures, scalar curvature, and holomorphic sectional curvatures are negative.

. u"d,t v. (\ u" d, t-, d , /r \J7_ -- , d , to1 urroJ eql uI uelllr^{ sl (V)zV I d flera 'relnrtlred q'(y)zy ra8alut eltleSeu-uou.{ue.ro; srseq leurouoql.ro elalduoc e sI 0--J{"6 } t"qJ'u

(r'z)

,z (t,+ u)(z+uXr+ q: l\ - G)"d

3

1nd a111 'lcnpord rel"cs srql q1r,neceds e sauro?eq(y).y t"qf treqllg alqe.redes = v (,/r'6) npxp(z)rlt(z)d)z-e)y" [[ J J fq(V)zV uo v( . '. ) tcnpord rel€cs uossratedeql augep eM'y {slp llun aql uo clrlatu erecurod eq1 * "lzplr(")V = zsP''a'\'("ltl - t)lZ = (z)y areqm

'a > npxp ,l?)al"-Q)u"l[ = )l^tt l"ql qrns V uo d suorllunJ crqd.rouroloq;o areds roltel xelduror eql aq (V)eV +"1 *;. eueld-;pq remol aqt uo J ol lcedse.r qll^r sruroJ crqd.rourolne ctqdrouoloq lo (J'.H)zV pue uo lcnpord relecs uossJeled eql eugep eAl'r(errr aures aql uI '(tJ' V)eV uo lrnpo.td rel€?s uossreled secedslraqllg rltoe esn eA\'(/J'V)zV (l'n)"V eq1 'flrepurrs eugap e$'V {srp ?run eql uo 3ut1ce E Io il Iepou uelsrlcnd € 3uqe1 'g aueld-;1eqraddn eql uo 3ur1ceU Jo J Iapou u"Isq?nJ B Jo peelsul '(l'n)zV u(''.) uossreled aql uo rDlms lcnpo.rd reuur uellpurell slt{l 7cnpo.r.d '1cnpo.rdJeuul uelllureH slql qtyr,r aceds treqllH e setuoceq (l'n)zV II€r el!\ '.H ul 'lceduroe $ l€rll s/rtolloJ1r u atuls J roJ ulsruoP lelueuepunJ e sI Jr eJaI{^{ Uf f Jf f 'fipxp(z)rfu(z)dtr-(z)xu = op(,/L'd) =a(,/r'6l' ll ll

fq paugap q u({1 'd) lcnpord reuur uer}runa11aq1 'r*o11 ar uo uorlcunJ e s€ pereprsuoc sI 1l ''a'-I '.y rapun lu?lrelul l? uo uolltunJ e sl (41'ai) uaqS 'H

) z

' ( z ) r f u ( z ) d u l y= ( z ) ( r / t , ' d t )

'(J'H)"V ) ,lr'o\ sluetueladue rog les e a 'J /n = Ur a?"Jrns ulrctuarll aql uo luetuala fq pacnpur frpap"(z)ny - ,p luaruele eere eqJ, €are ue se papre3a.r s\ lsp ''e'\'H uo rrrleur ar€rurod aqt aq - H"spp"I 'r_("*t) @)nU "lrplr(r)rU 'Il uo q?l^{ sturoJ crqdrotuolne ctqd,rotuoloq 3o o1 lcadse.r J '11e (t'tt)eV ecedseq1 uo lcnpo.rd rauul u€I?IturaH lecluousc € augep e,lr 3o lsrtg BImuroJ Euranpoldall

uorlra[ord

u€rutJag

puu +cnpord TBIBcS rrossralad

PUB ]rnPord

TBIBJS uossJalad

'1-'1-'L

'T'2

't'^L lf,nPord r"lef,s uossrelad

t8I

7. Weil-PeterssonMetric

184 We set

K a (z,C=) i r

p^( C) , z,( e A.

( 7.2)

n=0

Then Ka(2,.) formula

belongs to A2(A) for any z € A, and satisfies the rvproducing p(z) = (p,Ka(2,.)la,

z C A,

for any 9 e A2(A); that is, /(a is the reprodacing kemel for A2(A). Flom (7.f ) and (7.2), we obtain @

o

-= -K - A \7 - r !1 / , ( z= , Ci ) ? @ *| 1^ , /)\ '(r nI +2 3 , f)\ '(" nt + v ,3 r\') s ,( z d ^U_U"r'"

- 1 2 , , , . z , ( €A .

n(l_Z4!)4,

Simila,rly,we considerthe Hilbert spaceA2(H) of holomorphic functions ry' on .EIsuch that

= (,)1, dxd,y 1 x. ll,tll, Ilr^rr'r-,1,t The M<jbiustransformation?: H - A given by T(z) = (z - i)/(z f i) induces an isomorphism?* : A2(A) - A2(H) definedby T'(p) = (9oT). (T')2, I e Az(A). Hencethe reproducingleemelKs for Az@) is given by

= y+. - Ka(r(4,r|))T(ffr'111' Kn(z,e) 7r(z Note that I{6 and,Ks are inva^riantunder.4ut(4) For example, Kg satisfies

Kn(",Q= Kn(t(z),1$\1(St'(e)2,

C)n'

rz.( ' 5 e H.

(7.3)

and Aut(H), respectively.

z,(eH,

(7.4)

forallTeAut(H). Theorem 7.1. (Reproducing formula) Eaery9 e A2(H,f) satisfies

p(z)= [[ xrfCl-',p()rciQda€a,t, z€H. J JH

(z.b)

Proof. It is sufficient to prove that

,1,(r)=[[ ^rcfrl'G)@aea,t, JJA

zea,

(7.6)

for everyrb e Az(A,l-'), wherel' = TIT- 1. SinceR = H /t is compact,^-'lrltl is boundedon 4. Thus the integral in (7.6) convergesabsolutelyfor all z. By the mean value theorem for a holomorphic function, we have

'V)z

rJ)L

'r(z),1((')L)i =k){O 3

'V uo uor}3unJ crqdlotuoloq e ro; 'leraua3 u1 les am {

.v ) 2,"(z),1((z)L)/= ' ?? ) f i se ue??rr^r sr pu" 'v ernsol?aql poorlJoq fl Jo Jo v -q3raue uo crqd.rotuoloq q ./ uorlcunJeq1'V ur lceduroc,t1a,rr1e1ar sr dr e)urs

(o'r)

"fl .bpJp(),2)v:r())d._o)v = ev

'1t,r'oN las ellr I "J r , t ' J j ! = . "(z),1 QJ,Wz())4,_o)v I| ) 3 lorw

(8'z )

"|4 7 7 1 r J ) L lo)f ,-o)v | 3 = "(z),(,-r1lt'p1pQ' G),-r),>t )

t ptpzl()),Ll (A!:yZ(()),t)@ z- ((>lrlr"' I I'? = " = ttplp O,z)vx e),fur_e)u I I Z e),t, ( r\Ln

n rJ)L

s€ uellrr^rarsl (g'Z) elnuroJ ueql 'y ur sl d leql os y ur tJ rol3{ ureruopIe}uauepunJe e{ptr lceduoc d1arr,r1e1a.r sarras gJBcurod z.I'1, 'V ) z II€ roJ sploq (g't) 1eq1s^\oqssrqtr

O

' ( n ' z ) u; t @ ) f i"- - ( q r " [ [ \ " ( o ), r ( a p n -p

=

J J / vff

\

apnp ((^)g'(,)il, >r(@)il'h"_((.)g)v = @",fu "l(n),sl IJ uletqo a,rn', _,0- 9 3ur11ndpua ofi oI (L'L) Surflddy ' "6),L(r)fr = (g)"fr seqsll€spue'(L,,1,_L'V)ZV Jo luetualeue sr o4lueql '.(,.!) .(L",lr) - ofr ps e$ lI'z = (O)t qll/{ (y)tny 3 L luauale u€ esooqr'V ) z fue ro3'aao11 .0 - z roJ splorl (g'Z) elnuroJ leqt stress" slq;,

't p lp()'i l vx Q't)

Q)fr" -Q) u J"[[J =

= tnlee)fi,("|)l-r)"il f tolo lrnPord r"Ff,s uossratad'I'2

98I

186

7. Weil-Petersson Metric

We call this Of the Poincar|, series of / for l-/ of weight -4. Similarly, we define the Poincar6 series of a holomorphic function on I/ for a Fuchsian group l- acting on H. Theorem

7.2. Let f be an integrableholomorphicfunction on A, i.e.,

[[vaydndy<x.

JJA

Let Of be Lhe Poincar| seriesof f for a Fuchsian group l' acting on A. Then Of conuergesabsolutely and uniformly on compact sels in A, and belongsto

A2(A, r'). Proof. Denoteby B(z,r) the closed disk with center at z and radius r. Take any compact subset K in A. Since l-' acts properly discontinuously on .4, and since every element 1 e f' - {i.d} ha.sno fixed points in 4, we can choose a = 4 for any z € K and sufficiently small radius r so that l(B(r,r))nB(z,r)

1er'-{id}.

On the other hand, lf OQDy'(r)2 | is a subharmonicfunctionon 4, and hence the mean valuetheoremfor a subharmonicfunction yields that

G)'la'ea,t lroQ))t'QYl = # | l "u,ulr(t(cDt' = #r f f K)la€an' JJ.,rur,urlr Thus we get

dtdn e)'rs # D,rr(te))r' *r,,,,rrG)t F*,11., s 1r r 2 =

[ [ v r c y d , ( d , q
JJa'

Hence, @/ convergesabsolutely and uniformly on K, which in turn implies that / is holomorphic on 4. Next, for any 6 € l-' we have

A'Q)2 (z), = D f 0"6(r))t,(aQ))2 o f (6(z))6, 1el'

=I

fe"6(r00"6),(r),

1eP

= Of (z). Therefore, @/ belongs to A2(A,f').

tr

From the observation preceding Theorem 7.2, we have the following corollaries.

(rlz)

(orz)

Jf

f

trplp ()'z)nx())/._())sv = f,lu JJ

eJeqm

'H uo 6O {zd sa{st7os puv '(J'H)zV o7sfuopq lzd uaqJ

' H > z ' t t p l p O ' z ) n > t O ) / r _ O ) fHff vf = ? ) U r g ) I| J J '(J'n)Jl ?es

uDrol g'Z uraroaqtr, ) I fi.r,olpq.ro

'Q'U) s\ (,t'tt)zv uo u( ' ' . ) Jl ot Pepuetxa 'reqlrnJ 'Q'H) 'H)zv lcnpo.rdrelef,suossreledarlt Jl to acedsqnspesolce sl (J 'lcedtuocsr ecurs'rrrrou$ql q1ralacedsqr€usg e sauroceq(.f 'g)iZ ,r"ql Ar

'* > l(4!lz-Q) Hudfr?;"= -ll/ll qtl,llt I/ uo J ol lcedser r{}l^{ 7- lqSraaaJo sruroJerqdrourolne elqsrns€eu II€ Jo }es eq} eq (l'U) &l l"l .1 3 L,H ) z,(r)t =,e),t((z)t)t sausIlss pue 'Il uo uorlcunJ elqsrnwaru s sl 1l JI /i uo ..1,o1 lcedsar qlpa p- lq3rer'r go u.r,ol cttltl.totuolnv elqornsoaur, e pell"r sr H uo I uorlcunJ penler'-xaldruoc y .(n)zV Hy to! leure{ - g Surcnporda.req1 Sursn fq g uo slerlueresrp crlerpenb elqsrnseeu luuuoq 1f p eceJrns uueruerlf eql uo slerlueraJrp crlerpenb crqdrouroloq lrnrlsuor ileqs eA\ uorlcafor4

ueuErag

g'1'2

'(g'g ue.roaq;;o;oord eql 'Jc) 11 Jo sles lceduoc uo .{lurro;run pue dlalnlosee o1 se3rerruocJ rol I lo lO serras are)urod eql uaqJ,'((y zz/l - t)i - r)t)/6 - V) = Q)l fq uaLrS .;1,uo uortrunJ crqdrouroloq alqerSelur ue aq / le1 'H uo 3ur1ce dnorS uersqcnd € sr qrlq/'\ 'I < y qll,lr zy - (z)oL f.q pelereua3 dnor3 e aq J laT 'sarreserecurod e ;o aldurexe ue errr3e11y'a1du-tnxg '11 ut sles 'H u? Tcodu.toa J .tot utoutop loTuauopunt o s? ,I ataym uo filtu.totrunpuo fr.yaTqosqo safileaun eprs puoy 7t16t"t, e1l uo saxr?s eql puo

LH > z' Hu" 3 = Q)d Q'e)qr rr-Q)a "(z),1 z-e) ill luo*, tu.tol aq7 u, u?llu,rr"s! (J'H)zV

3 dt fi^raag '6 riregoro3

'V uo o\ tottt V?ns V to pootltoqrl|tau o uo peu{ap t@ D c?sweNeUl'(J'V)zV ro,tr 'T fte11o.ro3 ) dt fr^taaa { uorTcun! ctyd.r,oruoloy 'I'1, lf,nPord r"l"f,s uossralad

r.8I

188

7. Weil-Petersson Metric

and F is a fundamental domain for I in H. Proof. From ll/ll- < oo and forrnula (7.3), we see that integral (7.10) converges absolutely. It is clear that fzf is holomorphic on 11. From formula (7.4), we conclude that B2f belongs to A2(H,f).By an argument simila^rto that in the proof of the Corollary to Theorem7.2, we get formula (7.11). tr It is easyto seethat B2: Lf (H , f) - Az(H, f ) is a bounded linear operator. The reproducing formula in Theorem 7.2 implies that B2 is the identity mapping on A2(H,f). We call B2 the Bergman projection of Lf (H,f) fo A2(H,f). Rernark. We can also use the .Le-norm (C ] t) instead of the ,L--norm. Namely, let L\(H,f) be the Banach spaceconsisting of all measurable automorphic forms with to I on 11 such that respect /

llflli=Ilr^'r,r'-'olf (,)lodxdv1ooThen we obtain the Bergman projection from Llr(H, f) to A2(H, ,i-) (equipped with ,Ls-norm). For details, we refer to Kra [A-58], Chapter 3, $$2 and 3. The Bergman projection is a self-adjoint operator; that is, we have the following assertion. Theorem

7.4. Ang two elemenlsf ,g e Lf (H,f)

satisfy

(1 rf,sl n = (f ,1zc) n. Proof. Take a relatively compact fundamental domain F. for f in fI. By using formula (7.11), the transformation relations for )s, Kn, f , and g, and Fubini's theorem, we get

(02f,cln rt f- tr I =[[^\n(,)-2|Dtf^^H(o-2f(0Md$r1|7,Q)2gQ)axay rrF l tT rttr J

-

=

[ [ , r r t - 2 t r r t - " l - t r - e - - - , , l \se)-z@l,e)zKs(ve),edxayl aurt Jlr^n\s/ ,'..)|Ltert E/, t r i

fl

JJrls(o-'l(o X

(z)-2 s(z)ffi I l,\H E

4 a,av0-')'G)'d€dn

= (f,?zcln . tr

'

*(J'H)zv= ((t),D1t

'U)g 'ro1nc4.roduI ' ' *(J H)zV oTao (1) 71/(l to tustrlil"rourostuo sectuput nV = Qt)V fi.q uaar6 *(J'H)zV * (.1'H)g ty |urdtlour ?yJ g'L tuaroaql 'Il ul J roJ uletuop leluaur€punJ " sl d pue J/H '(J(H)z,V

'npap(z)dt(z)d"[[ =a(6'rt) ) d't TJ

- g"a.ra11

= (dt)dy

r€eull e fq paugap (l'U)zV uo leuollf,unJ '(t ' ''a'l '*(J ' /y luaurale us qll,rr (.t ' tt)zv p H)zV ) U)A ) r/ fre,re elsr?ossy '1xa11 eceds lenp eqt qll,rr ((.t),D',1fllecruouea fg11uap11eqsa,r,r, ,(J'H)"V

' Q)n I Q' H)s = (Q).r)'.r,

(eu)

el"q arn 'oOray = (.fhf Eurllas 'snql'((.i')ag)'Jol H uo J roJ sl"rtueraJrp rurerlleg p ('J'H)g aceds aq1;o Surddew .reaurla,rrlcelrns e sr lurod as€q aql ?" O uorlrafo.rd ,srag ;o 04; airrle,urap eql 'g raldeq3 ;o = ((;)ag)og aceds 6$ ur uaes s€it{ sv'1urod aseq eqt 1e (.t)alJo'(J'*H)zV p.rc3a.ra,r,r'(.7)ag qll/" (J)J Sutfgluapl lue3uel crqd.rouroloqaW se ((l)D",t '1urod aseq eql le (,f),2 fo aceds lua8uel ctqd.rouroloqat{} ((J)J)',2 fq elouaq lulod

aseg aqt 1e aced5 lueEue;

oqJ, 'l'Z'L 'aceds

rellnruq?ral e ;o saceds lua3uel eql roJ uo.Ileluasardar lrcqdxa ue a,rtE 11"qs o1t

sarsds rellnl'IlqclaJ, Jo f.roaq;, lBtutselruVuI'Z'

L

' H ) z'(z)l 4 " o z- = Q) ( d Hr u ) zd

(zrt)

uplqo em 'uorlreford ueurS.rageql Jo uolt$Sep aql fg '("f ' H)zV Io lueruele wr sr

.[[ "-= -(z)l4o t"plp19,..............., --rJr 1) 9 O)rl

tt

'(J'*H)"v l€rll eas e^1,snqJ,

'* H) z ' t p t u W " [ f

;-

o1 sSuolaq qarq.n

=? ) [ 4 ' Q

",'11'f$":l?t1trtff.'"#511 eqruro uorlce rord (sras ;;,lt Tjffi:T ro1,r1,a saredg ra11nruqrral;o froaqa 1eurrsatruyq 'Z',

68r

190

7. Weil-PeterssonMetric

For the proof of this theorem, we need the following lemma due to Teichmiiller. Lemma 7.6. (Teichmiiller) An element p e B(H, f) only if A, = 0, i.e., (tt,p)n = 0 for all 9 € Az(H, f).

belongsto N(f)

if and

Proof of Theorem 7.5. Cleaily, z1 is linear. Teichmiiller's lemma asserts that KerA = N(l-).Every / € A2(H,f)* is written as / = (.,rlt)n for some ry' € Az(H,f), where (.,.)n d e n o t e st h e P e t e r s s o ns c a l a r p r o d u c t o n A 2 ( H , f ) . Putting p = ^;rb,we see that p € B(H,f) and Au = f .This shows that z1is surjective. Hence, by the homomorphism theorem we conclude that ,4 induces an isomorphism of B@ , f ) /N (f ) onto Az(H , f)* . D Proof of Lemma 7.6. We consider this lemma in the unit disk 4 instead of fI. - 1), which^sends Take a Mobius transformation ^9given by S(z) = -i(z+l)lQ 4 a"nd A* to H and f1*, respectively. Here, 4* is the exterior of 4 in C. We set l-l = S-rfS, a Fuchsian group acting on d and 4*. Then p e B(H,f) corresponds to u € B(A, ft) defined by

u(z)=p(S ' "@ ) @ - , 5' \z)

z€A.

Further,O"lpl e Az(H*,1-) corresponds to iL e A2(A*,f') givenby

irr(z)=tb,1p1151'115,1272, z e A*. Now, formula (6.8) in Chapter 6 is rewritten in the form

=-* ll"ffiauao, zeA*. v(z) Thus f is expanded in the form 1

i l r ( 'z ) =- 1 t

€

7f u n=3

d n z - ( n + 1 ) ,z e A * ,

where

an= n(n- lX" -r)

f l

JJ^v(()("-'d(dn.

Hence, f = 0 if and only if an - 0 for all integers n 2 3, which is equivalent

to the condition [ [ ,tclrn

dutt = a

J JA

for all holomorphic functions / in a neighborhood of 4 . By the same computation as that in (7.8), we obtain

= @,of)n. [ [ ,rclro dtd,t

J JA

Therefore,by Corollary 1 to Theorem7.2, the lemmais proved.

n

Gz'D

'(t'u)a = ,t '14"o= l[,t]ul'o urelqo a,ta(qI'z) ruo.r; '.re1ncr1redu1

6z'D

'(t'n)a )'t

'[4o\= tt4n)a

'(tt't)'(gt'Z) 1eq1fldurr elnwroJ Surcnporda.reql pue

( O; r)

'raaoarotr41

' 1t' 11 )zy)d 'l dl,l- ffa ]A n 'elnur.roy aqt pue (gt'Z) '(lt'Z) ,(q 'reqlrng Surcnporda.r 1aBan '(,f),rf F/raX '(21'2)pue (91'2)uorg ?sq1apnlruoca,rn

(er.r)

.(,t'u)a ) ,t ,(dHru)zd = rlU l\n ot peel (tt't) p"" (91'2)selnutrog

' ( t ' n ) g u - ( J ' H ) s :H Surddeur reeurl elrlrelrns e urelqo an '.{€.trsrql uI '[r/]g qerluareJrp rurerlleg f,ruorureq asaql II€;o aceds ro?)al eql (J'n)g U fq elouaq'r/ fq pacnpurloquataStptu,rornegzruontrvq ar{t eq o} pl€s u [r/]g stqa

et t)

.(z)lr4dt"_(z)ry = ?)l,tlH

,(q (,f 'A)A > [4n lueurelaue eusepaa\'(J 'H)g > r/ drare .ro;'txaN ' 1 1tp ) z y ) d ' 6

= [[6]illdl aq1 eABqair 'e1mu.ro; Surcnpo.rde.r uorlcefo.rd ueu3rag eql sr z5la.reqm

pue (7I'l) urorg'(.7'H)eV ol(l'n)Jl3o

,(t,n)g > ,t ,(dg"U)zg = l4d (gt.r) sp1ar,( (61'2) elnurroJ'g'I$ Jo {reureg aql ur pal"ts s?.r\sy 'H)z '(z)lrt)"?7-=Q)ltt)d (qt'z) fq uanrS(L 'n)zV u€ q}-r^r(l ' r/ {.reaaaler)ossearrl'pueq raqlo eq} uO ) 3 lua{uale n)S 146 'd .{q pacnpu p4ua.taStptruDr?I?g?tuoulrvUeq1 l6ld $q} IlBc a14 'H)z p ) c ! " ( z ) H y = ( z ) l d l r l &;D fq .ro3'arag '(.7)g 3o srotcal (l 'U)A ) ld',lrteugapa^r'(J 'H)"V ) d lueruela,{.raae ol sprlualegrprurerllag ?ruorur"qasn a1yalor luelrodun ue lue3uel luase.rde.r pa,{e1d (l'n)ev Jo slueualeuorJ peuuepsler}ueraJrprtuer?lagrruoureq pell€l -os 'g'Z ueroeqJ pue (6'9 uraroaq;) ureroaqlilla6-sroJlqv eq1;o s;oordaql uI sIBrluaraJIC rurerllag cruourrlall'z'z' L sacedg raflnurq]ral 1o i(roaq; Furrselruyul 'Z'l

16I

192

7. Weil-Petersson Metric

Using (7.17) and (7.20), we also have

(7.22)

H2=H. With these prepa,rations, we get the following assertion. Theorem 7.7 The space B(H,f) of Beltrami differentials for I dirccl sum of HB(H,f) and N(f), i.e.,

B(H, f) - H B(H,f) o N(i-).

on H is the

(7.23)

The deriuatiue iDo of Bers' projectioniD al lhe basepoint iniluces lhe isomorphism tbo : H B(H, l) - T,(TB(-I-)). /n particular,

r" Qg g D = H B( H,r ) .

(7.24)

Morcouer. jt,p)n=

p e B(H,f),

(Hlpl,p)n,

p e A2(H,f).

(7.25)

Proof. Take an element p € N(i-). Since N(l-) - Kerf/, we have H[p] = 0. tf p e H B(H,l-), then by the definition there exists an element v € B(H,f) with H[v]= p. Hence, (7.22) leads to p= H[v]-

H2["]- H[p)- 0,

and we obtain

HB(H,l-)nN(l-)={0}. Every p e B(H,f

) is decomposedinto

p = Hlpl+U'- HU'l), and (7.22) implies that pr- IIljtl € KerIl = N(l-). Thus we have (7.23). It is obviousfrom (7.f3) that @o:HB(H,I-) * T"(?:BQD is an isomorphism. Accordingly,we have (7.24). Further, Teichmiiller'slemma (Lemma 7.6) yields

(7.25). 7.2.3. Tangent

Space of "(.f)

at a General Point

We shall give a representation of the holomorphic tangent space Q(T(l-)) of T(f) at an arbitrary point p = l-'1. Let T(f') be the Teichmiiller space of l' = w' l(w')- I . Then the tra.nslation mappinglw'l* ($2.3 of Chapter 6) of "(.l-) toT(f') induces a,nisomorphism of Z)(f(f)) b T"(T(f')). We give an explicit desciption of this isomorphism. Defining / -!. \

rc= F(.\)= [ % ! ] o(,,)-', . \ ( t a ' ) "r - ' ^ l

we have

'uorlJess?puoces aql a €q D a,ra'd;o rol e^rleluasa.rda.re Jo e?roqc eql Jo luapuadeput ere li Pu€ r4i acutS

'H ) z '(z)(l4"go 4)"-(z)u14- = (z)(14^t " "( ^9))z-Q)u ue- = Q)lttl(^1o ^11) '(tt't) pu" (gl'f) urog'raqtrng 1aBaar 'ursrqd.rouosl rresl (^J'tt)gtt <- ^6rcyf (J'H)g i n'I o nH

'anrlralrnssl aql 'acua11 1eq1seqdurrtualoeql ursrqdrourouroq nI o zll ]"ql pue "(^O))raX= (n7 o 4i)ray = (nI o ng)rcy 'ngrr>[,= (^go

'crqd.rourosr are

4 pu" a,I Wql pue'o(r4i).reX

'((^.t)a,t)",t

"(^o)T '

^

-

,IIraX

leql aes a.l!r 1eq1 3ug1ou'snq;,

((t)sJ)oJ,

Il ",

t

GJ'H )B .- ^ 1

( t' n ) a

:ure.r3erp elrlelnruruoc aql a sq am 'sarttleltlep Suqe;

'(^J)aJ

-^

Q)s,t,

"'l r(nJ'H)fl

1" I r(l'n)g

{,I

aql :urerEerp all?slnturuoc 3ut,ra,o11o; pue r(J'H)g Io suorlcalord (srag eq uplqo e^t ueqJ'flalrlcadse.r'r(nJ'H)g + (tr)aa i*\nnl Surddeur uorlelsuerl erll 4i ,(q elouaq ni1pue A p.I'(^J)aa ',(1a,rr1cadser'(^J),l,pue (J)JJo peelsul (^J)s1 pue (;)aS reptsuo?e14'{oo.r4 'd o to enoqc ay7to Tutotl ayqto nm ea4oTuasa.tdat ng 'taaoano141 'rt 7o uotTcato.td,s.tag Tuapuailapusl nI o ng lo eatToaueparfi s? nOny/(,t'n)g ataqn '((n1)J)'.t = (t'n)g H ot ((l),f)d,t = lo utsttld.totuost uD s, aI o ng |utddout eW'(J)J > l^^l - d'fitaaa rotr 'g'L uollrsodor6 'uollress€ Sur.troloy aql e^eq e1t{ueqJ,'Qt'D fq ua,r6 $ qstq^r'Il. uo nJ to! qelluereslP rurerlleg cruoureq Io (nJ'n)g U ecedsaq1 ol (,J'H)g Io uorlaalo.rdelq1nH fq alouaq

@z't)

r-(.').(+

= -&?rdiwl=t4^t 6r

#)

i(q ua,rr3usrqdrouost ue aq (r.i,'H)g

'(.r)z >

[tm'1

*

(l'n)A

: nI Ia'I

'[,t] = ([tr-r])'[,rn1

sacedg rallBurqf,ral ;o froaq;

t6r

Frursalruyul'Z',

7. Weil-PeterssonMetric

194

and ""("(f')) Remark.In the rest of this chapter, we identify fplgD and HB(H,f'), respectively.We also identify Q("(f)) B(H,f)lKer6, u n d e r t h e i s o m o r p h i s mH ' o L ' . HB(H,f')

7.2.4. Connection

with

the Kodaira-Spencer

Deforrnation

with with

Theory

The subject of this subsection is not needed for further development in this chapter. However, it is interesting by itself. We shall deal with the tangent space of T(,l-) from the viewpoint of cohomology theory. We recall the fundamental idea of Kodaira and Spencer on the deformation of complex structures. For details, we refer to Kodaira [A-57], and Morrow and Kodaira [A-77]. As was stated in $1.1 of Chapter 1, a Riemann surface r? is obtained by patching domains D1 = z1(U1) in the complex plane. The identification between Di and D1 is given by a biholomorphic mapping zjk of an open set D* j = zp(U1n U*) C Dp onto Dip = z1(UinU*) C Di. A deformation Rr of r? is considered to be the gluing of the same domains D1 viaa different identification /ir(.,1), where /i1(.,1) is a biholomorphic mapping of Dpi onto Di* with parameter I = ( 1 , , . . . , t - ) s u c h t h a t f i p ( 2 p , O ) = z 1 p ( z y ) .I f a l l f i x ( z p , t ) a r e C - f u n c t i o n s ,w e get a differentiablefamily { /?, }, of Riemann surfaces.In particular,il f 1yQp,t) are holomorphic, we have a holomorphic family of Riemann surfaces.From here on, we consider a differentiable or holomorphic family of closed Riemann surfaces. In order to know the actual dependenceof the complex structure of.R1 on the parameter t, we consider its infinitesimal deformation as follows. For simplicity, we assume that m = 1, that {Ui}i is a locally finite open covering of R, and that every Di is an open disk in the complex plane. Take the differentiation of fi*(rp,t) with respect to I at t = 0. This is regarded as a holomorphic vector field on Ui f\ Up, which is written as

,t 1 *=

}f;r, ^, 4 fi;(zr'o) ari'

The relationfir(fxiQi,t),r)

- zi on Ui fl [[ gives

? i n * 0 6 1= 0 Further,the relationfiiltt,t)

zk= zki(zj)'

on

U1flUp.

= f 1t(f *t(tt,t),t) on Ui fiU* i [/z yields

0i**0ul0q

=0

on

U1fiUPnU2'

Thus it follows that d = {01x } definesan element[d] of the first cohomology group //1(R, @) with coefficientsin @,the sheafof germsof holomorphicvector fieldson .R.For the cohomologytheory,we refer to Gunning [A-40]. This [d] representsin somesensethe derivativeof the complexstructure of of R. /?r with respectto I at I - 0, and is called an infinilesimal deformation. of ,R. We call Hr(R,@) the spaceof infinitesimaldeformations

Eur11n4'!17 uo pleg rot?e^ *C s sl qcrqn '(fz)fr,z/((lz)lrzSlto((!z)nz)rd tf - la ''e'ltlqeqd q3= lre/gfo a/vl ''tt)u ln uo pleg rolcel crqd.rouroloqe sr las lzg/g(fz)tlp - r!6 qtea ?sql qcns sr qcrqirl {'t!0I = B alcfaoc e dq paluase.rde.r (O'A)fl {ue a1e;'UJo { lp} Surreroc uedo eql o} el€urproqns > lueurala [g] flrun yo uorlrlred e eq { !6 } 1a1 'sr'ro11og se ua,rr8sl *9 Jo Surdderu asralur eqJ

'{o'a)tn ot ((t-r)0,03'a)ogQl(G-")r,03'a)oH;o ursrqd.rourouoq e se,rr3[p] = ([4)-g dq pauuep Surddeur aqr realcsr '((r-v)o,og'u)oug/((,-v)r,03'u)o1lul r€ql 1I *9 rl p ssel) acuel€Arnba eql uo fluo spuadap qq.f, '(O!)fl yo lrll [A] [6] luaurala .s fl rfB uaqJ, u€ se)npur{'t!0} = 0 ptrr-'rtng !2 uo plag rolce^ crqd.roruoloq

Qz't)

' q n U l.n

uo

ttzo. lzo - - 'tte - - - : - ( ! z ) ! a:" (cz)'ta

e

a

't2U ln uo seqslu€Alzg/gla - rzgf gta les aM fra,re 1eq1 fl U uo pleg rolra^ e sa,rr3{!zg/gla} teUt uorlrpuo? }uar?lgns pue f.ressacauV'U uo pleg rol?a^ *g 1eqo13e eugap sfernle lou seop {ftg/glo} 'rarreruoll '12 'f4 lrl = lggf lag uo pleg rolla^ uo -C € sl lrQlglo ueqa fo uorlnlos e a{€l '{fzpl[Zplrt] uotlenba = il.{ue rog eqt Flluareglp Jo '!ag/lag = frl qur* {lzp/lzpfd },(q paugap sr og pue'g' uo {lzglgfa} - o plng rotrel -C e f,g uarrr3sr ((r_")0,03'A)oH 3 a luaurale u y ' l I u o l € I l u e r a g l pr u r e r l l e g - C e ' ' e ' l ' U r o ( I ' 1 - ) e d , { 1p { ! z p / f a p f r t } r/ uno; IerluereJrp -C € sl ((,_r)r,03'A)oH Jo luetuele uB ?eq} eloN aqt ul ('tualoaql s(1ln€eqloq;o ecuenbasuo?€ sI srql) 'fe,r 3urmo11oy (('-v)n,03'A)oHg . ' ^

( o ' a, ') , u- # ( \

'-' ) ''n3W)oH

*t

usrqdrotuosleql lcnrlsuol eA\ ' ( ( " ) o , r O ' A ) o H= ( A ) z V

pue '(r_")O - O a^?rl allr ueql

'!, Jo suollces-ssolr er€ q)rq1rrsenl"A qllllr surroJ-I ctqdrouroloq;o sur.ra8Jo Faqs aW - (y)o,rC) pue 'r_v suorlres-ssorr er€ qcrq^\ senl€^ qlr^r Jo (1 'g) ed,t1 surroJ sur.raS;o eqt = (r_r)r,03 Jo lerluaregrp Jo Jeeqs *C 'r_y sura3 go aql = (r_r)0,03 Jo suorlf,es-ssorc Jo Jsaqs *C - (r_!r)O 'r_v

Jo suorlres-ssorccrqd.rouroloq;o sur.raEJo Jeaqs aql

:uorl€1ou aq1 ldope e1t '{n g lp uo 1l,r/t fq ue.tr3 sr rfv uor?cunJuorlrsu"rl sl-r ''e'l 'g, uo alpunq eurl Iecruou€r eql aq x p1 'too.t4 'yutod esogeqt p (A)J acoilsu11nu,ycpalaqTlo ((U),D"1eocds Tua|uot ayl qpn p?*l -uep, st A uo suorlout.totap (O'A)rn acodsaqa.6.2 urarooq.1, Tou,nsal.?u{u?lo sacedg rafl]urqf,ral;o

96I

froaq;

Furrsalusul

'e'l

196

7. Weil-Petersson Metric

pi = AailAzi, we get an elementp = {pidzildri} g Ho(R,t0'r(rc-1)).then the homomorphismof ^I{1(.R, O)to Ho(R,to,r(n-t))/6Ho(R,8o,o(,6-t))sending gives to the inverse mapping of d*. [d] [p] Next, we have a canonicalisomorphism , , 1 o( 6 ' ) - 1 : H L ( R , o ) * where

A2(R)*,

A : H o ( R , t o , L( n - | \ I 6 H o ( R , t 0 , 0 ( , i - 1) ) -

A2(R).

is defined by tl

tlp)@)= I I pQ)pQ)drdy, p e Ho(R,to,t(^-t)),e € A2(R). JJR

(This is a consequence of Serre'sduality theorem.Seethe proof of Theorem7.5.) Since7}("(r?)) is isomorphicto ,42(,R)*(Theorem7.5), it follows that the infinitesimal deformationspaceH'(R,@) is identifiedwith the tangent space f"Q@D of the Teichmiillerspace?(.R) of .Rat the basepoint. This completes D the proof of Theorem 7.9. of Now,we wish to seethat H'(R,@) is canonicallyidentifiedwith a subspace the first Eichlercohomologygroup I11(l-,I/2), which is definedlater, wheref is a Fuchsianmodel of -R,and I/2 is the spaceof polynomialsin one complexva,riable of degreeat most two. Note that IIz is regardedas the spaceof holomorphic vectorfieldson the RiemannsphereC. Further,If2 is canonicallyidentifiedwith the Lie algebrasI(2,C) of SL(2,C), which is the tangentspaceof S.L(2,C) at the unit element. Let (H,r, ,R) be the universalcoveringof .R with coveringtransformation group l'. We use the notation: B(H,r) = the lift of .Ho(.R,to,t(^-t))underer, V(H,r) = the lift of Ho(R,50'o1r-t))underr. Then 6(I/, .l-) consistsof smooth Beltrami differentialsit for I on I/. An element function on 11 such that 6o7f7'= 0 for of 6 eV(H,l-) is a C- complex-valued anyT€f. For every6 eV(H,f), we set 60 = 7itf 02, whichbelongsto B(I/, j-). Obvic to B(H, f ) I AV@, D. ously,f/0 (,R,to't (*- t)) / AHo(R,go'o1r- t )) is isomorphi For any it e B(H,l-), we put ^,\

(FQ), z€H

/r(zr=10, z€C_H. Let F be a continuous function on C which satisfies the differential equation

f f = , o nc

(7.28)

in the senseof distribution such that .F(z) = O(lrl2) as z e oo. For example, we seefrom Lemma 4.20and Theorem4.37 that the function

elaqAr

,lr_,il=' = (cr, J )|H !_1,s= (zII'J){

'mo1q 1es aair 'QZ'D,(q paugep o1 spuodsallel [/)/X leql slsaE3nsslqJ B

'r)L 'i=Wlf-h=@)t!ltY ultslqo a/rr snqJ

',IIil{r(=Lo[rfll aleq a^r'J > L f.rara rog rtotl-- l,o!l"acurs'1/ rog 1er1ue1od e sr [r/l/ 'sI '(AZ'D,tq uaar3 sl = lpql l l" I ol lcadse.rqlp y lo ltlll uorprluaragrp a{1 0 '7 '0 -raldeqg q /t'7 ueroaqJ ruo.1{ I Ie I o1 lcadser qll,t{ ,,! Jo e^rle^lrep aql L dq alouap 'J ) L frara rog '{(t'.yf1} o1 spuodsa.rrocr.7 qql 'g .re1deq3 Jo I'I$ ur pal?nr?suoc sr rlf,rqtt'r/1 luarcgaor r.ueJllag q?l^r C 1o dtm Surdderu - tL ,; pue (ff prrrroJuo)rsenbaq1 sr t/ 'a.ra11'{.1 > L , | *I = )T ,JIoLor! - rg17 qfli* {r,t/rU } ,(t1urnlelq"rluereJrp e eleq a^'(J'H)g > r/ f.rarrarog

'a = (a).L (t)(a)g lsrtl q)ns zil orul J yo Surddeureq?.{q uanrSsr q)lqr'rd p fi.lopunoqoc eq1q (a)g araqar'(4.)9-p.rX - cX sn{I'til ol sEuolaq reqloueq rJI I C d ueql'r/ ro;1er1ua1od

' 2 2) d ' J ) L

'#;=61-r e\^ zil uo slce .7 'a.ra11

.J

) zL,rL

,(zt).tx

+ ((rr)rx).(zt)

= (zLotL),tX

uorppu@ a1cfrcocaql sagfll"s rlorq^\ ,zII * ,1 :.{X Surddeur e urelqo ain 'acua11'211 3 (L).rX l"ql s^tolloJ l! '/ roy 1er1ue1ode oqe sr ,1"/l"og ecurg

'J)L ',t-

I

ii=-W)rx

,vo,{

'lroN las art 'J ) L {ue.rog saqsru€A -,Lf Log leqt sr (l'n)A, f d }eql uor?rpuof,luerclgns f, pue f.ressacauV'(J'H)ft ul peur€luor sfe,ra.p1ou sr / roJ dr lerluatod y

* zse("lzns:ili;"r6;lf,",Xj:Jt#Tj,t'"H'i,f.;fl s^rorror reqr r!,oo

-oloq e sl l€rl} si$oqsBtur.uals,1fa11'rl to1C prlualod reqtou€ rod .{ O 'r! ro11o4ua7od ue g u€ r{ensner eM '[gg-V] €rX uI 41 .ra1deq3Jo ''I "tuurerl osle aas '(86'l) Jo uorlnlos perrsape sarrr3 ttp?p

\62'L)

"[l

("-)Xr-))) 0t

GF-=?)t

saredg rafl]uqf,raJ;o

LOL

froaq;

Furrsalrusul

'e'l

7. Weil-Petersson Metric

198

BL(r, II2) -- 6(IIz), 2 1 ( f , I I 2 )= { x I y : | + I I z , x ( t n z ) = ( t z ) . ( x ( t r ) ) + x ( z z ) , 7 t , 7 2 € l } . This Ilr(f,

groupof l. nz) b calledthe first Eichler cohomology

Theorem 7.LO. Let B* bethe mapping B*: B(H,D/6v@,f) - Ht(f,IIz) defined,bg

0.(li'D= [xr],

where F is a potential for p. Then B* is an injectiae homomorphtsm. Proof. First, we show that B* is well-defined. If p,i e B(H,f) are equivalent, then there is an element 6 e V(H,f) with t, = ir + AilAz. Let F and G be potentials for p and /, respectively. We put

( F(r)*il(z), ze H,

G " ( z ) -o i , i , t

z ec - H .

Since 0o7 - 61t for every 7 € .i-, it is seen that for every ( € R, d(z) - Q as z + ( through I/, and that d(z) = O(lzl2) as z + oo through 11. Thus Go is a continuous function on C, and satisfies AG"lAz - /. Hence, Go is also a potential for l', which implies that X6 = XGo+ 6(P) for some P e IIz. Noting a n d h e n c e[ x 6 ] = [ X r ' ] i n H | ( f , I I 2 ) . that X6. - XF, we have16 =Xrl6(P), This implies that B* is well-defined. It is clear that B* is a homomorphism. Next, we verify that B* is injective. Assume that B.([/]) = [Xr] = 0. Then, there exists an element P e IIz such that XF = 6(P). Putting 6 = F -P' we see and 06 = p. that 6 is a potential for 1.r,and X6 = 9. Thus d belongstoV(H,f), tr Therefore,B* is injective. This shows that p] = 0 in B(H,f)I)V(H,f). We shall construct a canonical homomorphism

P t H t ( r , I I z )- B @ , r ) l A v @ , r ) . (It is consideredthat B* and B correspond to 6* and (6-)-r in the proof of Theorem 7.9, respectively.) Choose a smooth function p on H which satisfies the following three conditions:

( i )0 5 p S r . (ii) For each z € 11, there is a neighborhood U of z and a finite subset J of f such that p = 0 on 7(U) for every 7 € f - J. (iii) D?€r p"t/) - I on H. Such a p is called a parlition of unily for l- on f/. For a proof of its existence, see Lemma 3.1 of Chapter V in Kra [A-58]. For any [X] e //t(l-,I12), we set

'Q 'u)a

) zd 'rrl

'*

= (lerllH'ltrt)n)rt lltrtos'[zrl]d,)

p1ar,{ (19'2)pue (21'2)selnurrod'I{ ul J roJureuop l"lueurepunJlceduroc,{1arrr1e1e.l € sr dr areq^l

(re'z)

' f i p x p( 4 e r t( z ) r d r(z)uytfl = (ed,vt)tt

,{q paugap sl (J'77)g ) ztt'rrl s?ueueleo.tlJo lcnpord reuur eqf .(l,U)A uo lcnpord Jauur u€rlnu.ra11e SurarS r(q lrels er'\ acueq pue,(L.L uaroaql) q r p ( ( J ) J ) o , 6 p e g r l u a p re 1 4 . 1 u r o d a s e q a q t w ( 7 ) J l " (t'n)AU (Q)t)"t aceds lua3uel crqd.rotuoloqeql uo lrnpo.rd rauur uer?nura11e a,u3 a.tr,11eJo lsrrd crrlatr l uossralad-Ila

\ aql Jo uorlrugao

.I.g.z 'rrrlal'u

ralqgy e sr 1r leql aoqs pu€ (,1)Z uo crr?etu uossraledlra1\ aql augep II€qs e A

rrrlatrN uossralad-lla^. .t.z '(l'n)zV oluo 5/ray yo usrqd.rourosr D ue sr 12leqt pagrra^ = ([X])o tes eM.[X] ssep acualenrnbaeq1 uo,tluo 41I',,,t (//// spuadeppue'(J'H)zV luauela ue sr,sarurl aa.rq1/Jo uorlsrluareJrp aql Jo 'J roJ = W)X t€ql qcns l? uo uortrunJ 1eq1 parord q tI ) I II€ / ,L/Lol crqd.rouoloq€ sl g - otr - ./'"nql'(l'U)A ) g auros rct Zglgg = Zgl"lg segsrles (Og'Z) fq pelrnrlsuoc og uorlcun; aq1 (6/.ray I [X] .tue rcg.too.t4

culd.r,outolorl Io (.t'tt)ev

'J ro{ g uo su.tto{ctyo.tponb acodsaq7qTtmpa{cTuaptst j"tay .II.Z uoltlsodo.r4

'flaarlcedse.r'sasso1cfi6o7oruot1oc ralqcxfl pu€ sassDl, fr,6o1oruot1oc sragr pe1ec a.re6/lay pue *5/ru1Jo stuetuelg 'd n>L O *drul = (til'J)tH pue 'alrlcel.rns sr 6/ '(91'2 ua.roeql ]r) e,rrlcafursr *6/ pue uorlcnrlsuor eql uord leql s1l\olloJII'p! = Wql rselc sr lJ,*5/ *dod 5/;o [/] or [X] spues qrlr{,ra(J'n)A.g/Q'H)g - (21'J)fl:6/ tusrqd.rouotuoqe ure}qo arrr,ecua11 '(eII'J)rH sselc acuale,rrnbaeqt uo,{1uo spuadap (.t,n)Ag/e,tilg ) "l [X] o1 s3uolaq zg/"}g - r/ snql [r{ sselc acuale,rrnbaeq} t€rl} uees sr tI.(J,H)€l 'J

) L ' o , 4 - , L 1 L o o g= ( L ) X ?eql q)ns _iTuo uotl)unJ _C e sr od. uaql

(oe'z)

J)L

'H > z'(z)(L)x((z)L)d 3-

= G)otr f,ulaw uossreladlral|t'l

66I

7. Weil-Petersson Metric

2OO Lemma 7.L2. For anA pr,ltz e B(H,f),

thefollowinghold:

h(Hlttrl,nlyz)) = h(Hltttl,pz) = h(h,Hlpzl),

(7.32)

h(Hlpi,u[pzD = (Hlpi,plpzDa = (pr,plyz])n.

(7.33)

Prool. First, (7.18) and (7.31)lead to

h(Hlp rl,t'r) - (^hE, *, n 1p1) = (\'uE,9r(r ?,f,t )" . " Since the Bergman projection B2 is self-adjoint (Theorem7.4), we have

p), \'nE) n = h(yr, n lyzD. h(H lpt), p2)= (9 z(o?n using(7.22),we get In particula,r, h(HIptl, nlpz))= h(h, H'lpzD= h(pr,H[pzJ), which shows (7.32). Moreovet, from (7.17), (7.25), and (7.31), we obtain (7.33).

tr is induced by h under the Now, a Hermitia^ninner product on [(?(l-)) = H B(H, f). identificationf"gQD Next, we define a Ilermitian inner product on the tangent spaceQ,("(f)) of "(f) at an a,rbitra,rypoint p = lw'l as follows: using the identification = HB(H,f') (seeRema.rkin $2.3),an inner product on Q("(l-)) fpglD is inducedby the inner product h' on B(H, f') which is given as (7.31).Actually, the inner product of elements11' o L' lpr) and H' o L' [pz) in H B(H, f '), which are also consideredas elementsin Z)(f(f)), is given by h'(H' o L' lptf, H' o L' lpzl),

Ft,ttz € B(H, f).

In this way,we havedefinedthe Hermitian inner product on eachtangentspace rpQQD of "(f). of the inner product with respectto p. For Now, we study the dependence purpose, to it is sufficient considerdependencein a neighborhoodof the this point in "(f). base Take a basis{ pi}1;5" for A2(H,f), and set ui = trlgil, j =1,...,39-3. Then { ,illn=1" is a basisfor HB(H,l-). Put 3g-3

u ( t )= D r t r r ,

t =(tt,...,ft-s) e D,

j=l

where D is an open neighborhood of the origin in C3g-e.

fq paugapil d AV crrlaruuossJelad "^o uttol uoss-ral?d1l?/14 aql ro'ru"tot-4 loTuauopunt eqa lla l aql Io

'Q).r'

uo ?uleu uoNuuouety uossr?Ied1r?/fi aql aa6 11ece71t'eJoJequarr,r3uorleag -lluapl eq? repun srolcel lueEuel crqdrouroloq se pepre3ar ere aprs pueq 1q3r.r aql uo esoql pu€'slolcan lua3uel Isar ale aprs pu€q Ual eql uo { pue ;g'ara11 . ( A ' X)a ilrlsu Z = (A, X)"^6 urroJ aql ul uellrur q "^6 ueql 'tI-gI'dd '[Og-V] $rreH pus sqls[rp aas'spelap rog 'f, f.rera ro;'flaarlcadsat'd(!zgf g)p'o(!rg/d o7 d(ng/g)'o(!re/g) Surpuastusrqdrotuosr = f) lnp+lr = lz ! ql'dpunore (t-tgt"''1 l B a re q l , t q u a , r r 3 f l u o l l e f , g l l u a pe saleulprool 1eeo13ur4e1 :snolloJ * ((.1)'CXZ eceds lue3uel crqd.roruoloq aq1 rltlru Paglluapr sr d lurod e l€ (J)J go aceds 1ue3ue1Iser eqJ 'dlt{V culetu uossreledlle^\ aql ,tq Pecnpul (.t); "" crrlaru ueruuerueru eql aq "n0 le.I crrtotr tr uossralad-Ila/$,

aq?;o {11.ra1qg1 'Z't'L

'(t)poW dno"rf .tolnpourr?llnuqcreJ aqt lo uotTco aqq )uleur uossreled-INel1 eqJ 'tT'Z rrraroaql

repun luolroaut st (.1)a uo da!

'uoll -rasse SurmolloJ eql a\oqs usc aaa'crrlaru uossreled-lraAl eql Jo uorlrugep eq1 ,tg

'r) (qe

'llPttP(l)lfq

I={'f

3

?,= zd/YtsP

e-6e

se uelllud sr 1t',tgeco1 'auq fq palouap sl prrs'(J)J ro )uleur uossr???d-Ir?/! "qI peil"c $ (.f )Zfo elpunq lua3uel er{t uo lcnpord reuur eql 'Z'g$ ul I {r€ueg ur uanE $ uorlresse slq? Jo;oord y 'q uo *Cssolc lo n lfqqcoa'sacuoTstanc.tr?eaoqoe1tr?pun'gl'l 'rueroer{l 3urmo11o;eqt e^eq e,r,l,uaql

(vt'D'(r)u((3)rnld'(it,r1=

'6,1^J

rraroaql

/H = (l)ef araq^{

( [ ( l ) t a ] 1 ,1y1, ' [ O ! ' t ) g 1 ^ H ) 1=7 1( 7^)t !t ! q

,(q uanr3 sl [(r),orj - (7)d qlraa ((.7)g)(t)ag (l)!lq lcnpord reuur aql snql 'O 3 1 f.rarr,a > 't?g/e'!W/g srolcel lua3uel lo roJ (1r;,J'H)gH Jo s-rs€qt q teql s^1,oqs 8'Z uorlrsodor4 el;j{[(t)!A
) t'ltn)6y^I

= (7)t,t

?as e,1\'(t)al, u1 lurod e$eq aql;o pooqroqq3rau uado ue oluo 6,go Eurddeu rrqdrouoloqlq e q ((t)n)O = (t)"O ,(q peugap (.t)sl, *- O i o4i Eurddeur aql leql 8'9 uraroeql ,(q aurnsse feru em '6r leurs ,(lluarcgns e Sursooq3 rulaN

IO7,

uossraled-Iral\'8'l

7. Weil-Petersson Metric

202 = g*r(iX,Y),

,*r(X,Y)

X,Y efpQQD.

Here, iX means the real tangent vector corresponding to the holomorphic tangent vector iX under the preceding identification. Namely, if ./ denotes the almost complex structure on ?(l-) which corresponds to the complex multiplication by i, then fX means "IX. This c.r*, is also written as = -2Im hsrp(X,Y).

u*r(X,Y) is represented by

Locally, u*,

3g-3

e*" = i t

(7.36)

hj;(r) dti natx.

j,k-L

Note that u*, is a positive (1,l)-form on ?(f). We say that hsp is a Kiihler rnelric if the Weil-Petersson form tr-"

""

i'e''du*'= crosed'

Ut'];"t,;,r=,,.'",::T r"u'! \tr-

Ail'

is d-

(7.s7)

on D, where hi1 are as in (7.32). This condition means that hysp osculates to that is, order two to the Euclidean metric on C3g-3 at every point p efQ), we cim find local coordinatesZr ,. . . ,z3g-3 around p for which 3g-3

dr*r'=

D @1,+ aix(z))dzidT ,

i,k-1

where all partial derivatives of air of order less than two vanish at p. Now, we have the following theorem due to Ahlfors. Theorem

7.15. (Ahlfors)

?[e Weil-Petersson metric is Kiihlerian.

Proof.We follow Ahlfors [7].Bv a translation of the base point, it is sufficient to prove formula (7.37) at the base point. - Hf fv(t), and P(t) = We put fl - w'Q), f'Q) - *v(t)71.v(t))-t, R(t) f'(F).We set

I{(z,O= --l-:. Q c)2' We also set

Since/'(z) =TO

r , , 2 \ = U ' ) " ( ' ) ' ( / ' ) e( o ' t\t\z'\) _ J1q1y 7V1oy for all z € C, it is easyto seethat

EI;n =K{z,a), I { 1 Q , O- K r ( 2 , C-) K r ( ( , z )

(oe'r)'apnp (n)ltrt)r,r^, ffi"il

= +- @#*

'snq5'uars eqrrepun) o1lcadsa.r qrr^{pat"r}uareJlp "- ".T[Tf"T:il; 1er?a1ur leql ees er,l.'t?,'Vuorysodor4 ;o goo.rdeql ur leql ol r€grurs 1ueun3.re ue . g

' apnp (n)ltrtrrrr,, W'

il +-

=

,rM-()),n?"r# 'u3ts qrr^{pa}erruereJ-rp eq uec ler8elurdlt"Htt lerEaluraqt rapung o1 lcadse.r

- ^)l"tt ^)!Q)'I 'apnp(n)lrnJ,,r^t l# -i r +((4:!-- r l t Jr L r

a p n p ( n ) r r t , ( ( r n ) - (, , f/L)t ) ( ' ) ' f - @ ) J * I I

( 9 ) ; [ - - ( ^ ) , t ) ( Q-)@ ,! )J)]"tt u _= I lll r

- trJ)aq+ ((,),!

urelqo a,rl ,/t-t tueroeql o1 dre11oro3eql pu€ (z)rl = (z)r/ Bursl

'nln = .H ) z,) ,((r),!- ()),/)sor @,)),>t zU

.r,)o:r;Tii,Ti::;t5":LJlj::,'[;;?::il; srqr a{r1€a{r"ureu ropua (ee'z)

np,p (z)qn"(2,)),>r"l[ {up4p())ta *] t

[[

=frlrt,

uplqo ea,r,'{{q - lfi1 uro.r; snq;,

npap(z)ta e)qn"(),,t,x l[ *]"il {**

=e)!tt1

'(gZ't) pue 1aBe,u flrrl1lnI = (7)l,r ')'z r,o1 '1er3e1ur 3ut11eoa.r'ueqa'flaarlcadser $ql uI Q)rt'Q)rt elnlr?sqns ( 'nPxp(z)()lrt l,tplp (

Hff

v, ) (r).Irr

Q ) ( t ) q n , ( ) ' z )l:l r : l JJ 6L)

-

ll =(r)!!a

JJ

uroJ eql ur uellrr^\ s! ('g'Z) ur 1{r1}eq} aese/( '(gt'z) pue (0I't) urorg'rrro51 'O ) t Pu€ *FI n H > )'z 1P'ro3 t0z

f,rrlel I uossraladTalUt'l

7. Weil-PeterssonMetric

204

Here, this integral is defined as the Cauchy principal value. From (7.38), (7.39), and the definition of K1(2,O, we have 0h't

#(') daav v1 =+ il,{ I Lryr I{,(c, G)du,t} 4;ke)

( tl . = - - T i ? - - - - i t - T , u {t) r rzrr rr- f d4dn dxda, 7'( r )lvl(C) = - 24 lf o z)r1e, 0T6 l,J@ fr, # JJ,u.,\ JJ. I

(7.40)

where

Tt(r,C)=

I I,

dudu. I{ (w,z)K (o,C)L'@[ut](w)

This integral is also defined as the Cauchy principal value. In this step, we differentiated(7.3S)under the integral sign. To justify this procedure,we need to show that the integral in (7.a0) convergesabsolutelyand uniformly with putting H(r) = {z e H I lrl } r} for any positive respectto t. More precisely, number r, we shall provethe integral I(r) rr

= [[ -

llro

( rr

I II

[J/r1";l

-

|

' l z\r2(z,oT4Vtr ' ' ) dxdv & J \ / L'@l,;G)ldg,tl L l^(,,

convergesuniformly to 0 as r --+ oo for all 1. By Lemma 4.2I, the above ?2 is estimated as

llrv,t,,c)l'

alart aea,t = u IIrII{(w,z)r"@[v2](w)P dedn, s c2tr2[ [ Vr@,2)12 JJa

wherec is a constantsuch that llL,@lu2]ll- s c for any t andl. Thus we have

tt

g(,2)r2(z,ioldtd,t

J J H(r)

/ rr

1c* | I I

\"//41'1

\1/z / tt

^

\r/z

dtd,n) ( t t [i@,2)l2d'udu| lK(c,z)l' / \l l n' 1

If z stays in a compact set of f1, the first factor on the right hand side tends uniformly to 0 as r + oo, and the second factor remains bounded, which leads to the desired conclusion. Finally, by using again the fact that the linear operator ?' in Lemma 4.2I is isometric on trz(C) , we can show that

xg at$JI '(" '?'or) go Surddeu snonur?uor" sr (rn) t'rq leql easan ' , sgos'r! - t't1 Eur11n4'(s'lz);o Eurddeursnonurluoc e sraolloll!,gg.7 uorlrs lsrlt 4 Q)"t! 'C yo Surddeur tb-s.'ril -odo.r4 'tg't'tg = "'t! p.I l€f,ruouer eql eq ,/ leql e?oN

'*H)z '(z)(s), I E)z H>z

'O

l=@)n,rt

, ( r ) ( t ) nI )

'e '3 'lxatr1 les a^r ) s 11elo3 '(l'r) Surddeu Jo Surddeurcrlfleuelear e osle sr (^)rll esJe^urslr l3rll saoqs rueJoeqlEurddeuresraAureql 'o x H ) (l'z) lo Eurdderu ctlfpue-pe.r e u (z)rg' acurg'(1)a luercgeocrur€rllag " s"q pu" ,g raldeq3 ;o (Z'g) fq peugap s.rq?HAr'H Jo ,I Surddeu leuroJuocrsenbe sacnpuro1'uaqa l=!

'*H) z '(4ta ,r Zi- '

= @)dt

e-fe

,tq (J '.H)zV ) 6 lueuele ue eugepeM ,O 3 I fue .rog 'CI x H > G'r) ol lcadse.r{lle ooCsssloJosl gI', tueroaqlgogoordaql ur (z)rrf Surddeurl€ruroJuocrs"nb aql leql uollrasseeql;o;oord e ear3aM 'A qrvuey '? Jo uorlrunJ crl,tpue-par e sr qma aesetr (3)Ifr1 snql'(J)J uo cr1,(1eue-l"er ar" lt pue leql are sel€urproo?e{?rq acurs b tt"'(t),t acedsralpurq?lal eql uo cr1,(1eue-lear 'tg acedse{rlq eql uo suorlcunJcglr(1eue-1ear are .t, pue ll ilt'6'g etuural ,(g t=!

'!7pyltp3="^, e_6e urroJ aqt ur uallrJiir sr rrrlaru uosseled-lralt aq1 3o '-o ruJoJ FluauepunJ eql '(g'3 ura.ro -eq;) epu.ro; s,1rad1o11uor; '1ae; u1 'cr1,(pue-lear sr 1lr1 qcea (eJourreqlrng

'Qn'a'[1] srollqy

aas) I ;o uollcunJ-oop e sr (1)!fr1 qcee Ttsql ^,roqsu"c a,u ';oord qql u! teql se '1g sselc 'sr 'snonurluoc ale e.rc !f,/ luaurnSre etues_eql Eurpadeg 1eq1 Jo il* 'g1'2 ure.roaq;;o;oord aq} tuoq 'I ile t"q} ees ar\{ N/llqg'rlrQ/r{qe 1rout'ey 'gI'Z uraroeq;, ;o;oo.rd aq1 salalduroc srql

.a)#=

npxp (z)l'h)el^7 ornO 1n)lt,t) (n' z) ttr(4' n) y { 61,r - t

frpxp-(z)l,r,1l n}61ar(7,z)ta@, )) )r e),il { uOrO17)( t

"lnu'"il

#-= .lilu'"[[#-= a)# rulal I uossralad-Ira |t'l

902

7. Weil-PeterssonMetric

206

t and s, then 1r1,"is a conformal mapping of f'1(f1), becausefi,, and .F1have the same Beltrami coefficient v(t) on -Fl. Thus, applying Cauchy's integral formula to h1,", we see that all derivatives of h1,r(u.,)with respect to tu are continuous functions of (w,t,s). Hence, all pa"rtial derivativesof f1,"(z) with respect to z and Z are continuousfunctions of (z,t,s) e H x D2. On the other hand, the Corollary to Theorem 4.37 implies that for a fixed z, ft,,(z) is a holomorphic function of (t,s). Hence, applying Cauchy's integral formula to f1,"(z), we conclude that all partial derivatives of fl,"(z) with respect to t, s, z, and Z are continuousfunctions of (2,1, s) e I/ x D2. Thereforc, f1,r(z) i s a C - - m a p p i n g o f ( z , t , s ) € H x D 2 . I n p a r t i c u l a r ,f ' ( t ) = f t i z ) i s a C - mappingof(z,t)€HxD.

7.3.3. Alternative

Proof of Kiihlerity

of the Weil-Petersson

Metric

We shall give another proof of Theorem 7.15, which is also due to Ahlfors [6]. Here, we use the fact that the first variation of the area element induced by the hyperbolic metric vanishes (Lemma 7.16 below), which is interesting by itself. By a translation of the base point, it is suficient to prove the relations in (7.37) at the base point. We use the notation in $3.2. We set

, l ' i ( t ) ( r ) =e l u j ( t ) l ( f ' ( r ) ) { ( f ' ) " ( ' ) } ' , t e D , z € H . Then from (7.16), we have

,t,i1)e)= + [ [ Kk, 02-vl;)d€d,rt. r JJs By an argument similar to that in the proof of Theorem 7.15, we seethat ,/ti(t)(r) is a C--function of (2,1). From formula (7.38), we get

hln1)=

tl

dxdY. J J"riQ)'t'rQ)Q)

(7.4r)

Further, setting

pt(z)= (l(f)"(,)l' - lU'),Q)l') 'x'(f'(,))',

(7.42)

we see that formula (7.3a) is rewritten in the form

- l'(t-\')l')' = t t vrcra$xg)e)o a'av hin(t) r rF Pt(z)

g.4r)

We can differentiate (7.43) with respect to 12under the integral sign, because F' is a relatively compact set in -Il, and the integrand is a C--function of (z,t). Hence, we obtain

'raqunu

se uallrrtr sr ur3rro aql l" z uollcerrp eq1 ur "d Jo e^rlellJep aq? snqtr

'("!)l("1't'l - rrt(2:[-- (')'I) v- ,l(,)"(J)l) = (z)' d ("|(,)'Gt)l "((,),1)'v Ieer ll€urs flluatcgns

eA€rIetr 'uorlrugep eqt ,tB {ue sr s 'ereg 'r"otr = "./ pue ln = rt 1ase11 ]foo.la,

. t - f t . . . =. .! I, 0 = ' = ' \ f f i se{sz7os(a/'D

ur pau{ap td uor?cunl eqJ 'g1''L BrnuraT

'u"Irelqey $ d/t? tr feq| sarlo.rdqc-rqrh'0 - t le ploq (Zg'f) ut suorl€ler eq1 ',trerlrqre ere / pue 'q 'f, ecurg

'o= 6l !!,9 ,1tg 1

'(w'D pu" '(gt'Z) '(W't) uror; 'aro;araq; 1aEer'r

Uv't)

.o- npxp ,_(z)HuG)16(,)' lUgrr*"il el€q e,ll 'r1 o1 lcadsar qll/r{ |fi1 3o e,rrlerrrrep eq} 3ur4e1 '{l.rel.-,ols

(ev'L)

.o-npry "-?)ny1,1t^(4' ldfu"lI urctqoa,ra'(gt'l) pue (W'2) ,tg

(sv't)

=' . = n p x p _(z), u @' " w @g"| | @ffi Iv_Lr) e^eq osl€ a,r 'r? o? lcedsa.rqll^{ (I7'z) Surlerlue.raglp'}xaN

pueqlq'rr aql uourralpuo'asaqrreql aasa^r''.o1aqt;;Tffi:{?"i]J?i:l;

0,,, . { l(#) ff}et,^r,t'^'il

'l&\"ll (wt) npxp"-(z)u,{r"l'='l ffie)u+lzyralzf

=

7!e nt ,",!!qg )rrlel I uossralad-Ira^Ut'/

L0z

?. Weil-PeterssonMetric

208

iQ)=kl,="r,, = fi

ti @-ian- 6!1 i, tz)+-ik))

=-8Re{t+ a+\

(7.48)

=-sRELa,{&\, where /(z) = (0f"(z)/0s)1,=o.Thus we need to obtain more informationon

f(,).

First, we seethe followings:

(i) i: = v = \rf 9 on H, whete9.= 9i. (ii) / is continutuson C, /(0) = /(1) = 0, and lim,-(iii) / takes real valueson the real axis R.

i1r1112= O.

For the proof of theseassertions,we refer to Lemma4.20and Theorem4.37. SeealsoKra [A-58],Lemma1.4in ChapterIV, p'136. Now, we determine/ as follows.Sincerp is an elementin A2(H,f), it follows that Xig " bounded on I1. Thus, integrating g three times, we get a holomorphic function rlt on H satisfying the following four conditions: (u) ,lt"' - 9 on H. (b) / ir extended continuously to the real axis R, and satisfies that r/(0) =

/(1)= o'

(c) (Imz)r!'(z) -0 and (Im r)'t"Q) * 0 as lmz -' 0. (d) {(z)/22 * 0, (Im z){t'(z)lz - 0, and (lmz)2t/,"(z) *

0 as z + @.

Solving the diflerential equation in (i), i.e., (t-;\2-

he)=_ftve1, .

we obtain

i e ) = - 9 +4 T 1' .6. - t - 2 72 6Y; -t 'at - @2 *' F^ \( z ) , where F is a holomorphic function on I{. Let us determine F.. From_(ii), (b), and (c), we see that ]7 is extended continuously to R, and f = -rb/2 * F on R. F\rrther, by (iii) it follows that -rtr/2 + F = -{/2+ F on R. Thus h = F + rh/2 is a holomorphic function on Il, which has a continuous extension such that h is real-valued on R. Hence, Schwarz' reflection principle shows that h is extended holomorphically to C. Then from (ii) and (d), this h should be a polynomial of degree at most 1. - 0. This implies that Consequently, by (ii) and (b), we conclude that h

Gv't)

'tf

l=q,!

lta 3

= 6)ou

n

,

f

q

peugep sr / uorlcerrp eqt q d ry (1)oA arnloarw ,??NAeql.urrou lrun qlrrrr d 1e rolf,el 1ue3ue1crqdrouroloqe s! = o(tl,,,,tl qrns d p (W)dl /,.e.1,I ?eq1 ry W aceds lua3uel crqd.rouoloq e{t Jo luaruele ue eq d(r4g/d pr=j3 = A p.l

.lutrAatq 3_=

wttty

N

{q uerrr3arc w71!A sJosuelNnlDaJn) uDtuuoutery aq1 ,.raq1lng l=7

"'ra3-

lta

N

fq paugap ew !!A sJosuelernlDarw ,?cNA aq;. .*18 _-t4rzrtr

'l'te

ld

,(q uaar3

erc atq,g srosuel arnlDarnc aql ueqtr .1lfq) o1 xul€ru esrelur aq1 sr ("rg) areq.u t-u

, l?Q ,ffi*rrt J =r.lJ ^I

fq uaar3 ere zsp f,rrlau Jelqgx aqt qtl/tt Pel€rcosseuor?f,euuoc)rrlatu aql Jo 7lJ qoqnrfs Ia,uolsrrqc eqJ 'y'/ uorsuaurp Jo /t/ ploJlrr"u xalduoc ts uo crrlau relr{gy p ae qlpt:ipIlqr=ci!Z,= rsp 7e1 '(y1 '[qq-V] reldeq3 nzruroN pue rqsefeqoy aas) splo; -rueu relqex uo f.r1auoa3 ,1s.rrg xalduroc suorlou euros 11eca.r l€rluarasrp Jo 'elrle8au are crrleru uossraledlral\ aql Jo sarnle^Jnc leuorlces ctqdroruoloq eql pue 'alnle,rlnc relsos eql 'se.rn1el.rncrc?lg eql t?r{} ^4,or{seA\ crrlatrAtruossralad-Ila/$,

tr

'

aql Jo sarnlB^rnc

.t.8.^l

,aro;alaqa 'f.reur3etur flarnd sr 0 = 4 1eql apnlruoc am (9p.1) ruor;

rn"_

rk-r)\ . q ( z - . 2 )o _ . . " ( z - r ) * =- [ -Gt"(z)4, +(z),/,o Aro
'acua11 l"q? s^rolloJlf

' z = (z)! t - , t . 1 - z _ z- \P),,fr-:--=/',t + \-/.. ( r ) f r ?),frJ"(z_r) (r)fi

602

f,rrlal^l uossralad-Ira^{'9./

7. Weil-PeterssonMetric

270

It is said that the Ricci curvature of ds2 is negatiae at p if Rp(V) ( 0 for any direction 7. The holomorphic sectionalcuntature KoV) at p with respect to V is defined by ,u (7.50) Ri1mtuilotT. Kp(v)=D j,k,l,m-r

We say that the holomorphic sectional curvature of dsz is negatioe at p if Ko(V) ( 0 for a"nydirection I/. The scalar curttaturv R is given by N

(7.51)

R=Dftii. j=l

Theorem 7.L7. The Ricci curttatures, the scalarcurtt&lure, and the holomorphic sectional curaaturesof the Weil-Pelerssonmetric are negatiaeonT(f). Proof.It is sufficient to verify this assertions at the base point of "(f). We use the same notation as in $$3.1, 2, and 3. For simplicity, we set N = do(z) - dndy. We write a complex variablez = nliy,wepfi 3g-3.For d . o ( 2 1 , . . . , 2 n )- i l o ( 2 1 ) . . . d o ( 2 " ) f o r c o m p l e xv a r i a b l e s2 1 , . . . , 2 n . A2(f ,Il) with respectto the PetersTake an orthonormal basis {ei}l=rfor = 2 Dilo=, \l$)ati d-txthe Weil-Petersson son scalar product. Denote by ds21,yp preceding subsection, it follows the was seen in (7.35). Then, as in metric defined that 0h,t -

= o' i,k,l h 1 t(0=) 6 i k, ff{o )

1,.. . , N.

( 7.52)

Hence,from the definitionsand (7.52),we obtain

R, i t ^_=af 2f hi -t u , 0 2h , ,

Riltm= ffi(o),

(7.53)

Ri.=-i#r, rtu 7=''

at t = 0. Therefore, we only need to calculate 02hr7l AttAI^ at t = 0. From (7.40), we have 1h;-r 01fi(21,2;) , 24 f f ---# - ^ + ( t,,l . , = Ii{21,72)viQ1)a(22)do(22,21), I ... I dtl r J Atr Jptn where F is a relatively compact fundamental domain for f in 11, and (tz,rt) runs over F x H. Furthermore, from (7.39) and (7'26), we obtain \xt?1'a)

A*

- - 1 [[

r JJu

*,@r,zy)I{1(w1,2-)v2(w1)do(w).

(7.54)

salqerJ" uorle.r3alureql ?eql eeselr 'suorleurro;su€rlsnrqotrJI"aJ ol lcedsa.rqltm aqt,{B slerluera.Urp rurerllegpue'()'z)y'()'r)X'@)op roJsalnruorl€ruroJsu€r1 '(32' 3@)>I (r2' 2)x Gn' ilx

(32")x (n'z)N

(rz' tm)x+

(gz'))x(z'2)x (2'@)x (n'z)N (s'r)v (rz'tm)>r+ (?'r21, (9'))x (n'))x (rrl"'eol")x (z'z)y (3s'r)x- c areq^.t

(os'z) ' ( ) ' z ' z m ' r o t ' t z ' r z ) o p ( z * ) * n( r * ) t n ( z r ) c n( r z ) t n C

'HxJ[ J

.. [ + = J V 6

6)v#e a'!

zO

eq} uo dr pug eA{uaql '(lg'Z) Jo eprspueq 1qEr.r Jo urlrra? Prlql eql u\ (rz'lll)X'(zz'|m)N o1 Pus'urra1puoceseql ul (zz'rz)>I '(tm'cm)x o1'rura1 '(rz'em)N ol €lnuroJsrql ,(1dde a6 lsrg eql u (zz'rn17o

(es'r)

'H) a'n '(r)op(s'r)x @'z)>r"il +- @'n)>t urelqo am 'ureroaql enplsar eql pu€ slnruroJ s(uaeJC Sursn 'a,ro11

(tg't) . ( z * , r * ( z z ( r z ) e p ( z m )(urrnn) t n( " r ) r n( r z ) l ng " " " [

[ #=

(r,ffi)=6)f e^€q e/ll snql

'(9'@)x (rz'cn)x (9-"^1o Fz'tn)N+ (72'rz) (9'@) (rm' zm) (tz' tm)N * N X N (zz'rz)>I(zz'tcrl)>I (rm'zrrl)>I (rz'zm)N= g eHxrI ,(zm,rm(rz.z,z)op(zm)un g ( r z ) l r t ( r ^ ) t n ( " r ) , h I

eraq^!

.-.) #rltlzQ t * = @ I VZ,

plel,((gq'Z)pue'(99'l)' (Vg't) selnuroJ'acue11 '(32'tr1ry (zz(h)tx (rz (rn)tx - y erar{^{

"*"1 (gs'z) '(r*'rr'zz)op(^)tn ("r)'h(rz)lav

t #-

= @;!* 1aBer* snq;

(qq'r)

. *!8 .(zn)op -(zn)t,t (g.,zn)to (rz,zm)ty (zz'rz)tYg J[ J[ +1 aleq e \'uor1e1no1ea.repurtse fg )rrlal{ uossraledlra A 't'l

IIZ

7. Weil-Petersson Metric

212

in (7.59) can be interchanged in an arbitrary manner. Thus (7.59) is rewritten as 02h,t,

mintot .rA

f

I

= - 2 I ... I o5 J Jr.r"

c viQ)r{6

u{w) i^@}aoG,z,21,z2,wr,w2), (7.60)

where( rangesover ,F. We introducethe notation _ t t ) ( * r , z ) v 1 Q 1 )u * ( w r )d o ( 2 1 , w 1 ) , L 1 * ( ( , 2 )= I . . . I K ( q , C ) K ( z 1 , w v K J

JHz

f

f

L '; r ( C , z )= I . . . I K ( ' d , 0 K ( a , w z ) K ( * " , 2 ) v 1 Q 2 ) r n ( . 2 )d o ( 2 2 , w 2 ) . rH2 J (z'61) Then, from (7.60)we find

W,,

='#

|

z)' Di,tmdo(('

lr,u

(7.62)

where

DiEm= Lin(e ,z1T-;qg + L1/(,4me

z) . L^*( C, + L1/C,2)

Sincetrir((,2) = L1,i(2,(),andsincetrir((, z) = L1*0(O,lGD

i@

1tQ) for any 7 € f, we see from (7.62) that 02h.,

W#lol

=

1, r

... t Eirmdo(c,z), # J Jr,,

(7.63)

where

= Q1/(,2)+ L21((,2)lGrcA +Tffi) EiErm

+2L1n(C,z)TRe.

Now, we show that the Ricci curvatures are negative at the base point. In fact, from (7.49), (7.53), and (7.63), for any element V = Dl=rai (A/A$)o e "o("(l-)) with unit norm, the Ricci curvature ,Rs(U) at the base point in the direction 7 is given by

=-x R'(v)

N

21 1,," I

Egr*i& ao11,'1

i,k-L

t2 7fo

where

t=t1,,.

h do(C,z) S 0,

9.64)

2t4

7. Weil-Petersson Metric

By the same argument as in the case of the Ricci curvature, we can show that if KoU) = 0, then V = 0. Thus Ks(V) < 0, and hence the holomorphic tr sectional curvatures are negative at the base point. Remark. Wolpert [253] obtained the following estimates for the curvatures of the Weil-Petersson metric: (i) the holomorphic sectional curvatures and Ricci curvatures are bounded above by -l/2r(g - 1), and (ii) the scalar curvature is bounded above by -3(3S -\lar. We also refer to Jost [A-49], Chapter 6; tomba 7.3.5. Weil-Petersson

Metric

[235], and Wolf [2aa].

of the Teichmiiller

Space of Genus L

We define a metric on the Teichmiiller space fi of genus 1 which corresponds to the Weil-Petersson metric hsp on Q with S ?:2. As wa.sseen in 52.2 of Chapter 1, the Teichmiiller space ?r is identified with the upper half-plane I/. In fact, for every point r € //, we denote by I the lattice group generated by 1 and r. The torus R, = C I l, has a marki\g D, associated with the generators 1 and r. Then the above identification of H to fi is given by the correspondencesending r to lR, Er). where r € H, and,\, is a Let \lldzl2 be a metric on a torus fu = C/f, positive constant. Here, we impose a normalization condition on \zrldzl2 so that the area of .R, measured by this metric is 1, i.e., we put \, = l/tfirrn. Now, for any I € C with sufficiently small lll, we consider a quasiconformal mapping ft: Ro + Rt+t induced by a linear mapping

z€c.

i , ( z' ) = ( t * _ r -1r = / ) , + r- - \r 2 , \ Sincethe Beltrami coefficientpt of it is equalto -tlQ of pr1at r is given by

p , = l l l i FT = tr l ;i; =l

-

- i +t), the derivative

1

r_i'

This p, is regarded as a holomorphic ta,ngentvector 0 / 0r on T1 at p = lR, , Erf , and gives a basis for the tangent space ?p("r). We define the scalar product of 0/0r and itself by

1 1 . . p(, a a . = i l )'i dxdy= nw all^ ry ar,a) I l r.t1, _ 11, This metric is the desired metric on fi, and is written as I rt)-tz dsw p'2 = -2g^r1r9,l '

uo el€urprooc lts)ol e sr (z'4) 'ara11'(1 reldeq3 Jo I'g$ ees)g Jo ern?onrls l€turoJ -uoc eql seuTurrelaPqllqr!\ u uo e erle; '1utod ,lzPlTd zsp 3lrlelu usruu€ruetl{ es€q aql le (gr)J lo ((U)"f)".2 eceds lua3u€l eql replsuo? ol lualcgns s-IU '(U),2 "" rrrlaru uossrelad-llal\ eq1 saar3 @)W uo lcnpo.rd reuur I€rnleu e fq pe?npul (g,)Z "" clrletu e lsql aes aA\ '.{1r1uepreq1 o1 crdolouoq ere qcrqa JIastI o}uo Ur go susrqdrouroagrp Surlresard -uorleluerro dno.I3 eql ,tq Ur uo s?Irletu ueluuetuelg Ie Jo (A)W (U)0//16, IIe Jo eceds eqt yo eceds luarlonb " qtyla pagttuepr $ U aceJJnsuuetualg pesolc e Jo (g); eceds rallnuq)reJ arll lsql ilres a^r '1 raldeq3 Jo A'g$ ur pelels se^r sy

uossra+ad-Ila 4, aqt Jo uorlBlardrelul

crr+auroa9 IBIluoro.SIC v

crrlatr l 'g'g'Z

'ZZ 6 qlylr tJ o1 spuodsarroc elil.uJoJ slqtr ro; g .raldeqC Jo g'8 tueroeql uI elnuroJ s,1.rad1o14 'VPV

dan W =

el€q e.$ 'eroruraqlrng

.I'Z'EIJ

I t :1,-

"56rF.

,l

t------------'

.z-6 qll,lr rJ uo se?€urprooruaslarNleqrual o? puodse.r.rocq?lqr!\ 'Ig uo seleurprooc ser'r3(B (/) uaql 'Vf lW, - d las en g'"lzpl[y fq pa.rnseaur'flazr.rlcadser'-r aU-'I o1 ur3rro eql tuorJ secu€+srpete 7'7 l"q1 ees e^{'I'Z'3U ul pe}e?Ipul se uaqJ llJdlf JEU

,J-vtl^ I

'pueq reqlo eql uo las eitr

'tpy tp ey

= dao

,tq uerrrSq ediltp Jo "^o turoJ leluaur€punJ 'rolreJ aql lu"lsuof, e o1 dn l? uo crrleru ere)urod aq? qtl^{ seprf,uro, qlq^t > tc

rrrlel I uossrelad-IralUt'l

216

7. Weil-PeterssonMetric

at dsz consistsof all symmetric R. The tangent epaceT = Ta"r(M(R)) of ll(n) tensors of degree 2 on J?. Every element c of 7 is written as a=Adzdz*Bdz2+Bdz2, where .A and B are smooth functions on U, A is real-valued, and B is complexvalued. This o corresponds to a real symmetric 2 x 2 matix

i(B-B)l o- = ,r l e + B-+eB1 A ' -B - ' B l | 4a Now, the inner product of two elements ei = Aidzdz * Bidz2 +4

ilz2,

j - I,2

in 7 is defined by (or,azln=

[[

t(dfi2)p2 dxdy.

JJR

Here, tr(d1&2) is the trace of the matrix d1d2 with respectto the metric dsz, that is, tr(&fi2) -

AtA2 + 2(Bt6 +E[a) 2p4

'

Thus we have

(ot, ozl ar,,nr)l +2(B1g n= i l LlA,,q, ry Note that the following two types of elementsin 7 correspondto zero vectors in [("(E)): (i) A vector induced by deformation of the scaleLactorp. This is an infinitesimal deformation r|,p2ldzlz which is generated by a l-parameter family {p'"'$ld"l'}re;- of conformaldeformationsof the metric ds2, where ry'is a real-valuedfunction on ft. (ii) A vector induced by diffeomorphismsof R to itself. Namely, this is an infinitesimal deformation (* 4rz * - oa '-'\ 'o' Aro'-) \dz inducedby {/i(ds2)hen of deformationsof ds2, where {.fr}ren is a Lparameter family of transformations of .R which is generatedby a vector field X = a(z)(0/02) on E. We shall obtain a condition on an element c = Adzd2 * Bdzz + Bdzz such that c is orthogonal to all elementsof types (i) and (ii) in 7 with respectto the given inner product (., .)n. First, in order that o satisfies

(,,g\a =;

I LA{t

dtdy= o

lurodalar,reql urorJparpnlsere sacedsraflnuq?ra; araq,rl'[776]1o7y1pue'[996] equro{,1'[62] equrorl pus reqsrd'[lg] arreuraTpue sllag,[6t-y] tsol o1 ra;ar 'aroturaqFn{ '[0gz] prrc '[6gu] '[976] 'snlncle?sse€I4leql a,ra. r.radlo1tosl€ aas '[996] uo pas€qsl WH,!r fredlo1t ,(q uarr€ sr;oo.rd a^r]€urall€uy '[/] pue [g] sroJIqV ot enp er" rrrlaru uos$eledlre \ aql Jo flrralqey .ro; .raldeqcslqt ul syoordo,lrl'[g]U] IIaAI /tq pecnporlurlsrg ssl{ )lrleur uossreladlre1\ eqtr '[gg1]epe4eNpue'h9I] r{eznsteW '[ezr] '[Oet] '[28] *ty "tx PIr" raulPr€g aas'sdnor3 ueuraly tllselt Pue ery go flqrqels IsuroJuoers"nbo1 ,(Solouoqoeralqcrg erll Jo suorlecqdderod '[gg -y] "ry ,tq looq aql ur punoJ sr ,iSolouroqocrelq?rg eql Jo leep 1ea.r3y '(gsr-qqrdd 'g le1deq3 ul '[tfV] aas)a1.regot enp sl '[6-y] sroylqy ur UI$ Z$ ul;oo.rd rng '(9'2 euwal) €tutuels(rellnuqtrel;o;oord e .rog g .ra1deq3 Jo I "uuraT aas '[tlt] ptt [971]olounselI pue'[lzt] ttx fq e.resauesereculod uo sraded,tueur;o auog '[gg-y] .rauqalpus '[89-v] e.ry fq s{ooq aq} w peulet -uoc are 1$ ur sarraserecurod pue $uroJ arqdrourolne slretep a1a1duro3 Jo 'ltt-v| €rnlpox pue,$,orro4 pue'Lg-V] srlepox '[qq-V]nzruoN pue rqsefeqoy '[69-v] srrreg pue sq]lgrr9 fq slooq aq] m punoJ sr ,trlauroe3l€rlueragrpxeldurocuo lerrel"ru .,t.ro1cnpor1ur 1n;d1aq;oleap 1ee.r3y

saloN '(U)Z uo daf rrrleur ueruueruarg uossrelad-llel\ eq1 sa,rtEqeH,u ((U)J)oJ uo euo eql qlr^\ sapllutoc lcnpord rauul s-Iql l€ql aas ern'.sp se U uo crrlel'u cqoq.radfq eq1 3q1et'relnctlred u1 af f -d

= a(za'tol 4f*pg'fr ll duz

fq ua,rrE sr '-6, 'rarroe.ro;41 '1urod !,fi !4t = fio slueurela o^rl Jo lcnpord reurn eq? ul (U'I = f) + espq eqt le (31r)Jl" ((U)Z),,2 aceds lua3uel arqdroruoloq aql ot spuodsarroc 1, o;, aaedsqns aqt ?eqt s^{oqs uolt"^rasqo srql lo {(U)zV ) $ | f +,fu} = '(A)zV ur ,f euros rc1 rlt * fi = o s€ uellrrrlr sr o ;r fluo pue;r (g) p* (r) sad,t1Jo sluauala 11eo1 leuoSoqlro $ L ul p luatuela ue 'acuag 'u uo EerlueJeJ 'crqdrour -Ip crlerpsnb crqd.rotuoloqJo (U)zy aceds aql o1 s3uoleq snq; ezpg -oloq fl g 's! t€ql '0 = Zg/gg leql aas aar 'frerlrq.re x (zg/g)(z)o - y acurg

's- npxp" " + nPxP #"[ I #"[ | e^"q e!$ slmuroJ s(uearc urog '(g) ad{1 ;o 6/ luaurala ,tue .ro3

o - n w P(At.#r)"il

=a(d'ol

sagsrl"sla lsrll repro ur 'lxeN '0 eq plnoqsy '(r) adfl;o 5/ fue ro3 Lr7,

saloN

'[776] '[996] lurod,uarr\aql urorJperpntsare secedsra[nuqrreJ a.raqaa 1o1q pue €quro{L '[62] equoqtr pus raqsrJ 'Lg] "r,r*"f pu€ slleg '[6?-V] tsol ol .raya.r aar'arourraqlrn{'[0gZ]pue'[6gU]'[gy6] 1rad1orlA osl" eas'snFrlec sse€I4leql uo paseqsr qerr{^r'[996] lradloA\ fq uer€ sr ;oo.rde^rleuralle uy '[/] pue [g] sroJltly ol enp are Jrrlaur uossJaled-lre11t eqt Jo fluelqey ro; raldeqc srql ur s;oord orlrl '[g?Z] IIel\ ,tq pecnporlur lsrg se^t crr]eru uossraledlre1\ eql '[gg1]epeqeNpu€ 'hgI] rlnznslelq '[tzt] '[0at] ttx '[48] tty 'sdnor3 uetutely tl{.eru Pue erx Pue reurPreg aas '[89 ;o flqtqels leuroJuocrs€nbo1 fSolouoqoc ralqrrg eql Jo suorlecqdderoJ -y] erx ,iq looq aqt ur punoJ sr ,(Soloruoqocre1lrrg arll Jo leap 1ea.rEy '(ggr-ggrdd 'g raldeqg ul '[tf-V] aes)a1.regol enp q '[6-y] srolpy ur ZI$ A$ ul goord .rng g raldeq3 Jo I "uureT eas'(g'2 euural) srutuals.ra1nuqcral;o;oo.rd e .rog '[tlt] pt" [971iolounseltl Pu"'Fzt] ttx,tq ale salraser€culod uo s.radedfueur;o auog '[99-y] rauqe1pu€ '[89-y] e.ry fq s{ooq aq} ur peulet -uoc ers ur sarras pus surroJ crqdrourolne slretep a1a1duro3 1$ ?J€?urod Jo '[tt-vl errepox pue ir,!,orrotr{ pue'[Zg-v] errepox '[qq-v] nzruoN pue rqse,leqoy'[69-v] srrreg pue sqlgrrC ,(q s4ooqeql ur punoJ sr f.rlatuoeSlerlueresrpxaldurocuo lerreleru i{.ro1cnpor1ur lnydleq;o leep 1ea.r3y

seloN

uosslarad-rra^\ aq1saarE q?Hi$ ((u)rl, ""

(""Ji Jr; #i""$;"trjiT"T:'#

rauul slql leql aes em'rsp se U uo tlrlatu cqoqredfq aq1 3uqe1 'relncryed u1

. f"df i " L ' f iaf f JJrz=u(zo,ral

,(q uarrr3sr ! '1urod q (Z'I = !) !,1,+ !fr = fra sluauralao/rt Jo lcnpo.rd raum aql '.ra,roe.rotr11 es€qeq?fe (U)'f l" ((U)"f),,-f aaedsluaEuel arqd.rouroloq eql o1 spuodsarroc,1, fo {(ff)zy ) ,1,I f + fi} = } ecedsqnsaql }eqt s^otls uotl€^resqosIqI '(a)cv ur aurosro3ril * 4l fr = a se uallrr^r sl ^oJI fluo pue;r (l) p* (r) sed,(1Jo sluetuele11eo1 puoSorllro sr -L u\ p luauela ue 'arua11 'u uo slerluereJ 'ctqd.rour -yrpcrlerpenbcrqdroruolorlJo(U)uy aaedsaq1o1 s3uolaq zzpg E\qL -oloq $ g 'sr 1eq1'0 = zg/ge l€rll eesaar 'frerlrqre il (zg/d@)D - X eculs

's t*" t*" npap npop " t 1[ + II el"q aar slmuroJ s(uaerC uro.rl '(g) adfl;o

g/ luaurala fue ro;

o-np,p (#r.#")"ll =a(s,o,) segsrlesla lsql rapro ur 'lxaN '0 aq plnoqs y '(r) ed,t13o 5/ fue .ro; seloN

Ltz

2I8

7. Weil-PeterssonMetric

of ha^rmonicmaps. Moreover, see Takhtadzhyan [219] and [220], and Zograf and Takhtadzhyan 12641,[265], and [266]. tomba [235] showed that the sectional curvatures of the Weil-Petersson metric are also negative. It is also known that the Weil-Petersson metric is non-complete; proofs are found in Chu [49], Masur [143], and Wolpert [245]. For the subject in $3.6, we refer to Fischer and tomba [72]. Such a diferential geometric interpretation of the Weil-Petersson metric is closely related to the Polyakov integral in string theory (see, for example, Polyakov [176]). See also Nag and Verjovsky [164]. For the Weil-Petersson geometry on moduli spaces of higher dimensional complex manifolds, we refer to Besse [A-17], Fujiki and Schumacher [77], Koiso [118], [119],Schumacher[190], and Siu [208].

'ttr*9rrart>oo-I u>r)="7r4 se cAA ssardxa u€) ell\ 'Z/u ) od > 0 qq^{ 0, elqelrns e rod'01 lo oLV sr)rc eql rt,"iuor qrlqi{ (J o1 lcadsa.rqlrm) - (z)oL H uo 3A4Jo 1JIIe eq,cA Ial'C sraAor pu€ J o1 s3uolaq (t < V) zy '{pp} -._i' leqt pue ;o fuaurela atuos go lurod paxg e sl I }eq} etunsse detu eM 'fI eueld-g1eqreddn aq1 uo 3ur1oe uersq)nd e ar1e1'1srtg A J Jo Iapour (rl4 'cA1 A uo p? o1 pnba sr pue ul /tq g 3uo1e aql sluasardar q?rqa g,;o Surdderu l€ruroJuocrsenb e I ,,Eur1sra,r1,, ', ''a'r 'pooqroqqErau lcnrlsuoc e^r uaql Jo pooqroqqErau palreuuoc flqnop e r€lnqnl € seruoceq cAA Wrn llprus os p luelsuoc aarlrsod e esooqc ern 'era11

'{, > (C'd)d ) d} = ctr4 lA cqoq.red.ilq aq1eq d 1e1 aql 'U uo ecu€lsrp les 'ltsp fq pacnpuracuelsrp

'sEurddeur leuroJuocrsenb Sursn fq uorleur.rogepe qens luese.rdar e11 'I'g'3ld pue g raldeqCJo ees'sraproq eq? SulnlSarfq uaql pue U$ 'p 3uo1eg' 3ur11nc{q peurelqo ? r{13ua1cqoq.radfq fq Eurlsranlfq Q. saceyrns uueruerll pa{r€urJo {U > I I tg} fguey aql sueew 'uo ataq rnoquorlDulto{ep Ntr eql pefiec fldurrs sr q)rq^r 'p o1 lcedser q]l/{ A lo uoNlDu.totepuesptTlleyouetr eqt t€rll ilerdg '{sp cr.rlaur crloq.redfq eql ol lcadsar qrlta C crsapoe3pesol? aldurrs paluarro ue x-r.{ '(Z <) f snue3 Jo er€Jrns uuetuarg pesolc € eq U larl

suor+errrroJecuaslarN-Iarlrued'I'8 'seleurProoc ueslerNleq?uag ,{q uroJ l"}uau€punJ uossJaled{aM eqt Jo uol}s} -uasardar alduns e 'flaureu 'e1nur.ro;s,1.rad1o14 e a,rord aiu 'g uotloeg ur ',{11eutg 'fu ur uotleuuoJep uaslalN{eqtueJ e fq paurur -Jalap Jolce^ lua3uel eql etelnrpc en'4 uollcas ur '1xag 'sEurddeur l€ruroJuoc -rsenb Sursn fq suorleuroJep uaslerNlaq?ueJ eqrJtsep am '1 uorlcag ur '1srtg 'salsurProoc ueslarN{aqrueg Eursn fq 6) 6a uo uroJ leluaruepunJ uossre}adjla1\ eq} Jo (Z ? 'raldeqa sql uI '1red1o11'S ol enp 'uorleluasardeJ InJlln€aq e a.,rr3IFqs e^\

JrJlatrtr uossJa+ad -lla/y\ aql PUB suol+BIIIroJa( uaslalN-lallruad

g ra+deqc

220

8. Fenchel-Nielsen Deformations and the Weil-Petersson Metric

V

vN deto'*ation;t

V

*,,-'-H*..**-

-/

l_L

ll

i+

\qc mapping \

Fig.8.1. Next, for every t € R, we define a quasiconformal mapping ut of 11 onto itself by

* ' ( r )-

0 < d
Here, d = a;tgztand e = -t/(2eo). This tor givesa surgeryof I/ along the axis Aro. Note that the sign of I is differentfrom that in Wolpert'spapers[247]and [251].(SeeFig. 8.2.) Now, denoteby 4 the complexdilatation of urt. A simplecomputationgives

r1(z)-f7x,@i,

z€H.

Ilere, 17 is the characteristicfunction of -I = [" /2 - 0s,Tf 2 * d6]on R. FurtherITlor€, 11 satisfies

rt o.yo. (T'o/t0 = ,r. Thus, 4 is a Beltrami coefficient with respect to the cyclic group (7e). On the other hand, it is clear that the set ,l-c consisting of all elements in lw h i c h c o v e rC i s { T o t o o l - r l t € f } . B y l i f t i n g t h i s F N d e f o r m a t i o nt o . I l , we have a family of self-mappings of I/ which give surgeries along the axes of all elements in l-c. Thus, we can construct a family of quasiconformal self-mappings of ^t{ which induces this FN deformation as follows. Denote bv (zo) \ l- the set of all right cosets of l- with respect to (to), and set

'(o='lJg-\ "P = 'a \ l,,r,g/ ,p e rnc eqtJo o1 lenba sl (J)sJgo lurod es€q aql 1e {U 3 ll(rrt)d rolral lua3uel aql l€rll apnlcuocu?, a^r'61'9 ureroeq;;o;oord aq1 ur se'uaq; 'rrl drerra rog "dm = tot '9 .reldeqg las en Jo I'I$ uI peugap sl'd(n eraq,n 'ploJrutsru leer e s€ pereprsuor (.t)sJ ur uor?euroJap NJ slq? Jo rolcel 'lxeN 'C 8uo1eg lua3uel aq1 alnduroc IFqs ar* Jo uorleuroJep Ng eq1 sluasarder t{cr{rlr 'r(._;r'H)g ul {U > I | ,t/} "nt.,t e peul€lqo aAsq a.{r 'relncrlred u1 '909 -ggg'dd '[1y6] tradloA\ aas 's1te1aperotu roJ 'Surddeur 3ut11nsareq] azll"turou ueqt Pue 'J \ (0.t) 3 r-1,(01,) {.rarra .ro3eroJeq PeqlnsaP se (c1-)L uo 1 fq (",V)L 3uo1e,tlartrlonpur ((lsrrhl,,'i(11en1cy'ff ul t fq o'V 3uo1e ,,s1srall,,ql.rq^r 'ror Sursn fq rtlpcrrleruoa3 pue fllle{p eroru,,rrn lcnrlsuof, u€e aJ1yIJDuu e[ 'g 3uo1e Ur Jo uorleuroJep 'snq; N1I eqt sluaserdar qrlqa (J)J u! {U f I | [,7rrr]] ,(lTueJ s ul"tqo eirl 'slstxe 'Q)l > ['arn] lurod € seunuralap pue H lo ,an Surddeur cb-rr/ Iecruouec aq1 '3 fraaa rog 'aoue11'1(,1 'U)g o1 s3uolaq trl ltsql uorlrugep eql urog reelc st lI ,

J \ ( o L )) r

L(tort\1 7 '

-4t

!-l

'z'8'ttJ

,,rrT

suorl"ruroJeo uaslerNlaqf,uaJ' I'8

tzz

8. Fenchel-Nielsen Deformationsand the Weil-PeterssonMetric

222

(Here, recall that such a tangent vector is consideredas an element of A2(H* , f).) Furthermore, Theorem 4.37 gives an integral formula for ti = (0q/0t)11=s as follows. We set

where

x , ,z uslz)= --yllargz)=. 'lvo

z

It is easy to see that

-/ll- =o' ]13tt? Thus we obtain

it(z)=-+ ilr,G)ffid€dr,

zec.

(8.2)

As has been stated in the proof of Lemma 7.16, we know that (i) (d)" = L, on C in the senseof distribution, and (ii) d(0) = tt(1) = 0, and b(z) = o(lzl2) as z -----'oo. Conversely, these conditions (i) and (ii) characterize tir in the class of continuous functions on C, which can be easily shown by using Weyl's lemma (Lemma 4.6). Now, to get a simpler representation of gc, we rewrite formula (8.2) as follows. Lemrna 8.L. The deriuatiae b is written as

i(z)= -,

( l a

\1,*" #

(8.8) o,* L*^r,) *.,.,,,.,F,, rhotr,(,)

for eoery z € C. Here, argz lakes aaluesin l-r,r),

and

=-#U,*",'' o# o,+ fibs1et\ F.,(z) * fir.,1,1, wherc P.r(z) is a polynomial of degrve aI most two. This P, is aniquely determined by the conditions that Fr(O) = 4(1) = 0 and that Fr(z) - o(lzl2) as z + @ .

Moreooer, the series on the right hand sid,eof (8.3) conaergeslocally uniformly on C. Prool. First, substitute the right hand side of (8.1) for p in (8.2). Then we have

(q's)

'*rr uo

/

t r

J\(o[]l't

(t - ,r"r;) 3

)17 ! serrr31'g evrute1 'loo.td

'*H uo fryu.r,ofiunfipoao1safitaauocycnlm ,,.

, l L \

J \ ( o L )) L

"\r) I

v ( _ c d t

?

'sacuoTsrunc.nc 6uto6atol eql repull 'Z'8 uraroaql 7oq7snollot 7t .s^rolloJ uorlresse aql E uaqa'r,61 L4l- s€ e?uereglp slql ?asa1yo^rl lsoru 1e ear3epSo leurou,tlod e st acueraJrp slql teqt saqdun qctqn 'oo + z w (rlrDO sr pue 'g uo ctqdrouroloq st

( ,tf:_\_(,),1 o),r. " ' - X; g ) , f (o)r)oa ,"" ioplpQ'))a tlr l J Q)Lo1 l?q? u/rroqssI 1I snqJ 'J

) L',L'

( L o o n )- t ( L " t )

e^eq eir{ 'uorlnqr.rlsrp Jo esues aq1 ur 0n = z7 acurs '1xe11

ooz

. { reoto1'*,n' Y G Y x " T\ r * t' - ! t I'-":= '?t?,} tP{z?o1-(3)({"e"''3}xeru's)x ffi ""1

'yor=)'tp{.

=tr _"[}ffi "'[

onz

-))

,"

:2(r t'p?p666ffi,xt(t_4,

,

":= ",

[[ JJ ?

= (,),

eleq ad\ ueql '(?'8) Jo aprs pueq lq3rr eql uo urrel lsrg aqt (z)1 ,(q aloueq '9 uo fprro;tun 'fpealC f11eco1pue flelnlosqe sa3raluoc (f'g) f" eprspu"q 1q3u eql uo selresaq1

. (z . )XI . ))) = G,))a G-z)t

(r'e)

les at\'eJeH

"fl +

r/".J\ tL

't plp(z'))a

611&(())r)0z

'[[:--(z\m

ttplp (z '))uO)0"

"'

IJ I

suorl?ruroJao ueslerN-Iaq)uad't'8

tzz

224

8. Fenchel-Nielsen Deformations and the Weil-Petersson Metric

where P;7 = S. On the other hand, a direct computation gives the following, Bol's equation:

1.7'"ttl =- (i,l /t

\''

/r'\2

Thus, differentiating both sides of (8.5) three times, we have the assertion.

tr

In general, for any simple closed geodesic C on r? and any element T e f which covers C, we can construct a similar basic series as in Theorem 8.2. More precisely, let o and 6 be the two real fixed points of 7s, and set (a - b\2

- ' t o/ \ .\----l----------te_a)2(z_b)2.

Then

oc=

t

(rr,ot)'(t)'

re ('vo)\r

converges locally uniformly on .[/, and belongsto A2(H,l-). We call this Oc the Peterssonseriesfor C. Usingthis series,Theorem8.2 is reformulatedas follows: Corollary. Let N(l)

be as in 52.1 of Chapter7. Then i._ru = --)fi'Oc

modN(f).

Prool.Usingthe notationin $2of Chapter7, Theorem8.2and (7.15)in Chapter 7 imply that glvl = *Oc. Hence,(7.17)in Chapter7 gives

H l u l = -*^ o' U; Thus, the assertionfollowsby Theorem7.7.

o

8.2. A Variational Formula for Geodesic Length Functions Fix a simple closed geodesic C on ,R arbitrarily. For every point p = [^9,/] € ?(.R), let Co be the simple closed geodesicon ,S freely homotopic to /(C), and denote bV tc(p) the hyperbolic length of Co. Recall that tc is a real-analytic function on T(,R) (see Remark 1 in $3.2 of Chapter 7). Here, we compute the variation of 16 at the base point po = lR,fd]. More precisely, take p € B(H,f ) arbitrarily, and let wtp be the canonical tpr-qc mapping of f/ for every sufficiently small real l. Then rptP determines a point, say pr, of T(.R) for such t. Under these circumstances, we compute the value

( z - zY uotu \(-*t,o, I*fut)

*=",-,

1 ",vj

!'tnoa17t

())"

I L-

= (rt)"d(c7p

uretqo el,l.'(z)n - (tU)" acurg '{og ) z og sureruop eqt uo asoqt se qer3alul aq} l r\ = !.q pu€ {y > lrl > ll H ) z} = ',uo11 alrraer pue '(/'8) Jo aprs pueq lqErr eql ,(q ,t1r1enbaslql ul ,14 aceldar

'o+z'&-

dn=(o)ffi @)"d(cg) :"lnuroJ freurt.rd eq? e^€II au 'ecue11

. ? D 'Q)4tu+'(o)\p: = ?u)41

1€ql s^\oqs(9'8)

(z'e)

'pr"q reqlo aqt uo

cf f v 't'p?p = (z'))uQ)n I I ;- @)4 :uorleluesarda.rp.r3alut eql seq

'c>z

,L121 rhi=@4t

'fl 1eq1sa11dur1 zt'? ueroeqJ = 1'aoDI e^{ ueIIJ'9;oSurddewtb-rt| I??Iuouecaql eq nJ +e'I vo l?r{t nrt rltn '(z)rt '*H ) z

lrfz H ) z

'0 '(r)rt 1esaal'1xa11

' r 1 E o 1- ( d ) o t l€r{t slllolloJ l\'c7 1o uor}IusaP eq} /tg

(g'a)

'zty=Q)r_Gr^)ooLo

ttn

'1 1€rl1uortrPuoc eql ,tq paurur.ralePq (I () r1 luelsuoc e .r(laaa

ro3 'uaq; 'g sra^oe(t < V) zy = (z)oL 1"ql eunss? feur a,t 'I$ ul sY 'loo.t4 '

a/

)L

| \

"Oj,rt 6

\

/

= (d)oaQ7p) I tU

'g'Suraroaqr

'([62] rautpr"C 'Jc) u,laourl-ila,ll fletluassa s! sIqI

.

o=ll

lrt

l(rqcti I

P

= Qt)od(c7p)

suorlounJ {ttua1 f,rsepoeCro} "FruroJ [uoIl"-Ir"A V 'Z'8

9ZZ

226

8. Fenchel-Nielsen Deformationsand the Weil-PeterssonMetric

Furthermore, since

)"ft(c,4=-+(+-*) we see that

i

,?*

^ r " { n (l., l i ( , r-zR) ( z\ c , z )=} - 1 i |

)

{ . , 1 " _ .s n) -: t.(' - z})

C,?*L)"(-,

=-i^u*#=-b

Hence. we have

=- + (dtc)o"oi (-!) ut, t |^,,,"ret

=+11."(g.F)** =?Ru 11,"8**

Finally, divide F's into domains {Z(f) domain for l-. Then we conclude that

| I e f},

where F is a fundamental

=,,.H,,. (#)" oro, I1,"8*'o Il,urct =[[ u{0 ,H,"(#)'0,0,

=i,,2",." This completes the proof of Theorem 8.3.

8.3. Wolpertts

D

Formula

We have computed the tangent vector (represented by z in $8.1) of the FN deformation with respect to C at the base point, where C is a given simple closed geodesicon ,R. By a translation of the base point, we ca.ncompute the tangent vector field on (the real manifold) ?(r?) associated with the FN deformation with respect to C, which we denote by 0f 7rs, and call the Flf ueclor fieldfor C. Namely, 0/7rs is the vector field obtained by applying the FN deformation with respect to C with unit speed with respect to the hyperbolic length. Note that 0/016 is a real-analytic vector field.

.'c!8 - = ('"t9,3\

dtao- -

cte \B e/ (%-,Y.) "rn,,= (::g\ (,4,\- cto ,c78 I / \s \8 /'"" el€q e^r 'fre11o.ro3 Euro3ero; eq1 ,Lg .too.r.4 .,c!g _ = "rg clg ,clg 'A uo .g.g uorlrsodo.r4 ,C puo C sctsepoa|pasola a1dtutsIIo Jo[ 'ur?roeql fr,7tco.r,dtca.t 3ur,uo1oyeql sl t.g ueroaqJ o1 zt.re11o.roc reqlouv

.(.)ctp=(.,"19-\*^.

\

s /

'zt.re11orog

'uollrasse aql e^eq ea\ ecuaH '(rt)"o(tlp) tr o1 sr aprs pueq eql sa,rr3g'3 ueroeqr 'pueq .req10aql uo lqErr l3rll lenba '(t'H)gU 3 r/ frana .ro; " ( """.,t) "g 6= \ [ /

( , ,t",-"v \

-"

i) d htu e1zI/

(, '!#r\ d^ qeaz=(,1,'"-nr\ "^u s / \ 9'l \ eleq e^r'61'2elouwe1 pue r.ueroer{Jo1 ,(re11o.ro3aq1 uro.r;,1s.rrg Z'g '(U)Z.lo od lulod eseq eqt te flrlenba JeruroJeql ^{oqs ol seclsns 1r '1urod es"q aqlJo uorlelsu€t1 e ,Lg.too"t4

.(.

' / " ^6 )orp= ( .,'"-e-r\ g \ ,.a.? (d/vr6oI qpn pnp aq7 6uu1o7 fiq uaa$.toTnradoeq?il * puo '(6'6'7$'lc) Tcadsa.t (g)a pyotguou p^t eql-uo etnl.rnrls ralilutoc lsouqo lDrnpu ?Tl suDeu c ?r?qn

'crp= (%r\' / .\ I

'A uo Jo,{ .V.g ruaroatll C ctsapoaf paso1calilutts fr.r.aaa

ur peugep aleq ea'cl

'urero?rll fi.4t1onpSuruolo; eql ^roqs a1il 'clp ppg rolcal 1ue3ue1occrlfleue-par eq? U.g$ uorlcunJ {fue1 crsapoe3eq} urorJ ,pueq reqlo eq} uO "InuroJ s,1rad1o11'g'g

8. Fenchel-Nielsen Deformations and the Weil-Petersson Metric

This shows the assertion.

Finally, we havethe following, Wolperl'sfonnula, which is provedlater' euraesL = {Ci}?c=13on R arbiTheorem 8.6. Fb a systernof decomposing coorditrarily.Denoteby {tcr,... ,lcsg-s,0cr,"' ,0c"o-"} the Fenchel-Nielsen with L. Set rs, = (tgrf2T)|st for eaeryj. Then natesassociated 3g-3

uwp=DOr"oAdl6, i=L

Corollary.

For eaerg simple closed geodesicC on R,

, * , Il fau , '\) = - d r c ( ' ) . Proof. Take a system of decomposing curves which contains C, and apply Theorem 8.6. Now, to prove Theorem 8.6, we take a base of tangent vector fields

={#,, ,#,#., {X,,...,Xu'-u}

,#;}

on ?(r?), where ?(.R) is considered as a real manifold. Further, we set { t r , ' . . , o o g - o }= { t c r r " '

, t c " " - " , T C r , " ' , T c " " - " }.

Then arpp is written in the form uwp =

a;idx; A dxi. I lj
First, we shall show that every a;i is invariant under the FN deformations with respect to C3 for every /c. Lemma 8.7. For eaery i,i, and k,

- 0 on r@).

Proof. To prove the assertion, we use some basic notions and results from the differential geometry. See for instance Matsushima [A-72]. Let /(X) be the interior product with respect to X. Then the corollary to Theorem 8.4 gives

Surddeur l")ruousc crqdrouroloq-llue u" auuae 'aue1d-g1eq reaol aql sr *lT areqa\ 'S Jo sJ / *H aSerurronrur eqt ,S ,tq alouaq .g uo 3ur1aeS'Jo Iepour uersrlcnd" aq sJ lel '(U)J > [/ ,S] lurod f.reaa roJ 's,lrolloJ s€ Jlastr oluo (g)g 3o ' t fus'Surddeur e secnput / ueql 'f uorlcegar e s?turp" U 'uorssncsrpsnll 1eq1flrlerauaE;o ssol lnoqlr^a erunssefeur ell Jo tser eql uI ('flrxalduroc ,(lesseceuunpro^e o? 'slurod pexg Jo sles eql go lred etuos 'g'g '3rg aag) 'W uo scrsapoaSpesop aldurrs lTrrro allr araq,ra Jo raqunu alrug e Jo sl$suoc d leql etoN 'f ;o slurod pexs Jo 1as e{} sr Jr eJeq^i!,,ry fre,ra .ro; rtg - 14 Ud l"q1 pue'oaa1 raproJo og;o Eurdd€ur-Jlesl€ruroJuoe-rlue ue'0U Jo 1 uorloeuer e Jo uorlcrJlsar e saruo?eqY uot?ceUarq?€e leql ees el!l'ueqJ 'Ieqlo qcee qlr^{ }uepnuro? ale (6'1 = il fl uo 7'![ Jo lI v''rrg slurod pexgJo sles aql ueql'/ r{?BeroJ r'!4 p uorlcager eq} aq (A'I = il llf pue'!7 Suop luacetpe f11en1nu slued aq z'!4 pue I'id fal 'J ) lI fra,,re .ro; :uor?rpuoc3ur,r,ro11o; eq1 Surfgsrles OUrer"Jrns uusruarg e pug oiu '(l >) f'r7 frena JoJ uorlsr.uJoJapNd aq1 Surfldde 'snq;'q13ue1 cgoq.radfq eru€s eq? qlra\ scrsepoa3o,lr1olur({g;o (t'Z'l = t) f't7 fes 'luauoduroe frepunoq qcse sepr^rp {1 slurod paxg;o {rg ;o '.rarroatotr11 '(g'g ura.roaq; o1 frelloroC aqt 'Jc) {d go 1asa{} f uorlcagar aqt s€r{ {2, slued ;o rrcd f.re,raueq; '7 uelrS aql o} Eurpuodsa.rrorUr Jo uorlrsodtuocep '1ce; 'uorloegar slued eq1 "q = dpl u1 l€ruroJuo?-rlue ue Surllrurpe "51{'dl1 'suorleurJoJap 'pueq Jar{?o aqt uO Qr ue aleraua3 uec an NJ elqelrns rage 'pa3ueqaun sr lapy txpllp3"^, uolleluasarde.req1 'lg {ue o1 qll,lr uo-r}?urroJap lcadsar '1eq1 sarldurr NJ ar{l Sutfldde fq og' raqlou" o} U aEueqr e^r ueqlr l'g €ururerl 'PeJrsepse 'lcJo

g=

.6

r3!o

= (!X(!y)dan---

--

ul€lqo ailt(6'8) pue (8'8) tuo.r;'fy - Z pue'!X = A"ctg/g = X Eurllas'relnarlreduI'0 = lZ'Xl = h'X] r"qt 'n|;j{lX} of splegrotra^ Z pue 'l'y se eqel a,ray '1eql aloN 'Z pue 'r{'X sples rolf,el Jo las frela .roy (Z 'X|A)a n, - (Z '1,+,.Xl)d ^h - (Z .A)d/$oX = (Z (a)a noxT (O'e)

el€q a^r 'se,rr1e,r,trepelT arll Jo uorlrugep eql urorJ 'f11eurg

(e's)

'$ -

d,746(1c'e/e),

leqt apnltuoc aa,r'91'2ruaroaql dq 0 = dary

dilr y (# )

r + (a n . (# )

''a'r'uerralqey sr dzllo acurg

,) , = d no ( c' e ' d ,

\ sarlr3elnur.rog s(uelreC'H 'ueql 'y o1 lcedsarqll^r elrl€Arrapel1 aql eq x7 7a1

.o=(,cm)p= ((,#) "^,)n= ("^,(#),), "lnruroJ s,lrad1ol4'6'9

6ZZ

230

8. Fenchel-Nielsen Deformations and the Weil-Petersson Metric

Fig.8.3. j 5 : , S-

S*

by setting

js([zl=lz], lzleHlrs. Then we have a mapping

7'(R) f,: r(Q -----*

given by J ( l S , f l ) = [ S * , J so f o J ] ,

[S,/] €"(rR).

This .7 is an anti-holomorphic automorphism of "(.R) which fixes the base point lR,idl.we can easily show from the definitions of J and the weil-Petersson metric that gvp is invariant under "7. Furthermore, we have the following: Lemma 8.8. Denole by J* the pull-back operalor induced by J. Then

J. (dlci) - dtc;, - -drcl +\ab,, J.(drc)

(8.10) ni QZ

(8.11)

for eaeryi (i = I,...,3g - 3). Further,Qwp satisfies J*(uwp)

- -uwP.

(8.12)

Proof. The assertion (8.10) follows by taking the derivative of both sides of lcioJ=lt(c)-lci. Next, for'every j, we see that, though J(Cj) = Ci as points sets, the orientation of ci at p is the converseof that "t J(p) for every p e T(R).II is clear that rs, is determined modulo tcj12. Hence, we have

rcioJ - -rci+?4 with a suitable integer ni, which implies (8.11):

'r - dr; | >E! r qrr,u u,r[u]"rTir?,{#ll.',$';':;fi""tf,':"rt .{+E-te'!1e-6to _ (

"tg

'"g) ,

dtuo - =

I

\ 0

/

(2:g-,-'"p)

FLtn+t)\ e e)' ( (ru\ ,ro" , ('"'s) ."-) " dtu \\ s / \ e) ) , bg\

('"tg

I l

\ s

dam-

,tre-oe,!*e-6to

'(61'9),tq ecuag'(g - dg' ...'I = teql apnpuoran f) ! frera ro3 f

!c.tp\

-c=t o - = 1 /- ; l - t g

\e,

ou"

,o "te g (t"tg \ ., = *t" vQ e l u e \ e / " '1xag ul€?qoe,rr'(II'8) pue (0I'8) ur aq1Eu11e1 lenp 'g - tg q'[ ) ] I qry'{ {'f il" roJ

,rs=

d/r,o = q'!*t-6to

= (#,?)

ffi

eleq ea 'p'g uraroaq; o1 ,(le11o.ro3aq1 ,tq '1srrg '71 go lutod paxg P sl qctqa '1urod as"q eql le pereplsuoc eq o1 erc ^roleq suorlelar 11e,acue11.1urod a$srqeql le "1nturoJ eq? ,lrorls ol seclgns y 'aro;eq sv 'g'g ut?ro?lJ lo loo.t4

'(Zt'g) saqdurrsrqS '(7''X)d^ro - -

(a,ylau6 _ _ (.t.t,(Xl),t)da6 - = (a,y)(ann*S) leql epnpuoc eu'f, repun lue 'crqdrouroloq--rlu€sr -Irelur sr d/116acurg '(;61)*/ - = X*tg leql ees ein ll ecurs

'(I*t'X*t?)dtur|( A*t' x *t) d u r= (A' x) (d ilo *,C) e a uaql '(g);

urelqo

uo splag roloel luaEuel frerlrq.re aq ,4 pue y 1a1'fgeurg

rtz

"FruroJ

s,lrad1o11'g'g

Zg2

8. Fenchel-NielsenDeformations and the Weil-PeterssonMetric

aih =@wp

l a

a \

, 6t",

) \a/c, l A n ; A 0 n x ? \ uwP T-ar^) ar",'W* \atct+;

- - aik.

H e n c ew, e g e t a i r = 0 f o r a l l j , f t w i t h 1 < , t < i < 3 s - 3 . Therefore, we have proved that 3g-3

@wp=Ddtro-"*iAdr i, i=l

as desired.

tr

Notes This chapter follows wolpert's papers 12471and, [251].We remark again that the sign of the parameter t of the FN deformation is different from that in these papers. For someapplicationsof Wolpert'sformula,seeAppendix B.5' Several potential functions of the fundamental form uyp of the WeilPeterssonmetric have beenobtained,for instance,in tomba [236],Wolf [244], wolpert [254],and Zograf and Takhtadzhyan[264], [265],[266].Seealso Jost [A-49],and Takhtadzhyan[219].

1-yz-tz ) Eurddeur eql raprsuo?ea,r'esod.rndflql rod .dq uo fluo gr ,(lrl flluarcgns qlran) r .ralatue.red Jo arn?cnrls xalduroc aq1 Eurur.ro;ap,tq lpurs xelduoc e uo Surpuadap sacegrnsuuetuerll flrureg " a \ ??nrlsuoc Jo {rU} 'oO lo,{repunoq a ^ l ? € l a ra q l d C f q e l o u a q . ( g ) r _ , - o O p r * , { I > C ) z} = g ?eS l r l l '{Z> .A d ) ltll C > r} = (n), pue 0 (d)z ryqlaunssv lurod uar,i8€ punore (z'2) pooqroqqSraual"utproo? e xrd 'ac"JJnsuueruerg f.rerlrq.re ue eq U laT 'asec 1ecrd,t11nq aldurrss ur uorl"rr"A rotrelul s(JeJrqcsurc1dxaIpqs aA\ .(rhrlp pcol (a:ou .ro) auo;o rorrelur aql ur uorleruroJepe $ uorl"rre^ Jorralur s.rasrqcs 'frepunoq aql ,Buupads z(lq3nog Jo uorleuroJep " sl uorlsrrel s.prsruepeH 1nq

uorlBrJ€A rorJelul ssraJrqJs .I.v 'uorlerJel l"?rsselt l€luauspunJ go adrtl rar{}oue 'sacey.rnsuueurarg uorleraueEapssncsrp e^r ,g.V rl ,,(1eurg Jo II€qs 'flanrlcedser'e'y pue I.y ul (Z ?) d snua3go d; aceds rafinurqcreJ arll Jo arnl)nr1s xalduroc eq] ecnpor]ur o1 1uaun3.re (sroJlqy pue uorl€rJel Jorralul s.Jagqcs fgarrq ureldxa all 'uorlerr?A s(prerlrepeH uorlezrlereueS 3o raqlou" s€ paraptsuof, osle sr sq; .sSurdderu Jellnruq?lel ,(q pecnpur (suorleru ,s.ro;1qv .1 fq pacnpor?ur ,(11s.rg -roJep ser\{ U ac€JJns lpurs,, Sur.raprsuoc{q uu"tuarll pesolc € (U)J aceds .ra11ntuqclatr eq? arnlcnrls xalduroc eq6 Jo Jo '([ru-V] .racuadgpu" raJrqrs ,[g6-y] prc yo xrpuaddy tu"rnoC ;c) uollerr€A s(prsruspeH plueu€punJ pue alotu eql uorlezrl€raua3 e se l"clssep Jo 'uorlerle,r Jorrelur s(JeJrq?s sr uorlsrr?A PeJaprsuocil qcrqa\ ltscrsselclecrddl y 'sac"Jrns uueruaru suorlsrr"A se paleSrlsaAur uaeq p€q .(uorl€turoJap Jo Iscrss€lc 'dl.repdod pesn aq o1 ueSaq sSurddeur Il€rus,, qcns l"uroJuocrsenb alogag 'U a?€JJnsua,rr3 aql a?slnurJoJo1 fluo peeu a&r ,r(roaq1 ((uor?tsruJoJap Jo lletus,, '.re,ra,lro11'U eceJrns uueruarlf uaar8 e p (U),2 aceds rallnurqcra; Fcol aql .ro; aql uo saleurprooc FqolE ecnporlur o1 sfeat lere^as pessnrsrp e eq aA{

SaJEJJnSuuBr,rraru uo suorlErJB^ I€rrssBIC Y xlpuaddV

A. Classical Variations on Riemann Surfaces

on U for every €. When lel is sufficiently small, z,(Cp) is a simple closed curve (actually an ellipse in this case) in the z.-plane, which is denoted by C. and z, gives a conformal mapping of a suitable neighborhood .4. of Co (see Fig. A.1). Now, delete Do from ft, and paste the domain D. in the z.-plane surrounded b y C . . M o r e p r e c i s e l ys, e t V r = D r U z r ( A r ) , a n d g l u e ( R - D ) U , 4 . a n d I / . b y identifying z,(A,) and ,4. under the mapping 2.. Then we have a family {r?.} of Riemann surfaces depending on the complex pa,rameter e, which is a special case of Schiffer's interior variation. Here, note that, considering % as a subdomain of ft., we can take ze as a local coordinate on V..

'1",,ffi-, i t-----',..rsi

I".

t,

v

z(A)

z.- Plane

Fig.A.1.

When severalmutually disjoint points, say ?r, ..' ,pn, a,regiven on .R, we take a coordinateneighborhood(Ui,zi) for everypj so that zi(Pi) = o' zi(Ui)-It eCllzl<2|, AifiUp=$, i+k.

i =r,"' ,n,

Set Di = z;r(12 e C I lzl < 1)).For any complexnumberei with sufficiently small leil, considerthe mapping

,(q uarrEq (U)g > 6)(!tg/f0) pue ,.lA uo Eurddcurerqdrouroloq e s! d uaq; ''rl = (r),1 ,tq I(U)A +- ful:g Eurddeur€ eugaq.g_rgC uf u6r.ro aq?Jo ,,14pooqroqq8raulpurs f11uar?gnse xl,{ .I.y lul,e.toeqJ-loloo.t4 'ur3r.roeql ol reau flluarcgns r fra,r,a.ro;

ro r'-i[fi-a)d

'o

\ _,o.,,, \ /

t-69,...,I - g'fq)d,!rp/!zplt )

'a o+a;o Eurddeur e sr,/ reqr.^"#::rfr"rt".ffit:'i.1TJt+ l"uroJuof,rs"nb 'to g-ft'...'I

"'-1[11-s3 d, =X'!g3d

'o

_

\ " , ( d ) ! z o ) ' ! t o r _ 1 t ' \t r 1 ,o,,, 1

' las a t fg ;o ,(repunoq eql uo r' lz qlr^r luaprruroc pus , {g uo snonur}uoc '{f > l14l C) r} = lguo *Cssel)Josr,'f qceg.(g-Oe,..T -) f,fraaaro; lZb+lz=(!z)t'll '1'y uaroeql goord e aar3 o; 1nd aaa 3o '7 '[Og-V] E .raldeqp ll pue '[6/] reurpreg osl" eas 's^{olloJ se uorl"ruroJap e fq uorle leurroJuocrsenb -Ir€A rorrelur s(ragrq?S Eurluasa.rdar ,(q uaaoqs fpsee sr llnsar l"crsselc sllJ 'o.tazfr41ocryuapt st ld fi.r,aaa7o |utysruoa (U)zV ur-lueutap fruo tt fr1uopuo tl salou.proo?1oco7ctrltLtou.tolorl lo tualsfis o sea$ rol ['AJ +--< t |utdilout ay! lo ?sr?aul at17,l,aaoa.to14J ,t_=ol{!d} ('pa47nuoy.totacuaq s, q?nln '6uu1.r,otu pau{ap Qp.tnTou ?qt yryn paddnba sr,'g ,atag) .@),t 1" Tutodasoqay7 punorp se?ouNpron7nco1cttlil.toruolorllo ruaTsfisn saat6 b = (A)l oyu.t 14 lo

['u]*------,, 0uttlrlotuaqyto as.taauteq?ueqJ .g - t lo (e_7,eC))tA poorl.r,oqq,tau lotus fr17uarcfinso ul , puo ,t_=u[{tayu,o$ a.rolaqsD pal?nr1,lsuo) saeoltns uuowary lo filptuoteto ?q {'alt 7a7:ior\ipuoc |urmoylo! ay| sa{nlos ycryn ld qw^ sTuroil i sTsrzaer?qt uayJ 'i1uo.r7tgt, uaarf aq 't h) 1is lo ,\[{ra} fi.r,aaa.to! {a Io h pootl.roqq|nu o puo 'g "o sTurod..urrsrp fi1lonlnu ,t-;[{{a} g-tg IeT '(?,7) 6 snuaf to ecottns uuouLerypesop D aq A IeI .I.V uraroaql ('FZt] ll"d

aas 'acuelsur

.rog) 'leluaurepur-rJsl r.uaroaq?3uralo11o;aq1 ,,uoN .(rr, . ..,Il) = r s.ralatueredxalduroc ,tFu"J e lcnr}suoc uee a/$,,aro;eq sy

uo Surpuadap se"eJrns uueruarg Jo {'U}

'ts)d

''T *(d)tz=tr, uorl"u?A rorreluJs.ra$q)s '['Y

9t7,

A. Classical Variations on Riemann Surfaces

236

,\ ld4/dri, ritPl=f o,

p€Di p€R-D1.

On the other hand, by using the meanvalueproperty we obtain

ll_rr,=Il,,

tlt(21)d.21 AEj

- -2rir!(pi).

Here,writin e {, = ,lriQi)dzf on Bi, we set r!(p1) = /i (0). Now, to prove that F is biholomorphicin a neighborhoodof the origin, it sufficesby the implicit function theoremthat {p1}f!;3 gives a basis of the tangent space?6(?(.R)).Since7r("(r?)) is identifiedwith .42(.R)-by Theorem 7.5,this is equivalentto assertthat any complexvector("r, "' ,csc-s) satisfying

(8",,,,r). -0,

$eA2(R),

should be zero. The second assertion is seen easily by linear algebra. As {pi}}o=1" in the first assertion, we can choose a set of points such that det(g1(pi)) f 0, where

tr

ler\?,!=1"is a basefor .42(,R).

A.2. Period Matrices as Moduli As stated before, the first introduction of the complex structure of To (g ) 2) was based on investigations of period matrices. We shall show Rauch's variational formula for the period matrices, following Ahlfors [5], and explain how to get local coordinates of ?o by using this formula. First, we recall some fundamental terminology. Let R be a closed Riemann surface of genus s (> 2).Fix a set of 29 simple closed curves on ,R which induces a canonical homologg Dcse,i.e., a canonical base of the first homology group H1(R,Z). In this section, we use the notation {Ai,Bi}ni=, for this set (seeFig' A.2). of simple For every [S,/] e "(ft) = Te1 we have a set {/(.4i), f(Bi)}tt closed curves on .S which induces a canonical homology base on S. We denote this set by the same notation {,4i, Bi}oi=t. Next, on every ,S, there exists uniquely a set {di}f-, of holomorphic Abelian differentials, i.e., holomorphic l-forms on ,S such that

- 6i*, j,k = 1,"' ,9.

Io.t'

we call this {di}f=, the canonical base of lhe space of holomorphic Abelian differentialson S with respect ro {A1,Bi}oi=t' F\rrther, we put

('pt7ua,ta$tp ctTotpanb o so paptofie-ts9 t6lg Tcnpo.ttl aqt 'any)

'd . tgtg

aff

I I = lt[o(rfrlp)

JJ

qonbe puo 'sqstaaTutod esoq ?qt 7o rt uo4cer,rp eW u? rly frteaalo frllo(tftp) aatpauap aq1 '*g ssolc to (U)A a il fr^r,aaa.rogr(elnru.roJ lBuo.rlBlru^ s.qcnuig) .g.V uo111sodo.r4 '1urod aseq eql ,1urod es€q aql 1e qql e^oJd ol sa?gns 1r Jo uorl€lsuerl e Ag 'il yo itlgqellueraJlp xalduoc raoqs alu ,uorllasse lsrg eql elord o; 'zalo '7, - 6 uaq74 Tutotlfinaa 7o t lua.r f)u,wotil eqt soq IIp 'sacopns a4fu11andfrqo7 iutpuodsa.uoc esoql ut TilacxeEalo Tutorlfruo 7o g- 69 tlurv eql soq ea4oauep eql u?!1 ,?, < 6 uayn ,taaoatoy4l lorutxou II,Io Up 'ctyil"toutoloy tg *- oJ t |utildnu eVJ .Z.V uraroaql ry II 'ruaroaql 3uraao11o; erll Jo;oo.rd e all3 all ('[gZ-V] ery pue s"{reJ pue '[g-y] orres pu? sroJIqV ,acuelsur roJ eas 'slF?ap arour rog) .acods-t1ot1.tediln p,ary a{l sI (z/(r+r;aC )) rg ,e.ra11

'(u)-r> [/'s] '(s)z= ([/'s])z ,tq paugap og *- (a),J,n Surddeur € urelqo am 'eaue11 'elrugap a,rrlrsod q (S)Z yo {reurteur aqt pue ,cr.rlaururr(r q G)Z 1.red t"r{t ol lcadsar 1eq1 sarldurr uorl€lar por.rad lecrss"lc eql lsqt il"rsg 't=[{l7,lv} rllll'A ,S lo rulotu pouail loe,ruoao?aq1 (rfz) = (S)Z x-rrpur 6 x t $ql iltsc e,1t

' 6 (. . . ' r= r ' f

"[ =,,o

'z'v'ttJ

'z'v InPon s" sarulel{ PoFad

Ltz

A. Classical Variations on Riemann Surfaces

proof of Theorcm A.2. From Proposition A.3 and Ha"rtogs' theorem (cf. Bers [A-14]), we can see the first assertion. Assume that g ) 2. Then the classical theorem of M. Noether gives that, if ,s is a non-hyperelliptic closed Riemann surface of genus g, we can find a base of Az(s) among the set of products of two holomorphic Abelian differentials. Hence, by the same argument as in the proof of Theorem A.1, we have the second assertion. Finally, when g = 2, we can see directly that the set of products of holomorphic Abelian differentials spans .42(R) for every closed Riemann surface S n of genus two. Thus, we conclude the third assertion. Furthermore, Theorem A.2 implies the following: Corollary. The complex slrttclure of To introduceil in Chapter 6 is lhe unique one uniler the condition thal the canonicalperiod matrix nloaes holomorphically on To. Remark. Besides {r1x}, Ahlfors consideredintegrals of holomorphic Abelian differentials along suitable l-chains, and succeededin introducing a system ofhole' morphic local coordinates at every point of ?o (cf. Ahlfors [5]). This was the first introduction of the standard complex structure of ?r.

Proof of Prvposilion /{.9. Fix a smooth Beltrami differential p € B(R). For every complex number e with ll.pll- ( 1, let f , : R ---- E. be a quasiconformal mapping with complex dilatation ep,. Let {|i,r}t=t be -the canonical base of holomorphic Abelia,n differentials on .R. with respect to {Ai,Bi}J=t. Fix j arbitrarily. We set w = ( f , ) * ( 0 1 , , -) 0 i , o , where 1{]..7.@i) is the pull-back of 0i,,bV /.. Then c,ris a square integrable closed differential on .R, and we have f

t

l

I , = J|A r e i , , -JIA x 0 i , o = 0 ,k = 1 , , . ' , 9 . JA* ffence,the period relation impliesthat (r,r*) = [[

-uAa=0.

(A.1)

JJR

Let zrbe a generic local parameter on .R.. We write 0i,, = ai,r(zr)dz, with holomorphic function ai,e.Letting z be a generic local parameter on -R, we decompose c,.ras (A'2) u)= ur + @2;

',{larrtlcadsarrzU IUr l="1u11+t0g,q+t6v} pue uo selrnc pasolr alduns Jo sles aq ,1srrg pue t pue'sacegrns zgr uutsrueql-pasolc oml eq pueig, rf{fS,(fy} 1a1 'g xpuaddy eeg 'aceds rTnpou aq1go {repunoq eql o1 Sur8rarr,uoc saouanbas;o adflolo.rd e serrr3pue 's3urddetu IeuroJuo?rsenbo1 enp asoq?tuor; }uareJlp fll€ll -uassesr uorlerr€A uq; 'paleErlseaurfldaap uaaq a^€rl 'sace;.rns;osuorle,rauaSap pelF)-os 'sace;rns uuetuaru Jo uorlerJe^ Ielueu€punJ pu€ Iscrsselc raqloue sv

sareJrns uueruerll Jo uorl€rauataq

tr

.g.y

's^tolloJ uotlress€ eql snql

-ll'/'ll- t '(rlrl)o= . ll0'rBll . > llo'{dll

TIitiEl

Ptul l,tr''ro'rsa[[l; JJ I I

urelqo aan'(g'y) urorg .qtr.t il . o't6o'tt6" + [ [, J J

=16;rfi - e),rt:t s? a)uereJrp srql ssardxa all

.zov 0"t0"[[

=(g*

,o'rr;- = (9)r{r - (a)rt:t

J J

eleq atr 'uolleler porrad eql ^q ure3e ,,t1eurg

(qv)

#ffifu>l,,ll

'(g'V) dq '.re1ncr1.red u1

. _ l l r r r l l _/ rl;l r o l l llo'toll

(r.v)

?eql (Z'V) uorJ ^\oqs u€? e,lr ,{lrlenbaur el3uerrl eq? pu€ (g.V) fg 'zorlt - co a^?q e^l uerlJ 'zp(z)"('l). )'lp = zO @)'I o ' lxeN las a^t 'f, qcee rc1(fo'!o) = ,llroll eraq^\

'"llznll = ,llrrll

(s'Y)

o1 luep,rrnbeq (I'v) l"ql slrorlsuorlelnduor eldurrsy .zp(z)'('l) . )'!o- zo ?)'t o ,zp ((z)o'{o- (r), (l) . (t)'l o '' fo)= ro setr"Jrns uu"ruerg1ouorlerauateq.t.v

6tZ

A. ClassicalVariations on Riemann Surfaces

240

which give a canonical homology basis on Er and ft2, respectively (see Fig. A.3). Here, gi (i =1,2) is the genus of Ri, which we assume to be positive.

R" (s":2)

R, (g,:1)

Fig.A.3.

For each j, fix a point p; € Ri, and a coordinate neighborhood, (U1,zi) - Bi = around p;' such that zi(p)= 0 and ti(Ui) {z e C lkil < 1 }. For everv ( 1, we set complex e with 0 < l.l

( J i , , = U-i { z e C l l r i l < l . l } ' i = 1 , 2 ' Then, identifying U1,6 and Uz,, by the mapping ZL' 22= e,

we obtain a Riemann surface E. of genus g = gr* 92 (see Fig. A.3). Note that

s>2.

when 6 = 0, we take as .Rothe closedRiemann surfacewith a node (which comesfrom the identificationof p1 € ftr and Pz € Rz'Also cf' Appendix B)' Thus we haveconstructeda family {8. I l.l < 1}, which we call a degeneration to Ro with respectto (t/r, u2). on every -R., we considerthe canonicalbase {0;.rlsr:, of holtmorphic Abelian differentialswith respectto {Ai,Bilt=r, and' canonicalperiod matrix II(e) = (r1r(e)). Furthermore,the following ;#;;;h" va,riationalformula is known. (For the proof, seeFay [A-30], and Yamada [261].)

'[I9Al spelus^ uI I fre1o.ro3 ees 'Erelap aql otut a^lep lou op ain 'rarraiuog .r ol lcadse.rqtur turel rapro ?srg eql pue '0a pue X Jo senlsA eq1 dllrcqdxe urlrop alrr^\ ppoc eA\ 'lur,lsuoo elqDpnso s, b puo '.toycaa tg uo tt"tToutpouad ,ata17 f)uorsu?utrp-(t -n) o c! X ID)tuouD? ayq st,oy

. ( o * l , l )e ) o + f o e + ' j q + = t , l z # - u l) ^t L

.g.Y uraroaql ([196] epeu"A pue '[Og-V] feg ya) ur\ou{ sr €lnturoJ l€uorterrel 3urno11o;eql ueql '(r)7 secrrleur por.rad aql auuep erlr ,7.V .3rg ur se lerruouec ug pw ty Sursooqg 'f snueS;o e?eJrnsuueruarll pasolc€ sl ,gr uaql ,g t;1 f .?.V'EIJ

'(zn'rn) o1 lcadsa.rqll/'^ ry oluoNllreu?fiapelpt osls e!$,q?!q,!\'{I > lrl | ,U} dltureye lcnrlsuoc uec e^ 'a.ro;aqs-efem etu€seq? ur 'uaq; 'Q = zn U r, leqt Pu€'f qceeroJ{r > | tzll c ) z} = (frilt, pue 0 - (!d)!z wqtqcns 161 = .r; fd puno.re (!r'12) spooqroqq3reu alsurproocpue 'g uo zd pue ld slulod 1?urlsrp 'eseq oral xld ,(Eolouroq Iecruousre secnpurqclr{rt U uo se^.rncpesolceldturs;o pue '(t <) t - f snuaE;oaceJrnsuuerueqlpesolce eq Urlel las e eq FI{\A'fy} 'flesreard erotrq'feal Jelrrurse ul zU = IU leqt aspJerll l€erl uet a,u,,1xa11 'lg uo qo4ua.tafitpuDrIeqVcryd.totuoloy asoq to r=,ro{t'!61 a.teyn'(!6' ...'I = {) l2 uo lzp(lz)r'to - t'!6 yt!,n f)cruoun?aq1st

.'.'(o ) t' r o=) r x ( (o )'o 'rr' p u o ' f g u o e u l o u t p o t . t a dl o r r u o u o ?a q 1 s 9l y ' 2 , ' T =

'(o-lrl)

t qcna.tol'ate11

t X z X'l't"z'1 | z r r n l =(')r ( . r'u lo+f- 9 Lzxtx, o L ; '|rl

'7'V rrraroaql

rt7,

saf,"Jrns uu"urarlf ;o uorleraua8eq't'V

242

A. Classical Variations on Riemann Surfaces

In the describeddegenerationto ,8o,if we take aspecial (Ut,Uz), and restrict e to the set {qtz | 0 < t < 1}, where 4 is a constant with lql = 1, then we obtain the so-called Schiffer-Spencer's aariation (cf. Schiffer and Spencer [A-94]). A variation corresponding to the second case is called the uariation by attaching a handle, which we shall explain more closely. Let .R be as in the second case. Fix two distinct points p1 and p2 on .R. Fix also a point po € R- {pt,pr}. Then there exists uniquely a harmonic function G ( p ) o n R - { p t , p e } s u c ht h a t : (i) G(ps) = [, (ii) G(e) -log(I/lz{p)l) it extended to a harmonic function in a neighborhood ofp1, and (iii) G(p) - loglz2@)l is extended to a harmonic function in a neighborhood of P2, where zi G(p) the negative Now,

is a local coordinate with 4(p1) = 0 at pi for each 1. We may call this Green function on R normalized at ps with positive pole at pl and with pole at p2. for a sufficiently small positive t0, we put

u,= and

a c6y' toe! r,o,r, } {oe I

Uz= {pe n I G@)< logts}u {pz}.

Then [/1 and U2 are both simply connected domains. Fix such a t0' and take these domains as U1 atd Uz in the second case. As a local coordinate, say (j ' on Ui, we chooseone such that R"(i(p) =exp((-1)iG(p)),

j =r,2.

Fix a complex constant 4 with hl = 1, and consider e = qt2 for every t with 0 < t < ts. In the sarne way as before, we can construct a family {&,r, | 0 < l ato\. This is Schifer-Spencer'suariation bg attaching a handle.We know the following classical Schiffer-Spencer'saarialional forrnula. (For the proof, see for instance Schiffer and Spencer [A-94].) Theorem A.6. Let Q1 and q2 be arbitrary d,istinct points on R- \po,pr,pzl,Let g1(p) be the Grcen funclion on R tz normalized,at ps with positiae pole at q1 and with negatiuepole at q2. Then 1

ct(p)- go(p)= #*"tol(c(c')

- G(qr))+ l2Re(qc)+ o(tz) (t * o).

is locally uniforrn on Hew, c is a constant independ,enlof t, and lhe conaergence R- {Pr,P",Qr,9z}.

'g xpueddy osp ees 'sacedsllnpour peg -rlceduroc aq1 fpnls aql ur ;o Iool plueuepunJ ? sr sa?€Jrnsgo uorleraua3aq .ec€Jrnsuuelueru fre.rlrq.re ue .lvzzl pu" [966] rqcnSrueJ eas roJ prle^ IIIIs sl elnuroJ s,racuadg-ragrqog'rarroaront '[qUZ] pue IVZZIrqcn3ruea aql roJ araq pe^rJap esoql ol relnuls selntu eas 'sacrJleur porrad l"f,ruouef, 'suorle.raua3appue suoll€turoJep -roJ leurro;uoctsenb leuorlerJel urelqo u€f, pu€ SurleureEleure dq suorlerrel raprsuoc u€) e/r\ 'sace;tns uueruelg frcr1tq.re rog ';c) '([gZA] rqcn8ruea pue f11uercgns,, ,,a3.re1 [IfI] e{e}qo q ar l€ql erunsss plnoqs a,u '(ar)J;o lurod aseq aql Jo PooqroqqErau e sra^oc {A

-U uo leuroJuor q.f (U)-r > | [/'S]]

'eldtuexe 'ueqA 'U leql epnl)uoc ol repro ur Jo lesqns e eq g 1a1 roJ 'ploq fgressaceu lou seop I'V rueroaql se uotlJesse u€ qrns 'leuorsuatutp flalrugur sr U a?€Jrnsuu€tuerll uado uefo (U),2 eceds rallnuqrletr eql uaq1\ '[696] rqcn3etue pue 'lt1Zl'lgZel rqcnSrue;'[eOt] '[fOt] eqlqs'FtI] Iue]IetrAtr alrc osp aal'sace;.rnsuusruorg le.leuaEJoas€c eql ur suorlerrel roJ sy'[661] ruel '(p)g -r"tr{ pue l{ounsny errr€lsur roJ aes eceds Jellnurqclal aql uo i(lerrqdlour 'sl€rlueragrp u€ITeqv ctqd.rotuoloq -oloq eloru ..secrtrleruporred Iecruouec,,eseqJ Jo ror^er.leqdrepunoq aql uo uorlrpuo? elqelrns e Sursodurr .{q ,,xtr1eur pouad 'g u" roJ ualg ace;.rns uu"ruerp frerltqre Iecruou€c,, eql Jeplsuoc uec eA{ 'rl 'ralano11 'rl lerlua.ragtp le.rauaE" roJ sploq llrls llnser eq1 rurerlleg qloous e fpo reprsuof,a,r,r'{lrcqduls Jo a{ss aq} roJ 'g'y uotltsodord uI

seloN w Z

saloN

Appendix B Compactification of the Moduli Space

Following Bers [32], [33], [34], and [40], we shall construct a compactification of the moduli space Mo of closed Riema^nnsurfaces of genus g by adjoining to Mn the set of biholomorphic equivalence classesof closed Riemann surfaces of genus g with nodes.

8B.1 Compactiffcation

of M1

As an example, we construct a compactification M1 of the moduli space M1 of tori. As was seen in the remark in $2.1 of Chapter l, My is identified with the complex plane C, and every point in M1 is represented by the biholomorphic equivalence class [.91] of the torus ,S1 defined by the algebraic equation urz = z(z - l)(z -,\) for a complex number ) with ^ + 0,1. A compactificationM1 of M1 is the Riemann sphere e = CU{-}. Ilere the point oo in e corresponds to an algebraiccurve given by the equation w2 = z(z -l)(t -,\) for ) = 0, 1, or @.

)-n

degeneration -------) (l-0)

Fig. B.1.

For example, taking I - 0, we see that the algebraic curve ,9o given by the equation w2 = z2(z - 1) is the one which has a sole singular point at po = (0,0).

'(Z'g'31,{ aas)g uo epou qrea Sutuedo fq sapou lnoqltm d' snuaS;o og acegrnsuueuralg pasolce 1aBe.n l"ql sueau (rrr) uorypuoJ'qrou.tey

r=!

' { - I + - * ' ' ". 7 r .- o uaql '.{laarlradsar '9, yo slred pue sapou sraqunu er€ pu" u/ JI (II) Jo { 's1 '.i? ec€JrnsSuuarr.oc 1eq1 IesJeAIunaqt seq fg, i0 < lu + Z - !62 segsrles pue '16 snua8 go e?€Jrnsuueruelu pesol? e uro{ fu Sutaoruar,{q paurelqo sI qclq.t\ eceJrns uueuratg e ''a'r slurod l)ullslp '(!u'!0) ed.{1e1rug.{11ecr1,{leueJo e)eJrnsuu"urarg € sl Ugo fg, 1.reddleag (u) (g 1o 1.tode pellec sr sepou slr II€ snurur g;o luauodtnoc palf,euuoc V 'ellug sl U Jo sepou Jo u, raqunu aql 'lf,edruoc sr U ecurs 'Ur > ltrl P,tn t > lrzl I C x C lo epou € pell€r sr d'asec ratt€l aq?uI)'{I ) (zz't7) ) rasa{t to {I > lrl I c f z} {slp }Iun eqt o1 crqdrouoauoq er€ sluetuale esoqn\ spooq.roqq3reul€?ueuepunJ Jo ure1s.{ss seq U 3 d ,{re,tg (t) :suorlrpuo, aa.rq1Sur,r,rolloJ aql segsrl€sA I sepou q?!n 6 snua| to acn!.tnsuuDurexv pasop p, palpt $ gr ecedsJropsneH pelf,euuot lceduroc y '(Z 7) 0 snua8;o sa?eJrns uueuttrergpesol) Jo 6W aceds qnpour eqt Jo uorlecgrlceduroc e ltnrlsuoc eA\

( z= )

6 "roy 'W Jo uorlergrlceduro3

g'g

'og se papre3ar st uotlecgtlceduroc lutod-auo s1t pue '{ o1 fqecrqd.rouroloqrq sl -J/C e)eJrns uueruelg eql luelearnba O C } 'g rcJ rl q3norql oo r se lerll ees a.!r ;o uorleraua3ap aqt ,(g paurelqo sl '3lJ uI €ar€ pap€qs eq1 3ut1e; '(Z'Z)lSa roJ J urerrropl"lueur€punJ e se 6'I 'H ) r aruos roJ I + z - (t)'g't * z = (z)',L suorlelsuerl o.rl r(q pale.reua3 dnor? acrllel aql eq | 1a1:ssacordSurarolloyeql fq peurclqo sr 6,_;rdnor3 srq; 'I + z - (z)9 uorlelsuert eql .,{qpele.raue8(C)l"V yo dnor3qns e sl -J eraq^r '*J/J eceJrnsuu"uraru eql ol spuodsauoeC f oo lurod aql ueqJ,'I .ra1deq3 e.tr '1xetr1 eceds luarlonb ei11qtlm I,ZV,ty11uap1 Jo I'Z$ ul s (Z'dlsaln '0 od Y5'uo * y e olul se epou a rl s)lool (y)Iy a^rnc pasol) eJo uotlerauaSepaqt,tq paurelqo sr 1r pu"'I'g'3lJ 'uorlerr.rasqosrql uord 'epou to oS: l€r{l ees a^r 7ur,odalqnop fuoutplo ue pellel sl ,9Jo odlurodreln3urse qcns'peyluapr are 0 - ,Mpue 0= Z slutodorr,rl areqr'n'{ t > lUl I C > U} p"" { t > lZl I C > Z} rr{"lp o^rl Jo uolun eqt s€ papreSarsl o5lslql'od;o pooqroqq3reue ur'snq;'?uelsuof, e,rrlrsod" $., ereq^\

'{t >

l u l ' t > l z l ' o - - t t 4 zI c x c > ( u ' z ) }

,(q paluasardar sr og elrnc cleJqa3p aq1 'I - zfz + n = .1A pue I - zfz -m = Z uorlsruroJsuerl eleurprooc eq?.,(q'od;o pooq.roqq3raue u1 614Jyo uoqetyrleeduro3 'g'g

9VZ

246

B. Compactification of the Moduli Space

node Fig. B.2.

A homeomorphism f : R --- S between Riemann surfaces with nodes is said to be biholomorphic if / induces a biholomorphic mapping of Ri to a part of ^9 for every part .Ri of .R. If there exists a biholomorphic mapping of .R to ^9, then rR and ,S are said to be biholomorphically equiaalent.We denote by [E] the biholomorphic equivalence class of a closed Riemann surface r? with nodes. As a compaclification Mo of Mo with g > 2, we take the union of Mo and the set of all biholomorphic equivalence classes of closed Riemann surfaces of genus g with at least one node. Now, we define a topology on Mo by using the Fenchel-Nielsencoordinates as follows: let r? be a closed Riemann surface of genus g with rn nodes and ,b parts. As stated in the preceding remark, we can take a closed Riemann surface .R, of genus g without nodes and a system of decomposingcurves L = { Cr, . . ., C"o_"} on -R, so that rt is obtained from fto by degenerating each element of a subset {Cir,...,Cj^}of,Cintoapoint.Denoteby(t.,0)=(4,...,Lss_3,01,...,|sc_s) the Fenchel-Nielsencoordinates on the Teichmiiller space?, of genusg associated with .C (see $2.1 of Chapter 3). From the proof of Theorem 3.10, for any point (1,0) e (n+;sr-s x R3e-3, we can construct a closed Riemann surface R2,sof genus g such that ,Ra,einduces a point in ?o whose Fenchel-Nielsencoordinates are(1,0). Here, we admit the casewhere some/r', ,. .. ,lj^ in { /1 , . . . ,lsg-z } vanish. In this case,we get a closed Riemann surface R2,9of genus g with n nodes by a construction similar to that in the proof of Theorem 3.10. (We consider that each elem e n t C 1 r , . . . , C j . i n , C d e g e n e r a t e s i n t o a p o i n t o n R l p .T)h e n . R h a s t h e F e n c h e l N i e l s e nc o o r d i n a t e s( l ( R ) , 0 ( R ) ) = ( 4 ( R ) , . . . , 1 2 c - s ( R ) , 0 t ( R ) , . . . , d s c - r ( f t ) ) with /i,(ft) = .-. = lj.(R) = 0. For any (.,6) - (er,.:.,€Bs-2,61,...,dec-s) with positive €i and 6i, the (e, 6)-neighborhood of [^R]in M, is given by the set of all biholomorphic equivalence class [.R1,e]of closed Riemann surfaces r?z,awith or without nodes satisfying the following two conditions: forallj-1,...,3s-3,

(i)

Wi-li@)l(ei

(ii)

l9j - 0j(R)l ( 6i for all j with 4@) + 0.

'.(a)pow fq rDlnpoutJellnun!?rry aq1 dno.r8srql l! alouap pue (g)41o dno.t0uorTonuotsuo.tT 'A IIetr e1ydno.r3 e uroJ A : rn srusrqd.rouoeuroq 3ur,r.rasa.rd-uorleluarro1p ,fq pecnpur (i[nl] sEurddeureql'[U - S:looll ot [,U - g:/] spuasqrlq/" (U)d * (U)A:-[o/] Surddeu € sa?npul U * ,A:ot uotl€ruroJep3uo.r1s,,(.rarrg

'a

Jo (a)J

eceds rallnurqclaf eql qlra peyluepl q (U)C 'sapou ou seq U JI 'g' t- S I t suorleuroJap Euolls Jo [U - g:,f] sasselc acualenrnbeIJe Jo las aqt sl U Jo (g)g acods uotTounotap 6uo.t7saq; 'flairrlaadsar ,f1r1uapl aql ol pu€ Surddeur crqdrouroloqlq s ol crdolouroq suuqdrouroauoq alrlce[q "* orl pue q ereq^d et

'a*zS

" vf l

^

l'/

u*v

ts

urer3erp elrlelmuruot e sr eJer{}l\ ryel -oatnba eq ol pres ere F zg :zt pue .- r.9 :I/ suorleur.rogep3uor1s om; U U 'f ualr3 e ro; f snuaS;o sepou qlr^{ saceJrnsuueurarg pesolc leurur.rel fueur flaltug fluo 's8urddeur crqd.rouroloqlqof dn 'ale ereql leql eloN 'sepou g - dg ,,(1aueu'sepou Jo raqunu alqrssod 1se3.re1 aql seq ll lr lvuttuel sr p, fes e11 'rusrqdrouoeuoq Surarasard-uorleluarro ue sr Ur Jo sepou 11eyo sa3eturesrelur or{l Jo }ueuelduroc eqt of /;o uorlcrrlsa.raqa (rrr) 'sepou 11eSurpro,reg uo alrn) pesol? aldurs e ro apou € sl apou e;o a3eur esrelur eql U S'Jo Jo 'Ur epou € st Jo S'Jo apou " 3o a3erur aq; (r)

(rr)

:suorlrpuor aa.rq1Sura'rolloJaql sagsrl"s llJl U lo uotyout.totapfuatTs " pallec sl U * g :/ uorlcatrns snonurluoc V'(Z ?) f snue3Jo sapou qlllr sec€Jrnsuu"ruerlr pasolt o^rl eq,g pu€ A p"l 'sePou qlrAr sal€JJns uuel'ueru ,{q pacelde.r er€ sereJrns uu€tuaru pesol) 'sapou qlr^r ec€Jrns uu€tuarg e qf,rqa ur aceds .ra11nurq?latr Jo uorlezrlereuaSe go aceds uorleruJoJeptuorls eql lcnrlsuoc e,n'1'g tuaroeql errord o1 Jepro uI

sacedg uorlBruroJeq tuo.rls 'uorlras

g'61

lxau oql ur paureldxa $ ueroaql slqt Jo;oold e Jo eulllno uy 'acods $.topsnog 6) 6W acodsaqa .I.g uraroaqtr,

'(77 Tcoiltuoco st'paquasep so pe?ctunsuoc

'tuaroeql Surr'ro11o; eql e^eq arrr/!!oN ''W uo fEolodolSropsntsH € sa?npur qcrqr*'t1,g ur [g] yo spooqroqqErauleluaurepun; ;o ruelsfs e a,rp spooqroqqSreu-(g'r) aiaql saredg uorl"urrolaq tuorlg'g'g

Ltz

B. Compactificationof the Moduli Space

248

We define a Hausdorff topology on 2(R) as follows. Let S be a closed Riemann surface with nodes, and C be a closed curve on a part S; of S. We set

.

tslCl= iglts,(C'),

where C' runs over all closed curves on S; freely homotopic to C, and lsr(C') is the length of Ct with respect to the hyperbolic metric on ^9i. We also set IIIP] = 0 if P is a node of S. b e a f i n i t e s e t o f c l o s e dc u r v e so n p a r t s o f S a n d e a Let C = {Cr,...,Cr} positive number. A strong deformation h: S' - S is said to be (C,e)-smallif it satisfies

(i) (ii)

l t s , l h - L ( C 1 ) lt-s l C l l l < e f o r 7 r , . . . , r , lls,[h-1(q)]l< E for all nodesq of S.

We say that a set [/ in 2(r?) is open, if for every [/: ^9* /?] € [/, there exists a finite set C of closed curves on parts of ^9,and a positive number e such that' whenever h'. S' -,9 is (C,e)-small, the point lf oh: St'.- R] € 2(,R) belongs to [/ (compare with $3 of Chapter 3). There is a canonical projection np: D(R) - Ms which sends [/: S * R] to [S]. It is seen that the canonical projection I/6 is a continuous open mapping. Let us introduce the Fenchel-Nielsencoordinates on a strong deformation space. First, assume that rR is a terminal Riemann surface of genus g with nodes * n] € 2(l?). It f-t@i) is a simple {ei}10=1t. Take an arbitrary point [/:,s closei curve on a part ,91of S, then we can choosea unique simple closedgeodesic Li on 516which is freely homotopic to f-t@i). lf f-t@i) is a node, then we p,rt f,i = f-r(Pi).In this way, we have a system L = {fi}1n=1" consisting of all nodes of S and some simple closed geodesicson parts of S. -t@i) is not a node, we set lf f

,j =

*rs*(L)e;ei,

(B.1)

where di is the twisting parameter with respect to tri such that 0 1 01 < 2t (cf. $2 of chapter 3), and ts*(Li) is the length of the geodesic.Li measuredby - 0. the hyperbolic metric on S1. If f-r@i) is anode, put zi It is shown that the numbers (rr,' . . , zzs-s) depend only on the equivalence class [/: S * r?] and the mapping of D(R) to C3g-s sending [/: S * R] to (rr,...,zzc-z) is a homeomorphism,which is called the Fenchel-Nielsencoordiis also proved that for every strong deformation fs: Rt + R, nalesoni@).It is a universal covering map onto its the induced mapping lfsl-:D(Rt) -D(R) is not a those image, the image being the set of [/: ^9 * -R] for which f-'@i) 1(p1) is not a node. F\nthermore, D(R') is homeomorphic to node whenever /o C3s-2. Next, the Fenchel-Nielsencoordinaleson the strong deformation space2(R') of an arbitrary closed Riemann surface ,? of genus g with nodes are defined as

pa)npureqrpue(a)auoaln?f, nrrs'",0*'ilT',Tfi ;T,':T"r:".1r:t'#rXTiit

'orqd.rotuoloqsl (("U)A)-[oI)U n ol uor]crr]ser stl Jl ctrld.tou.roloypa11ecsr (g)g ur 72 les uado ue uo uorlcunJ snonu?uoc e ''e'r 'eln1ln.r1spa3urr e seq (g)4 ueq; '('U)A Jo ernlrnrls xalduoc eql tuoq pecnpur arnlf,nrls xelduroc Iernleu e s€q ((A)A).10t] 'ecueg 'deru Eur.rerroc l€sralrun * q ((,U)C).[0/] - (A)At.[ol) 'rarroa.rotr11 'esuep eraq.nou rt (('U)C)-[ot] @)A pue (lI)@ ur ureruop € sr (A)A ,- ("A)A: *[o/] Eurddeu pacnpur eq] Jo (("A)A).loll a3eurr eq] ueqJ 'A - oA : 0;| uorleu.ro;ap 3uor1s e e{€I 'arn}rnJ?s xelduroc e seq (og)4 acuaq pue '('U)J aceds .ra11nuq)lel aql qtr^r pegrtuapl q ("U)@ 'uorlces snorrlard aq1 uI palels sV 'sepou lnoqlr^r ro r1lrrlrsnuaS aures eql Jo e)€Jrns uuetuarll pasolc .req1ofue eq U lel pue 'sapou lnoqlr/( 6 snua3 Jo eceJrnsuueruarg pesolo e eq oA p.l '3urmo11o; aq1 sr ty,ggo ern?rnrls xalduoc eql Jo uorltnporlur lsrg aql 'sfeiu, o.tr1uI 6W Jo "rt1rnr1. xalduroo eql ecnporlur ol ^roq,tgar.lq aqrrf,sapaM

otr_ll ernlrnrls Jo

xalduo3

p.g

'lceduoc ,\ 6try WqI rtfi suorlcafo.rd apnlf,uo) e^\ snqtr '6141rerot uW * (a)A; lecruouec aql repun "N'"' 'I;l7 Jo sa8erureql l"qt qrns (fy)4 3 fy slas lceduroc erc araql leql satldrur Z'g "rutuaT 'saleurprooc ueslarNlaq)uag Sursn 'ueqa 'd snua33o sepou ' " ' ' IU sa?€Jrnsuuetueru qll/tr e4ef ino11 lualearnbe-uou ? leurturel II€ '([6p] stag '3e) rasng .,tqua,rr3 sI eurural srql ;o yoord l u a r a J r pV ' [ 0 g I ] r { q e l e r y p u e ' [ 8 1 ] € r l r € s n r r qi 8 6 ' d ' I I . r a 1 d e q 3u r g e u u r a l ' [ 1 -v] "Uo{lqv 'eldurexe ro; (aag'Durutalronoc eql Sursn,(q pe.rlo.rd sr eruural srqtr

17o.rol 7 j

' E''I f t ' " ! = j saarw |uzsodu.rocap (13)7|ugfitstlos to {t-0tg'"''rCl

ula4sfis pql qcns 6 uo fi1uo o sorl 6 snuaf aco!.r,ns uuDurezy pasop fuaaa lo |urpuad,ap.J luolsuo? aatTtsodD s?smea.taq7'676 fi.to.t7tq.tD ro4 ,Z.B BruruaT 'euurtuel8urrr,lo11o; aq? peeu eM'6W;o ssaulredruor aql /t\oqs ol rapro uI 'usrqdrouoeuoq

e

s r ( e - r e , 7 , ' " ' ' r r r l ) o l [ r U * g : q ] S u r p u e se _ o e C o l ( , A ) A g o S u r d d e u e q ] ? t s q ] u^,roqsq lI '(t .ra1deq3ur 6'g €urue1 'gr) ,_rgC o1 crqd.rouroeruoqsr (,g')4 acurs '(,A)A {z uo q)uerq snonurluof, panpr'-a13urs € serl uorlcunJ aql 3ol }eq} saqdurr '(1'g) ,tq uear3 uraroaql durorpouotueql uoqt 'gyo apou e lou $ (ld):l;r'ara11 dql ar€ (t-68,2'"''rz) eret{.n [ t t - S : t t o o t ] J o s e ] e u r p r o o ru a s l e r N - l e q ) u a e ',U Jo "po.t e lou fl (fd)r-o/ ft 'g;o apou e sr (fa)r_o;"r,

seleurProof, uaslarN-laqrueJsll'(U)A

> [,U *

lzSol= lm lz - lm 'tq uartr3 ale (e -fe61 ' ' ' ' 'rm) g:r1j lurod {rara ro;:s,llolloJ 6.;ig arnlcnrlg xalduro3 'p'g 3:o

6VZ

B. Compactification of the Moduli Space

250

Using the Klein-Maskit combination theorems (Maskit [A-71]), we have the following theorem (cf. Bers [40]). Theorem 8.3. D(R) is a comltler manifold and is rvalized as a boundeddomain in C3c-3. Now, we get the following two results. Theorem 8.4. The Teichmiiller rnoilular transformation group Mod(R)* is a discrele subgroup of the analgtic automorphism group "f D(R). Moreouer, the subroup Modo(R)* induced by the biholomorphic mappings of R onto itself is finite and is the stabilizer of fid: R- Rl in Mod(R).. Theorem 8.5. Therv erists a neighborhoodN of lid: -R- R] inD(R), inuari' ant uniler Modo(R)*, such that the quotient spaceNf Mod"(R)- is homeomor' phic to a neighborhoodof lR) in Mo. By H. Ca.rtan's theorem, the quotient space NlMod"(R)* has a normal complex space structure. Thus M, becomesa normal complex space, and it leads to the following, the main theorem in this Appendix. Theorem B. 6. The compaclificationMn of the moduli spaceof closedRiemann surfaces of genus C e_ D has a normal compler space straclure of dimension 3g-3. The second introduction of the complex structure of Mo is the following. For a given closed Riemann surface R with nodes of genus g, considering both quasiconformal deformations and degenerations, we construct Riemann surfaces which represent a neighborhood of [n] in Mo us follows. A s s u m e t h a t r ? h a s r n n o d e sp 1 , . . . , P m a n d & p a r t s f t r , . . . , - R 1 s u c h t h a t each.Ri is of type (Si,ni). Then it follows that

Dni=2^, Dor*m-k=9.

j=l

i=L

For each node po on rR, suppose that pa corresponds to a point obtained by identifying a point oo in E* -.Ro, with a point 6o in E* - fto, for some d1 and denotesthe natural compactificationof Ror. Take a o2 in { I,. . . ,k }. Here, { local coordinate neighborhood (Uj, zo) at oo on E[such that zo(ao) = 0 and z.(u:) = 4, the unit disk. similarly, choose a local coordinate neighborhood (U3,J") at 6o on R', such that too(bo) = 0 and ."(UZ) = 4. Further, take a relatively compact open set Vi in Ri for every i = l' .. . , /c so that I/ = l)!=rV m e e t sn e i t h e r [ { n o r U 2 "f o r a l l a = 1 , . . . , f f i . w e c a n f i n d B e l t r a m i d i f f e r e n t i a l sP ' t - t . . . , p 7 yo n Setting N = 39 -3+rn, pi vanishesidentically outside I/ for all j = such lhat R' R- {pr,...,pm}

' u . L"' '

(".rnn)n (""u f)

- ta = "':a

'{ l'"1t l(d)''"*ll''zn) '{

"' ? tl l" " l 5 l (d )" '" r ll )

d\ d| -

o's'an o's'nn

"''ro) I = 1 2 I I €r o J I > l " r l l e q t q r n s u c u r l u r o d e a q ( u o '

1nd a6 - D !3-I

'8'g'ttJ

\

sapou

A

x"uo uotleurro;ap cb

€-lsrd f uo!l€f,grluaPr

'g uo Dd epou eql o1 Eurpuodsa.rrocsgr uo epou eql t'Dd fq elouag 's1red I prre sapou u, qlr^t f snua3 ;o sgr ecey.rnsuueuerg pasolc e u p t q o a t ' l . ' u t r " ' t 1 = r c q ? " a J o J( " 9 ) ' / q l l ^ ( " D ) ' / E u r , t ; r l u a p(r8 ' { U ( " ' ' ' ' I U ruoq '8'zDUuo ("9)'/ yo (r-toorn'7n) = ('"n'?n) pooqroqqSreualsurprooc € 'flrelnurg 's'rDU uo ("a)t e^eq a/rd, Jo pooqroqq3raualeurp.rooce sr (tjto"z'"n) - (''or'?n) trqt ees e,lr'l.nlu'o leuroJuoc s "/ ecurg'"'fU ol fUSo Surddetu e o1 spuelxe lA ol'[ Jo uorlcrrlsa.raq1 pue'1!u-.nl) adfl elrug I€ruroJuof,rsenb "/ .repun {g, - ''!A a3eur f11ecr1f1eue lred e lo (!A)"t Jo ec€Jrnsuuetuarg " q aql '( {U)J t=jU ul lpl' ,Aho pooqroqqSreue oluo 6'go Eurddetu orqd.rouoloqrq ot s Surpues(ig)t t=j[ of,q g;o Surddeur aql t€q] erunss?feur e16

€ sl ["/'iU]

I=f 'lr1 lsf : c71 n, rua,saoc rur*rlrag qlr.ryr : Surdderu e slsrxa e.raq1'g ) (t"' "' .I") - s fue 7A *,A J FruroJuorrssnb ro;'uaqa 'n,C ul ur3r.roaq1;o pooqroqq3rau uado ,flluaralgns€ aq O ?eT lpurs '(y xrpuaddv ul I'V ureroeql t=jLJ "rnot :.:c) W?',U] ?ulod eseq aq? w (!A)l saceds rellnuqtlal f" (rU),f rN' ( ' ' ' (I lcnpord aq1;o eceds luaEuel aql Jo srs€q e ecnpur ,taq1 1eq1q?ns pue o7g;o arnlcnrlg xaldurog 'p'g

19Z

B. Compactification of the Moduli Space

252

Identify any two points c and b in .Rl,o if o and D a.re contained in Uj,,,o and(J!.r.o, respectively,for some a, and if they satisfy za,swa,s ao. By this identifiiation, we obtain a closed Riemann surface rR",oof genus 9 with n nodes, where rn - nis the number of o's with oo I 0 (see Fig. 8.3). Now, we set

6 = { ( s , a ) €D x C ^ | l o i < l 1, i =L,...,m}, D = { [ f t " , o €] f u o l G , Q e b ] . Then, 2 is a neighborhood of [R) in Mn' However, the mapping of 6 onto D sending (s,a) to [.Rr,"] is not always injective. By changing D suitably, we may assumethat the biholomorphic automorphism group Aut(R) of .Rinduces a finite group G, consisting of analytic automorphisnT "jD, sl'ch that the quotient space DlCts homeomorphic to aneighborhood of [n] in Mn . For details, we refer to Bers [33], XIII in $7, and Masur [143], $2, and Wolpert [249]' $4. As before, H. cartan's theorem implies that the quotient space Df G has a normal complex space structure. Thus, I4, becom"r a normal^complex spaceof dimension 3g 3. Note that this complex space structure on M, is equivalent to the one given in the first introduction.

8.5 Weil-Petersson

K6.hler Form on the Moduli

Space

Kihler formwylp on the Teichmiillerspace?(.R) of sincethe weil-Petersson genus g (= 2) is invariant under the action of the Teichmiiller modular group Mod(R), it is regarded as a form on the moduli spaceMo. This form is denoted by the same notation^uwp. we are interested in the behavior of u.,ryp near the boundary of Mo in Mo. From the construction of Mo and Wolpert's formula (Theorem 8.6), uvvp extends smoothly to the boundary with respect to the coordinates (t,r) = - 1,-..,39 - 3 and ( l r , . . . , l s g - s , T L , . . . , r s g - s ) ,w h e r e r y = t i 0 1f 2 t f o r X -(h,. . . ,tss-2,01, . . .,flsg-t) are the Fenchel-Nielsencoordinates asso(t,0) ciated with a system of decomposing curves on ft. In particular, M o h* a finite volume with respect to the Weil-Petersson metric. On the other hand, the boundary behavior of uw p with respect to the coordinates (s, o) given in the previous section is studied by Masur [143], and wolpert '[ 2 5 1 ] . Next, curyp induces a cohomology class [c.,szp]on Mo such that lusr p)lt2 is a rational class (see wolpert t249]). Thus multiplying [c..,szp]lo'by some integer, we get a line bundle over rt[o. wolp"rr l252l proved that this line bundle is positive, and consequently Mo is embedded in a complex projective space.Hence, we have the following result, which was first proved by Knudsen and Mumford [117] by using algebraic geometry. Theorem 8.7. The compactification Mo of the moduli space Ms of genus g (] 2) ts a projectiae algebraicuariety.

'[221]punaq pu" urqeg pue '[991]uosleg'[191]uralsqtogpu€ unrgerl'[09] tlqtg pu" auerC'[96] luedrg uu€tuerg .radns;o pu" roleqrl€g aas 'aldurexeroJ :parpnlsuaaq aA"rI se?eJJns sacedsrlnpou pue sacedsreflnurq]lel (/tJoeqlEurrls radns qlr^r uorlreuuocuI '[296] '[gzzj rqcn3ruea'[oo] pq!"n uI palPnls are eseql 'aceds tradtom pu€ rTnpouaql;o f.repunoqaql reau suorlrunJueerg eql pus uolltunJ elez Sreqlag eql Jo ror^€qeqaqt ^rou{ o1 fressacauq q erer{tr,'[911] ,ro1ef1o4pue '[291] uosleN'[92] ueparrg'[91] uoqap pu€ aun€C-zers^ly '[9I] ,ntnr'1y '[att-V] .,"L o1 .rap.raru 'crdo1Surlsaralurqql rod 'a1orluelrodtul ue feld sacedsqnpou pue sacedsrellnuqcrel froaql aql 'scrsfqd alcrlred f.roeql 3ur.r1su1 ;o ;o '[09I] '[qOZ] as€tntr pt" l [tOA]"tolqS pue '[16] elpr'neaS'[ft] o[ereqrY 'aldtnexato3:'aaslcafqns $ql rod 'sse.r3o.rd 1ear3 e epetuuaeqs€q e.raq1'sarlau€Auerqocsfuo uralqold s(Iqoc"f uo 'f11uacag '[qqa]'[oqz] pu€ '[9ZI] rauua4 '[gg1] 1.rad1o7y1 '[691] proyrunry'[UtI] u€lqc"le€I '[96] p.rolurnntr pue slrr€H '[qO] '[fO] slrrsg I '[gg] '[96] rar3eg prr" rarlag '[ZO]'[tO] '[66] re.reg '[99] 1.rad1ol\pue sSurpprC srrr€H pus pnqua$g '[gf] uralsdg pu€ rplrp^roS '[gt] €ql"uroC pue ollersqry se q?ns s.radedfueru ere areql 'ecedsqnpotu eql Jo frlauoa3 aq? rod

'[ee-v]

eare{ru"N pue '[ggl] uaprcN '[gg1] gqcnqey4lees'a.rourJeqtrnJl'[gat] nrX ol raJar

osp e \'lzgzJ'l'vzl1.rad1o,11pu€'[t?I] rnsew'[ot]'[te] '[eg]'[ze]srego] enpsr stql lsrs sealxrpuadde luerul€erlrno 'hq] proJurn4pueau3rleqfq pacnporlur ur uar.r3 saceJrns uue[uar]I pasolc go aceds rlnpou aqt Jo uotlecgtlceduoc

eqtr,

seloN saloN

t9z

References

Books and Proceedings [A-1] Abikoff, W. : The Real Analytic Theory of Teichmiiller Space, Lecture Notes in Math., Vol. 820, Springer-Verlag,Berlin and New York, 1980. [A-2] Ahlfors, L. V. : Lectures on Quasiconformal Mappings, D. Van Nostrand, Princeton, New Jersey, 1966. [A-3] Ahlfors, L. V. : Conformal Inaariants, McGraw-Hill, New York, 1973. [A- ] Ahlfors, L. V. : Complet Analysis,3rd ed. McGraw-HilI, New York, 1979. [A-5] Ahlfors, L. V. : Collected Papers, Vols. 1, 2, Birkhi.user, Boston, 1982. [A-6] Ahlfors, L. V. and Sario, L. : Riemann Surfaces, Princeton University Press, Princeton, New Jersey, 1960. [A-7] Ahlfors, L. V. et al. (eds.) : Aduances in the Theory of Riemann Surfaces, 1969 Stong Brook Conference, Ann. Math. Studies, No. 66, Princeton University Press, Princeton, New Jersey, 1971. [A-8] Ahlfors, L. V. et al. (eds.) : Contributions to Analysis, A Collection of Papers Dedicated to L. Bers, Academic Press, London, 1974. [A-9] Arbarello, E., Cornalba, M., Grifiths, P.A.and Harris, J.: Geometry ol Algebraic Curues, Vol. I, Springer-Verlag,Berlin and New York,1984. [A-10] Baily, W. L., Jr. ; Introductorg Lectures on Automorphic Forrns, IwanamiShoten, Tokyo, and Princeton University Press, Princeton, New Jersey, 1973. [A-11] Beardon, A. F. : The Geometry ol Discrete Groups, Springer-Verlag, Berlin and New York, 1983. [A-12] Bers, L. : Topology, Courant Institute of Mathematical Science, New York University Press, New York, 1956-1957. [A-13] Bers, L. : Riemann Surfaces, Courant Institute of Mathematical Science, New York University Press, New York, 1957-1958. [A-14] Bers, L. ; Introduction to Seaeral Compler Variables, Courant Institute of Mathematical Sciences, New York University Press, New York, 1964. [A-15] Bers, L. et al. (eds.) : A Crash Course on l{Ieinian Groups, Lecture Notes in Math., Vol. 400, Springer-Verlag, Berlin and New York, 1974. [A-16] Bers, L., John, F. and Schechter, M. : Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, Providence, Rhode Island, Providence, Rhode Island, 1964. [A-17] Besse,A. L. : Einstein Manitolds, Springer-Verlag, Berlin and New York, 1987. [A-18] Birman, J. S. ; Braids, Linles,and Mapping Class Groups, Ann. Math. Studies, No. 82, Princeton University Press, Princeton, New Jersey,1975.

'1161 'uopuoT 'ssard 'ecuatatuog f,ruap"rv lDuorlnlqruJ aflpuqutog g26y tsuotycung cttldtoutolnv puD tdnotg ep$st1 , (.p") .f .L4 ,rtarrreg [If-vJ '9961 'fasral a,ra11'uolarurr4 'ssar4 dlrsraaru1 uolaf,urrd 'ncoltng uuotxerA uo salnl?a1 : .C .U ,turuung [g7-yJ .gl,6I ,{ro^ a,rag 'i(a116 'fitlatuoag cntqa1ly {o sa1dnut.t4 : .f (sur"H pu" .y .d ,sqfUIrC [6t-v] '6961 'pue1s1 apoqg 'aouapr,ror4 ',(1aoog uerrraruv ,gl.lo1 ,sqderEouo141 lpf,ll"rueql"I I lecr ,santng -l"uraqlpl^l cntqa0yy ol uoq)n[nryul : .V a ,sqfUIrD [gt-y] Jo suorl"Isu?rJ 'p161 'fasral alalq 'uolarurr4 (ssar4 r(lrsraarufl uolarurrd '62 'oN sarpnls .ql"W .uuv bcuataluog puol -frron EL6I 'eacoltng uuDuarA puo tdnotp snonurluocsr1 : .1 .traquaarg [lS-v] '886I 'pu"lsl 'acuapr,ror4 ',(laroog 'ruor1 apoqg uecrraury Frrl"uaql"I4l .U 'V ,pp"tr[ pu€ .I^i .r14 ,ueurplog -oluatardag dnotg to frtgauoag : (.spa) [9t-vl 'zg6I 'l"artuow 'l?arluow ap ?1lsra^run(T ap sasserd sa1 'rlnrprronfi to taqndot4 ctlntalcDrDUC : .M .J ,turrqag [9t-v] 'lg6I '{ro rrag'da11q't1otlua.laStq cqotponfi puo ffnaqa rellnuq)EJ: .g .g ,raurprng [le-Vl 'ZI6I 'I06I '168I 'lr"qllnts 'rauqnal 'D 'g '(Z 'I) '1'1o71 'uauotpung II'loA : .g ,ura1y pu" .U ,arprrg [tt-vl 'I86I'{ro ua{dtoutolny r?p uoeqtr eW r4!!

ua0unnpoll

rt eN puta uryrag '8epan-ra8urrdg 'saco{tng uuoraery uo sarnpel ; .g ,ralsrog [ZS-VJ '196I '{ro1 natri 'easlaqg ''pg po7 'tuotlcung ctydtouto1ny : 'U 'T 'prod ht-v] '{ro ,rleN puu uqrag '8epan-raturrdg 'Zgt 'lo4 tl6I ''qleIt ur saloN ernlf,aT 'tacottng uuDuatq uo suo.Icunl optlJ | .O .f ,r(eJ [gS-V] '6tr61 'stre4 'aruetg ap anbr1eur9q1e141 'l,g-gg 'slo1 'anbsrrglsy 'arreururag fesr6 'tacopng n7 ?lg_r)os 2261-9 261 rns uo$rnqJ ap xnvaoil : .n ,nreuaod pue .g ,qcequapnsT ..V ,Hl"J [62-V] '086I '{ro^ l|^aN pue urlrag '3egan-raBurtd.g ,taco{tng uuDraatA : .1 ,rary pue .I I .H ,szrlreJ [gz-v] '(asaurdel) 9961'o{4o;'oqso;-of1o1 'tcttfit14 .1 ,uurrfoy pus .H ,pt l-zg laal?ouaqpry lo quautdoleoa6, : (.spa) ILZ-V] 'g961'atprrqure3'ssar4'lru11 atprrqureg 'acodg cr1oqtadnn .g .q ,uralsdg {o qcadty ".4auoeC puo 1oct1fi1ouy: (.pa) [92-v] '9961 'a8puqure3 'ssar6 ',rru11 a8prrqurcg 'tdnotg uDrurelx .g .q ,u1e1sdS puo fi6o7odoa : (pa) louorcueurp-noq [SZ-V] '886I '{IoA 1!leN Pu" urgag '3epan-raturrdg 'sarras lggyq tepan-raBurrdg 'acuara;uoC IUSI I 9g6I 'II p,t" 'slo1 .I" .O ,urser( llnpory puv cuoq)un.r utld.toutoyog : (.spa) ?e I [tz-v] '(lz6l'{ro^ ,{ro1,ta1q ,aruaosrc1u1,ncot n aN pu" uqrag 'tegan-ra8urrdg 'lurrdar) 096I ta1dnuu4 s.plqc.rr1 ; .g ,1uernog -rns IDurruW puo's0utddo141 Touttotuog [SZ-VJ 'Lg6I '{ro1 'sacoltng ,uqog malq'ra,roq uuvutetq uo iutddo1,y Totu.ro{uog :.11 [ZZ-vJ '086I 'uoPuorl Pu? {ro^ ,rra11'sser4 urnuald'fitaaq1 aatng ca1dutog {o qooqdotcs y : .H .g ,suaura13 [IZ-VJ '986I')IIo i aN pue uqrag 'tepan-ra8urrdg '(taun7o11 lDtrouery q?noy 'g 'g) stofilouy xaldutog puo fi4auoag ptyuetagtg : (.spa) .14i .11 ,seryeg pue .1 ,1a,rzqC [02-V] '9961 'a8prrqureg (ssar6 ',rru1 atpuqureg'uo1ttnq7 puo : 'V .S 'rapalg pu? 'f .v ,uosseg [6I-VJ uesp,N nlto

saco{tng {o tutsttldtouolnv

sef,ueralau

992

256

References

[A-42] Hirsh, M. W. : Differential Topologg, Springer-Verlag, Berlin and New York, 1976. [A-43] Hitotsumatsu, S.: Theory of Analytic Functions of Seaeral Complex Variables, Baihukan, Tokyo, 1960 (Japanese). [A-aa] Hubbard, J. H. : ,9ur les Sections Analytiques de la Courbe Uniaerselle de Teichmiller, Memoires of the American Mathematical Society Vol. 4, Number 166, American Mathematical Society, Providence, 1976. [A-45] Iitaka, S., Ifeno, K. and Namikawa, Y.: Spirit of Descartes and Algebraic G eometry, Nihonhy6ron-sha, Tokyo, 1980 ( J apanese). [A-46] Ito, S.: Introduction to LebesgueIntegral, Sh6kabo, Tokyo, 1963 (Japanese). [A-47] Jenkins, J. A. : Uniaalent Functions and Conformal Mappings, SpringerVerlag, Berlin and New York, 1958. [A-48] Jones, G. A. and Singerman, D. : Complex Functions: An Algebraic and Geometric Viewpoint, Cambridge University Press, Cambridge, 1987. J. : Tuo-Dimensional Geometric Variational Problems, Wiley, New Jost, [A-49] York,1991. [A-50] Kato, M. : Topology, Science-sha,Tokyo, 1978 (Japanese). [A-51] Kawada, Y. : Theorg of Automorphic Functions ol One Variable, Department of Mathematics, Tokyo University, Tokyo, 1963, 1964 (Japanese)' [A-52] Kawai , S. : Algebraic Geometrg, Baihfrkan, Tokyo, 1979 (Japanese)' [A-53] Klein, F. : On Riemann's Theorg of Algebraic Functions and their Integrals, Dover, New York, 1963. [A-sa] Kobayashi, S. : Hyperbolic Manitolds and Holomorphic Mappings, Marcel Dekker, New York, 1970. Yols' [A-55] Kobayashi, S. and Nomizu, K. : Foundations of Differential Geometry, l, 2, Wiley, New York, 1963' 1969. [A-56] Kodaira, K. : Complex Analysis, I, II, III, Iwanami-Shoten, Tokyo, 1978 (Japanese). [A-57] Kodaira, K. : Compler Manilolds and Deformation of Complex Structures, Springer-Verlag, Berlin and New York, 1986. [A-58] Kra, I. : Automorphic Forms and Kleinian Groups' W' A. Benjamin, Menlo Park, California, 1972. [A-59] Kra, I. and Maskit, B. (eds.) : Riemann Surfaces and Related Topics, 7978 Stony Brook Conference, Ann. Math. Studies, No. 97, Princeton University Press, Princeton, New Jersey, 1981. S. L.: Quasiconlormal Mappings and Riemann Surfacec, Winston, Krushkal', [A-60] Washington, D. C., 1979. [A-61] Krushkal', S. L., Apanasov, B. N. and Gusevkii, N. A. : Kleinian Groups and Uniformization in Examples and Problems, Tra.nslations of Mathematical Monographs, Vol. 62, American Mathematical Society, Providence, Rhode [A-62] [A-63] [A-64] [A-65]

Island, 1986. Kusunoki, Y. : Theory ol Analytic Functions, Hirokawa-Shoten, Tokyo, 1962 (Japanese). Kusunoki, Y. : Theorg of Functions (Riemann Surfaces and Conlormal M app ings), Asakura-Shoten, Tokyo, I 973 ( J apanese). Lang, S.: Introduction to Complex Hyperbolic Spaces,Springer-Verlag, Berlin and New York, 1987. Lawrynowici, J. andKruyz, J.: Quasicontormal Mappings in the Plane,Lecture Notes in Math., Vol. 978, Springer-Verlag, Berlin and New York, 1983.

'g96l '{ro1 'ra,roq 'rr1tor14 ,uag pepenoC : 'g 'uueurary [26-v) '9/6I '{ro^ ^ aN pu" urlrag 'telran-raturrds '/8t '1on ''qle11 uI saloN ernltrarl 'uotgo1ltzsg raraln.u-t rafporqctaq ueuoqlunl : '; 'rauaqc,tg pu" ';41 '11 'uueurlag h6-V] '996I '{ro^ AraN pu" urlrag 'Began-ra8urrdg buorlcung uoltq)nl uo uedo4 : '11 'grracuro4 [06-VJ '(asauedel) 'od1o; 'ueddnqg-ns1ugf,y'tacottns uuDuety : 'X 'p,rel{lo 2961 [68-y] '9661 'pur1s1 apoqg 'acuapnor6 '{lartog l"f,lleruaql"W u"f,rraurv '08 'lo1 'sqdcr8ouo141lef,rl"ruarll"I I Jo suorl?lsu€ta'rc1qouo1 xepd 'J 'IsIpO pu" 'f 'rqrntop -u,toC IDraaaS ut frtoaql uotpuntr ),rlauoe7 : [gg-VJ '986I 'uolsog 'rasneqlrrg 'Z puu 'qon 'r"lado2, '1 'uaqarg I IDcttouraqpn pepeiloC : [29-y] '6961'atprrquep'ssar4 dlrsrarrull a8prrqureg 'tdnotg a1atcstg to fuoaql ctpod-tg aqJ : 'l 'd 's11oq]IN [gg-v] '1961'a8prrqureg'ssar4 dlrsraaruq a8prrqure3''pg pu7's1uto4 {o qag auo14 to fi6o7odo1 aq1lo sluautag | 'V '11 '141'ueuraraN [99-V] '9961'dasral lra11'uolacurr6'ssarg dlrsraarull uolaf,urrd 'acuata{uop 'tuo4tung ct1fi1ouy : ('spa) 'p 'U 'euuqueaap uolnurrd la []g-vJ '086I '{ro1 /r.raNpup uqrag '8egan-ra8urrdg 'Z18 'lo1 ''qten ur satoN arnl)a.I bacodg p6a1g {o uorlocyftycodutoCpptoil : '1 'ea,reryur"N [tg-y] .'86I ,{ro^ ara11'ra11aq laf,r"W 'eeotng cntqaily aaqcaCo-t4to fi4autoag : 'W '€qureN [28-v] '(asauudel) 'od1oa 'urddnqg-c1r4rroy41'raaopns uuDuety '1,'g'ro1ep 6961 lo finaq1 : [t3-v1 'gg6I '{ro 'fep16 'ncodg rallnurq?,eJ ,raatr1 tuoayl ct1fr1ouy xaldutog aqJ : 'S'8on [Og-V] {o '(asauedel) 'or(1o; 'ueddnqg-nslug[y'sp1olzuary : 'S'ruruqernyq 6961 [6/-vJ '926I 'IoqrY uuY 'ssar4 ueBrqrrl4l;o dlrsra,rrun aqJ 'tuotqoco1 rrell puD se1rnC : 'q 'prolurnyl [8]-vl 'Il,6I '{ro^ ,r aN 'uolsull!\ 'rrrepox pue '1 ',r,rorro1q [12-y] 't86I'uoPuoT pu" lr"qaurg

'l1oy1 'tpyo/luoy4yxaldutog : 'y

'ssar4 crurapety 'atnlcatuoC tttluts arg : ('spa) 'H 'ss"g pu" 'lA '1 'uz8rolq [9^t-V1 '116l '{ro llraN pu" urlrag '3e1ran-raSurrdg '5'puo 'g 'g 'asroyq 6 suor.r' ueurt1ut fi6o1odo1crrleuoeg : [92-vl '(asaucdel) 'o{:1o; 'ualoqg-rureuenl'1olfaquJ an|taqaT : 'g 'uleqozlyq 9961 [ll-VJ '(asauedel) 9961'od1o; 'ualoqg-rurruenl 'tuotlonbg 'g 'eleqozrltl Totyuata$rg IDrlrDd to fitoaytr : [eZ,-V] 'Zl,6I '{ro1 ,r,rag 'ra41ag 'tp7ofiuo741alqoquan$tg : '1 'eurrqsnslpl lerr"I I l [Zf-y] '186I '{ro ,r aN pu" uqrag 'tepan-raBurrdg 'tdnotg uDtunIN : '9.'fHsen hl-y] ''z6I 'uoPuol 'ssar4 crruapraty'tdnotg Jre1l puo suo.Jvlessal uoep\rneuoN ; '14 'snu8cyrl [g]-VJ 'Cr6I ':IroA /t{aN pu" uqrag '8ega1-ra8urrdg ''pg pt7 'auo14 aq1 ut siugddol,tgTouttotuoctson\ | .I.N .rau"lrr1 pu" .O ,o1qa1 [OS-YJ '186I '{ro^ ,r aN Pue uqrag '8e1ran-ra8urrdg 'saaodg rellnu!ilel puD suo.puntr yapayn : .O 'ofqaT [gg-v] '996I '{ro^ ^|aN 'uolsurd\ pu" lr"rlaurg 'l1oy1 'tuorlcung uqdtoarcpv ul etrnoC : '1 'rauqal [trg-y] ,roUS V '7961'puc1s1apoqg'aeuapr,ror6'flarcog l"f,-rl"uaql"nl u"f,rraruy 'tuotlcung culdloutolny puo tdnotp snonuquo?sr1 : 'f 'rauqaT [gg-V] sef,ueralau

L9Z

258 [A-93] Sario, L. and Nakai,M. : Classifcation Verlag, Berlin and New York, 1970.

References Theory of Riemann Surfaces, Springer-

[A-9a] Schiffer, M. and Spencer, D. C. : Functionals of Finite Riemann Surfaces, Princeton University Press, Princeton, New Jersey, 1954. [A-95] Schlichenmaier, M. : An Introduction to Riemann Strtaces, Algebraic Curttes and Moduli Spoces, Lecture Notes in Physics Yol. 322, Springer-Verlag, Berlin and New York. 1989. [A-96] Schober, G. : Uniualent Functions-Selected Topics. Lecture Notes in Math., Vol. 478, Springer-Verlag, Berlin and New York, 1975. [A-97] Shafarevich, I. R. Basdc Algebrcic Geometry, Springer-Verlag, Berlin and New York, 1974. [A-98] Siegel, C. L. : Topics in Complex Function Theorg, Vols' I, II and III, Wiley, New York, 1969, 1971, 1973. [A-99] Springer, G. : Introduction to Riemann Surfaces, Addison-Wesley, Reading, Massachusetts, 1957. [A-100] Stein, E. M. : .9ingular Integrals and Difierentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970. [A-101] Stillwell, J.: Classial Topologg and Combinatorial Group Theorg, SpringerVerlag, Berlin and New York, 1980. [A-102] Strebel, K. : Quadratic Differentials, Springer-Verlag, Berlin and New York, 1984. N. : Modern Function Theorg II (Theory of Conlormal Mappings), Suita, [A-103] Morikita-Shuppan, Tokyo, 1977 (Japanese). [A-104] Sunada, T.: Fundamental Groups and Laplacian Kinokuniya-Shoten, Tokyo, 1988 (Japanese). S. : PL Topology of SurJaces/, /1, Maki-Shoten, Tokyo, 1986, 1987 Suzuki, [A-105] (Japanese). [4,-106] Teichmiller, O. : Collected Papers, Springer-Verlag, Berlin and New York, 1982. Toda, N. : Riemann Surfaces, Science-Sha,Tokyo, 1976 (Japanese). [A-107] [A-108] Tsuji, M. : Potential Theory in Modern Function Theorg, Chelsea, New York, 1959. J. : Lectures on n-Dimensional Quasiconformal Mappings, Lecture Vi.isi.li., [A-109] Notes in Math., Yol.229, Springer-Verlag, Berlin and New York, 1971' [A-110] Venkov, A. : Spectral Theory of Automorphic Functions and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990. Weyl, H. : Die Idee der Riemannschen Fliiche, B. G' Teubner' Stuttgart, 1913. [A-111] S. T. : Mathematicol Aspects ol String Theorg, World Scientific, Yau, [A-112] Singapore, New York and Hong Kong' 1987. [A-113] Yoshida, M. : Fuchsian Differential Equations, Friedr. Vieweg & Shon, Braunschweig, Germany, 1987. [A-114] Zieschang, H., Vogt, E. and Coldewey, H'-D. : Surlaces and Planar Discontin' uous Groups, Lecture Notes in Math., Vol. 835, Springer-Verlag, Berlin and New York, 1980.

'996f i(a1aryag 'ttg-gzg 'dd ''qlDW 't6uog 'lDur?7uJ 'co.r7 (,{411oqrsep eur?lqord np anbrrlauroa8 aqrordd",T : 'y.'alprneag

'992-tt?'(886r) ?rr

[IZ]

'tfrt14 't11oy 'untuuroC 'sace;rns uu"rrary pap"rg : 'd (lu"drg pu€ 'I 'rolaqcleg [ [02] ' g g 6 l ' t z - I g . d d ' 1 . 1 o n , l v Z - V j u r , s r a d d r z r " l r u r r s J l a s: . y , e p r s y [ot] 'Il,I-€9I'(fAOf ) 97 fr6o1odo1'salrnf, 'I['eq1uuro3 pu"'g'olareqry [gI] '986I

yo saoedsrlnpou aq1 yo sdnort pr"rrd aqJ:

'falarlrag 'LZg-tZg'dd,t't11o7,t1'.r0uog 'louralul lord rur.'ed{1 npy;o suorlenba prluaraglp pue 'suorlrun;: e1aq1 'sler8alur u"rlaqy Jo sporrad : 'g 'ogareqrV [Zt] '9961 'tuoy 'arodrturg 'ryrluarog pIroIA '0Ig tuo11 pue {ro ,$aN '1oot1cg -6It 'dd '('rp") 'y,I 'uesug pu" 'd '1aIeg ''g 'lI.&\ ap 'gg6t llrdv g7-2 'fiyaotflndng 'fitgauutfruadng 6uttdg aryeuJ aq1lo t6utpaacot4 'gg, t$uttltndng ur 'sauoarll 3urr1s pue sa)"Jrns uu"rrarg : '4 'uos1alqpu" 'T 'aurnrg-zarr,rly [gI]

'r8r-9zr

'(ggot) 'g 'zarealy 'g 'tfit14 'pn71 'sarrepunoq q1r,r,rturrls 9Iz ;o u(roaq; : [9I] '816-926'(ZSOt) 'V 'cord e|S'n 'suorlenba ru"rlleg roJ ureroaql ssauanbrun y : 'C 'pa16 pue 'n '1 'sro;1qV [lt] 't0t-98t '(OSOf 'uuv ) Z,L'tltoW 'slrrlaur alq?rr"^ ro; r[aroaql turddeur s(uu?ruerg : '1 'srag pu" 'n '.I 'sroJlqy [tI] '086I '{ro1 r$eN pu" uqrag '8epan-ra8urrdg '6-I 'dd'g62 'lo1's3rl"ureql?l4l ur seloN arnlf,eT '('pa) 'f 'zcrllrour(raae1bl6t Tuqnzox 'suo.puntr ct1fr1ouy ur 'suorleurro;su"rl snrqol{ ;o sdnort yo sarlradord rrpotrg : 'A 'T rsroJlqv [ZI] '9161 'nlurqag 't8-Il 'dd ''IIDW 't0uog 'IDureIuJ 'ao"r4r'sdnort uerurelx pue sacrds relnuq)reJ 'sSurddeur FuroJuo)rsenb , 'l '.I 'sroJIqV [II] .t96I ,Uo 'dd '11 '1on tffirtouegory ,r,ra11'fa1r46 '('pa) 'T '; 'f1eeg 'tgI-IgI uo urapory 'T 'sroJlqy sernpel ur 'suorlectlddv rreql pue s8urddelll l€ruroJuof,rsenb lA [0I] 'I08-16Z '(SSOI) 'suorleaper '1 'sroJIr{V 60T'qlory olcy FurroJuor-rsen$:.'A [O] '9961 'raga1-3"tlIW 'lsul '6-t 'dd ''t1qo741 '.tiuop 'lvureluJ 'aor4r 'sareds rall$uq)ral : 'A '1 'sroJlqy [8] '92I-r9I'(tg0t) 0 't19y aefi1ouy,p'7'ateds ralpurqlraJ;:o sarlradord arnl"^rnC: 'A'T'sroJIIIy [/] .t1toW .I6I_ILI,(tSOt) V2 'uuy 'sace;:rns uu"ruary 'A 'T 'sroJlqy 3:oacuds rallnurqf,ral uo $lrerrrar eruos : [g] '096I'99-9t'dd'[?e-Ir] ut'saf,eJlns uu"ruarll pasol) Jo aceds aql Jo arnlrnrls crlfpur xalduror aqJ : 'A '1 'sroylqy [S] '9961 ',{asra1,ra11 'uolarutr4 'ssarg d11sra,rru11 uota)urrd 'tl-t 'dd '('.p") 'p la 'A 'T 'sroylqy '09 'oN sarpnts 'ql"W 'uuv 'eacottng uuouatg {o frrcaqg eql o1 suoqng.rluoC ur'rfrnluar c q3norql sa)"Jrns uu"ruerlr pur sturddeur purroluor Jo i(roaq1 aq1 ;o luaurdole^eq : 'n '1 'sroygy [l] 'ZgZ-ltZ '(eSOf) 'edf,1 atrua? Ig1.'ytory olcy 'g 'prety -ra^rp Jo sdnort roJ uraroaql {lrprtrr s.lrolsol I yo loord crrlauroa8 y : [g] '269-?19 'ruaroaqt uortezlurrollun aqJ : 'a\ '.uo{lqv [z] '(rgor)

gg fi1t11uo741 llpn

'rauv

' Ltz-uz'(sror) vBt'!q110w DtcY

'111 'sdnor8 upruralx puc sacuds rallBurqtrreJ Jo selr"punoq

uO : 'lA '.UolIIqy h]

srad€d saf,uereJau

692

260

References

[22] Bers, L. : Spaces of Riemann surfaces, Proc. Internat. Congr. Math., pp. 34g361, Edinburgh, 1958. [23] Bers, L. : Quasiconformal mappings and reichmriller's theorem, in [A-g4], pp. 89-119,1960. [24] Bers, L.: Simultaneous uniformization, Bull. A. M.5.66 (1960),94-92. "spaces of Riemann surfaces as bounded domains,, [25] Bers, L.: correction to B u l l . A . M . 5 . 6 7 ( 1 9 6 1 ) ,4 6 5 - 4 6 6 . [26] Bers, L. : uniformization by Beltrarni equations, comm. pure Appr. Math. 14 (r961),215-228. [27] Bers, L. : A non-standard integral equation with applications to quasiconformal mappings, Acta Math. 116 (1966), 113-134. [28] Bers, L' : on boundaries of reichmiiller spaces and on Kleinian groups, r, Ann. Math. 9L (1970), 570-600. [29] Bers, L. : uniformization, moduli, and Kleinian groups, Bull. London Math. soc. 4 (1972),257-300. [30] Bers, L. : A remark on Mumford's compactness theorem, Israel J. Math. 12 (1972), 400-407. [31] Bers, L. : Fiber spacesover Teichm[llar spaces, Acta Moth.1g0 (1973), 89-126. [12] Bers, L. : On spacesof Riemann surfaceswith nodes, Buil. A.,U. S. AO (1924), t2t9-1222. Bers, L. : Spaces of degenerating Riemann surfaces, in [A-32], pp. 43-55, 1974. [33] [34] Bers, L. : Deformations and moduli of Riemann surfaces with nodes and signattres, Math. Scand.36 (192s), 12-16. [35] Bers, L. : Automorphic forms for Schottky groups, Adu. in Math. 16 (1925), 3 32 - 3 61 . [36] Bers, L. : on Hilbert's 22nd problem, in Mathematical Deuelopments Arising lrom Hilbert Problems, Browder, F. E. (ed.), Proc. Sympos. pure Math., Vol. 28, American Mathematical Society, Providence, Rhode Island, pp. 559-609, 1976. [37] Bers, L. : Quasiconformal mappings, with applications to diferential equations, function theory and topology, Bull. A. M. S. 89 (L977),1083-1100. [38] Bers, L, : An extremal problem for quasiconformal mappings and a theorem by Thurston, Acto Math. 14f (1978), 73-98. [3S] Bers, L. : The action of the modular group on the complex boundary, in [A-bg], p p . 3 3 - 5 2 ,1 9 8 1 . [+o] Bers, L. : Finite dimensional reichmiiller spaces and generalizations, Bull. A. M. S. (Neu Series)5 (19S1), 131-t72. Bers, L and Ehrenpreis, L. : Holomorphic convexity of reichmiiller spaces, Bull. [41] A. M. 5.70 (r964), 76r-764. [42] Bers, L and Gardiner, F. P.: Frickespaces, Ada. in Math.62 (1986), 249-284. [43] Bers, L and Royden, H. : Holomorphic families of injections, Acta Math. L|o7 (1986), 259-286. [44] Blanchard, P. : complex analytic dynamics on the Riemann sphere, Bull. A. M. S. (/fero Series) 11 (1984), 85-141. [45] Bonahon, F. : The geometry of reichmiiller space via geodesic currents, Inuent. math. 92 (1988),139-162. [46] Bowditch, B. H. and Epstein, D. B. A. : Natural triangulations associated to a surface, Topology27 (1988), 9l-117. [47] Bowen, R. and Series, C. : Markov maps associated with Fuchsian groups, IIIOS Publ. Math., No. 50 (1979), 153-170.

(ef,grns 'gtt-IIt'(rgot) Lgz'uuY'tlro]t[ uu"uarg leedruoce;o aeedsrlnPoruPeguu"runaql Jo uorsueurPPu? ernlf,uls aq1 ;o ;oord su?ruu?urary, i(1arnd" uO : '1 'y 'tquror; Pu" 'g 'V 'reqrs-rJ[Il] 'VgL-lgL'(ggOt)gZ'qtory 't orlDsO'spunoquorlelepptur -fre,r q1r,us8urddeurI"urroJuoJrsenbpurarlxa uO : 'X 'ue:les Pue 'U 'uuurugag [0ll 'l0I-t8 '(ssor) uo salrnC : 'V'g'q'uralsdg [691 gl'1.'tttory olay'srrdolos! Pu" sPloJru"ru-z 'W 'V 'nng 'ZtZ-902'(OSO1) 17 (sauagneN)'S 'sa,rrnrcprqatp xalduroc ,(roaqtaql u-rssarEord: 'f 'sur"H Pu? 'O 'pnquaqg [891 Jo 'utol^I 'tz9-98t '(eee t) oz'NS uopuoT'11ng'sdtur truorur"q uo lrodar raq?ouv : '1 'arreural pue '1 ''r1 '"U"9 [lg] 'sace;rnsuu?ruery '61-6t '(916i) Ztt'tltonl oley pasolf,Io sarlnu"JcrqdrouroloqaruosIo suollf,esuO : 'I 'ery pue 't 'g 'aptg [SSJ ' r 6 9 - 1 8 9' G n t ) w ' t 'I 'ery pue 't '3 'apeg 'qlohl : uo uea,rlaq sarrlaruosl ran]urq)ral erin.o'saltds [s9l ' t L Z - 6 t ? , ' ( S e OO t)LZ'uuv'ttlory 'sace;rnsuu"ualg uado;o sarpru"J crqdrourololl:'5'g'ra1,raogPu"'t'C talreg [tg] '[lo 't 'tt-6I '(696I) g'uoa) 'f ''rf 'q1ag pue 't '3 'ageg {froaql : raqy alpunq uorldgrsap rellsuqrraJ;o [89J V '1 '29-9t '(196I) at'utow asfi\ouv,p 'seredsrafi]urqrraJ]o i(rlauroatprluaraJlP aql uO : 'f ''rf 's11agpue '1 '9 'agug [Z9J '982-992'(816I) Z o r 't 'C 'apeg 'qtDIAI'uuv 'sal1aue,rlqof,"f [tSl Pu" saf,"Jrnsuu"ruarg Jo saIIur"J : 'r,6t-68t 'Ol,Ai 'l 'UIDNa1n6,'alqerlualaJlPs-Ief,u"lslPla[lulqf,Ial aql : 'f '3 'apeg [09J ?? 'tr6I't0I-66 'dd'[zs-V] ar{l uO.:,'t '9 'aFeg [091 ur rrrlaur i(.ropo9q1ereo rall]urqttal ur'saceds ' 9 9 r - r 9 I ' ( e / 6 I , 16 l ' S 'hl 'V 'nng (saceynsuu"urarg palulod sarJrur"J crqdrouroloquO : 'f 'C 'alr"g [89] Io '€9-'9 '(/2,96I)SZf 'S 'W 'V 'cuo{'sa)cds rel]urqf,ral Paf,nPall: '1 '3 'ageg [19] 'orl.1 '?6-I '(f86I) a^sq fruarcgaP urns gg1.'ttloN olcy 'O 'urs€rq 'g q)nl^asuorlf,unltururaruoc ?uurJueAaN [99] 1o arnlrafuot 3 Io Joord : 'dns 'utroN "larg 'tcs 'uuv 'trt-Lgz'(sgot) 8r 's8urddcura:111-prruouflod;to scnueufP aqf uO : 'f 'preqqng Pu" 'Y 'fpenoq [99) ' 6Lr-gLr'(slot) ot'tltout 'V 'fpenoq '1 'prraqqnll 'luenuJ 'sprluarag,rp Pu" []9] Ieqerls 1o flrsuap aql uO : 'alf,rrf, '8?-tz '(986I) aql Jo zst'tttoy'l D?ty 'Y 'fPenoC '1 '3 'ageg sursrqdroruoauroq Pu" [t9] Jo uorsualxal"rnl"u i(lpurro;uo3 : '9961'dalarga g,' 8t L-VZL'dd''t11o 1t1 't6uog 'lourelul'co,r4, 'saqdrourolorlsuorl"oqddesa1rns ar8nrrqS: 'y 'dpenoq [ZS] 'snua8 'Iqnd 'oN '60I-92 '(OgOf lttory 5gg7 ) 9g 'autgaq ua,rrt;o saarnf,Jo amds aq1;o {lqrqrcnparn eql : 'q 'prcilurn11pue '4 [tSl 'CZ9-T09'(ggOf t1'1''sfrUd'UloN 'unuruop '(,(roaql rallBlrrqr-Ial ) pu" uorl"zrurrolrun)saf,"Jrnsuu"ruarU redng : 't{ 'f 'ulqsg Pu" 'T 'auerC [09]

'rs-62'(el6r)

'[ eseull1C'areds rlnpow aql ur )ulaur uossralad-Ire aq'1,: 'J'nqC \ [6t] ?'UIDW '2961 ',{asral nalq 'uolaourrd 'ssar4 flrsraarull uolaf,urrd 'Z0I-06 'dd '('spa) '1y1 'y 'ra1cn; pu" ''C 'q 'raouadg ''H 'U 'xoJ 2pt1c{a1 'fi6o1odo1 'g u1 ctotqaily fi4auoeP puo uotqe0yy {o touog ut unnodutfrg y 'sursrqdrourolne,p dnort un red anbr1,{1euearedsa un(P luellonb : 'g 'ut1re3 [8]]

r9z

sa)ueleJau

262

References

[72] Fischer, A. E. and Tromba, A. J. : On the Weil-Petersson metric on Teichmtller s p a c e ,? r o n s . A . M . 5 . 2 8 4 ( 1 9 8 4 ) , 3 1 9 - 3 3 5 . [73] Fischer, A. E. and Tromba, A. J. : Almost complex principal fiber bundles and the complex structure on Teichmiller space, J . Reine Angew. Math. 352 (1934), 151-160. Fischer, A. E. and Tromba, A. J. : A new proof that Teichmriller space is a cell, [74] Trans. A. M. S.3Og (1987), 257-262. [75] Friedan, D. : Introduction to Polyakov's string theory, in 1982 Les Houches Summer School, Zuber, J. -B. and Strora, R. (eds.), North Holland, Amsterdam, pp. 839-867, 1984. [76] Fujiki, A. : The Moduli space of polarized algebraic manifolds and Kihler metrics, ,Srigalcu,42 (1990), 23L-243 (Japanese). [77] Fujiki, A. and Schumacher, G. : On the moduli space of extremal compact Kiihler manifolds, Publ. RIMS, Kyoto Uniu.26 (1990), 101-183. [78] Furusawa, H. : A remark on the hyperbolic collar lemma, T6hoku Math. J.39 (1987),29r-298. [79] Gardiner, F. P. : Schiffer's interior variation and quasiconformal mappings, Dulce Math. J. 42 (1975),371-380. [80] Gardiner, F. P. : On partially Teichmiiller Beltrami diferentials, Michigan Math. J. 29 (L982),237-242. [81] Gardiner, F. P. : Approximation of infinite dimensional Teichmiiller spaces, Trans. A. M. 5.282 (1984), 367-383. Gardiner, F. P. and Kra, I. : Quasiconformal stability of Kleinian groups, Indiana [82] Univ. Math. J. 2L (1972),1037-1059. [83] Gehring, F. W. : Univalent functions and the Schwarzian derivative, Comment. Math. HeIa. 52 (1977),56r-572. Gehring, F. W. : Spirals and the universal Teichmiiller space, Acta Math. L41[84] (1978),99-113. [85] Gehring, F. W.: Topics in quasiconformal mappings, Proc. Internat. Congr. Math., pp.62-80, Berkeley, 1986. Giddings, S. B. and Wolpert, S. A. : A triangulation of moduli space from light[86] cone string theory, Commun. Math. Phys. 109 (1987), 177-L90. [87] Griffiths, P. A. : Complex-analytic properties of certain Zariski open sets on algebraic varieties, Ann. Math.94 (f 971), 21-51. [88] Gunning, R. C. : Affine and projective structures on Riemann surfaces, in [d59], pp. 225-244, 1981. [89] Hamilton, R. S. : Extremal quasiconformal mappings with prescribed boundary values, Trans. A. M. S. L38 (1969), 399-406. [90] Harer, J. : The second homology group of the mapping class group of an orientable surface, Inuent. math.72 (1983), 221-239. [91] Harer, J. : Stability of the homology of the mapping class groups of orientable surfaces, Ann. Math. 121 (1985), 215-249. [92] Harer, J. : The virtual cohomological dimension of the mapping class group of an orientable surface, Inuent. math.84 (1986), 157-t76. [93] Harer, J. and Zagier, D. : The Euler characteristic of the moduli space of curves, Inaent. math.85 (1986), 457-485. [94] Harris, J. : On the Kodaira dimension of the moduli space of curves, II. The even genus case,Inoent. math.75 (1984), 437-466.

'90I-Il, '(ggOt) '1u?auJ 'sarnlonrls xalduror pu" s)rrletu ulalsurg : 'tr1 'osroy EL'rllr* [8IIJ '99-6I '(916I) 'PuDcS 't!lDJ4|' '1 saarnc 68 ,,t'tp, Pu" (laP, uo sarr"uuurlar4 alq"ls Jo areds qnpour aq1 yo {11,rr1ca[ordaq; : 'q 'pro1urn1,1pu" 'J 'uaspnuy [211]

'Lr-92

'(ooor) 'aceds ralnurqllal i(repunoq dnor8 reln (sreg l" ool''ttlaut luenuJ Jo - P o r ua q t J o u o r l f , " a q 1 i { l r n u r l u o r - u o N: ' d ' l A ' u o ' } s r n q J 1o Pu"'d's'goqPray [911] '(SgOf 'ttlDw 'uuy 'slarluaragrprrlcrpenb IIt-862 ) ?ZT d1r:rpo3rg : '1 'ar11urg pu" 'H 'rnsz141''d 'S '.Uoqr{cray [gII]

pu" s,rog pr"ilIq;o

'08-r,I '(SgOt)Og'npg 'qtoh[ 'lueuuop 'crlIpue ere salenbqlr"g: 'd 'S 'Joqrl]rax [tII]

'9961 'e,raezsr"l '8Lg-ggg 'dd ''t11o141 't6uog 'lDureluJ 'cot4 'eceds rallnuqrral 'd 'S 'JorOIrreX ;o frlauroat aqJ : [tII] '992-9tZ uorlezqraar uaslalN er{I : 'd '5 'goqpray [ZIIJ '(gSOr)

LTI

'tttory 'uuy

'ualqord

'It-sz'(086r)

rellnruqf,rel;o r(rlauroatcrloldrudsraqJ :'d'S'Jolpllrax hII] 61 fi6o1odo7'saceds 'IL6I'tZZ-gOZ'dd '[Z-V] ur 'qnpouralf,rrJ uO : '1 'uaay [0II] '99-It '(oeor) '\?DW 'nun 'o4g uorsualxg ; '4 'sauo1[60I] DuotpuJ roJ sruaroaql 6z't 'qyo741 '1,0I-99'(lgOf ) 7V'sfieatng uosssny'dnor8 relnpourrallnurqf,reJarll pu€ salrnf,Josaxalduro3:'n'1q 'aouu,rt [AOt] '886I '6le-loz 'dd'II 'lo^ '[?Z-Y] ut'sa:eFns uu"ruarU 'H 'etrqg 'a 'rqsofeurl puu Jo sarlnu"JcrqdrouroloqJo ruaroaql ssauelruyy : [191] '9lt-t0? '(ggOf 'rlrr1ry'splo1rueueqoqraddq-3 lcedurorpuols ) OLZ'uuv -ueurp oml olur saf,"Jrnsuu"urarg lrraduroclosdeurrrqdrourololl : '1 'rqsofeurt [SOt] ' g s I - Ig I '(sgot) xa1dutog(uraroarll s(rralasJoloorderli(1uue uV : 'A'rqso,ieurl [ggrJ z,selqotrDA '80t-t6€ '(ggOt) 'UloW 'uraroaqt : n 'qsodeurl []0I] aqn6r srr{f,uerJ apJo.suollezll"rauaC Og't '869-189'(t86I) Oz'ttlory'I DqosO'se)"Jrns crerqa8luarrrlcaford-rsrnburc1rac1o saoedsturra,ror l"sra^ru1 : '1 'rqsodeurt[SOt] .I86I ,oot_1,1,6 .dd ,[6s_v] uI 'sacrdsrelpurq)ral pu" saf,"Jrnsuu"ruarU sarlnu"Jcrqdrourololl: 'n 'rqsofeurl Jo [ZOt]

' v L z - r c | ' ( a mzt )t t

'\lDhl '11 'rnseyq '1 'preqqng D1?V'suor1ulo; pu" sl"rtuaraJrp : pue crlerpen$ [I0I] '886I 'Z6I-69I'dd 'IgtI 'yon 'scrleuraql"I ur saloN i ernlraT re8urrdg '('.p") 'J 'r1e,rrogpu" '.S 'ueurlcrg ''I 'nu1eT '296y nnsuao1 'ttsfi1ouy xa1dutog ur's1sraa1-uqaq paleadar pue dnortn reln8ag :'V'O 'IEqfaH [00I] '186I '6I 'oN '?Jodag 'qnl 'arun sraulotlC 'froaql prloads pue 'saoepns uu?ruarU Eurleraua8ap'sdnorS-g reln8ag : 'v 'O 'FqfeH [66] '99I-ttI '(szor) g|tttDw q 'apv'sacsds rallnurq)ral pue ,(411oqrg uO : 'y 'C 'I"qfag [86] 'ZI-I '(286I) I ffiIqDtrDA xaldutop ({s;rp eql yo s8urddeur rall}urqlreJ uO : 'g 'qorag pue 'y 'tr6 'ueu{eg [Z6l '98-tZ'(286I) Lg'Uto*'luenuJ'selrntr yo aeeds rlnporu aql Jo uorsuaurrp"rrcpox eql uO : 'g 'pro;un14i pue '1 'srrreg [96] '2961 'pue1s1apoqg ,aruaprnor6 dlarcog p)rl"uraqt€I4[ u"rrreruv ,gpl-66 .dd ,I 'g' '1on 'scrlruraql"I 't 'S 'qrolg tr"d I arnd ur ersodrui(g 3:os8urpaacora ('p") '996l uropmog 'fulautoag cntqe0ly ur 'qnpour rraql pu" saarnC ; 'g 'srrreg [96] se)uaraJe[

t9z

264

References

[tt9] Koiso, N. : Yang-Millsconnectionsand moduli space,OsakaJ. Math. 24 (1987), 147-171. Kra, I.: Deformationsof Fuchsiangroups,l, ll, Duke Math. J.36 (f969),537[120] 546, ibid.38 (r97r), 499-508. type of someself-mapsof Riemannsur[f Zf] Kra, I. : On the Nielsen-Thurston-Bers Ianes,Acta Moth. L46 (1981), 231-270. ll22l Kra, I. : The Carath6odorymetric on abelian Teichmiller disks, "I. d'Analyse Math. 4O (1981),129-143. [123] Kra, I. : CanonicalmappingsbetweenTeichmuffllerspaces,BuII.A. M. S. (New Series)4 (198f), 143-179. Kra,I. : On the vanishingof and spanning sets for Poincar6seriesfor cusp forms, 11241 Acta Math.153 (1984),47-L16. [125] Kra, I. : Non-variationalglobal coordinatesfor Teichmiller spaces,in [A-24], Vol. II, pp. 22I-249,1988. [126] Kra, I. : Horocyclic coordinatesfor Riemann surfacesand moduli spacesI: Teichmtller and Riemannspacesof Kleinian groups,J. A. M. S. 3 (1990),4975t6.

ItZll [128] [129] [130] [131] [132] [133]

[134] [135] [136] [137] [138] [139]

Kra, I and Maskit, B. : The deformation space of a Kleinian grorp, Amer. J. Math. lO3 (1981), 1065-1102. Kravetz, S. : On the geometry of Teichmiller spa.cesand the structure of their modular groups, Ann. Acad. Sci. Fenn. A1278 (1959), 1-35. Kusunoki, Y. and Maitani, F. : Variations of abelian differentials under quasiconformal deformations, Math. Z. 18L (1982), 435-450. Lewittes, J. : Invariant quadratic differentials, Bull. A. M. 5.68 (1962), 320-322. LeBrun, C. and Rothstein, M. : Moduli of super Riemann surfaces, Commun. M a t h . P h y s . 1 1 7 ( 1 9 8 8 ) ,1 5 9 - 1 7 6 . Maclachlan, C. : Modulus space is simply-connected,, Proc. A. M. 5.29 (1971), 85-86. Mabuchi, T. : Compactification of the moduli space of Einstein-Ki.hler orbifolds, in Recent Topics in Difierential and Analytic Geometry, Ochiai, T. (ed.) Adv. Studies in Pure Math., Vol. 18-I, pp.359-384, Kinokuniya-Shoten, Tokyo, and Academic Press, London, 1990. Maitani, F. : Variations of meromorphic diferentials under quasiconformal deformations, J. Math. Kgoto Uniu.24 (1984),49-66. Mafl6, R., Sad, P. and Sullivan, D. P. : On the dynamics of rational maps, .Ann. Sci. Eale Norm. Sup.16 (1983), lg3-277. Marden, A. : Geometric relations between homeomorphic Riemann surfaces, Bull. A. M. S. (New Series)3 (1980), 1001-1017. Marden, A., and Strebel, K. : The heights theorem for quadratic differentials on Riemann surfaces,Acto Math. 153 (1984), 153-211. Maskit, B. : On boundaries of Teichmriller spaces and on Kleinian groups, II, A n n . M o t h . 9 r ( 1 9 7 0 ) ,6 0 7 - 6 3 9 . Maskit, B. : Moduli of marked Riemann surfaces, BuIl. A. ]1. S. 80 (1974), I t.'-t

I t.

[140] Masumoto, M. : A characterization of the kernel of the Poincar6 series operator, Trans. A. M. S. 3OO (1987), 695-704. Masumoto, M. : Adjoints of the Poincard series operators, J. Math. Kyoto Uniu. [l4l] 29 (1989), 569-589. [142] Masur, H. : On a class of geodesicsin Teichmiller space, .Ann. Math. 7O2 (1975), 205-22t.

'(rgor) '91,t-1,t8 g 'oN '6?I tltodag tcttfril4'aceds qnpour pue sturrls uo sarnlraT : '4 'uoqep [191] '8-I '(886I) gV uodo1 'I 'rqsrue:lep ')oS 'qtory '1 'saeeds ralllurqllal Jo np"rlno aql uo rueroeq? V : [g9I] 'f 'WoW nqoq1oJ '89-st'(8t6I) oe 'n '"P"{"N 'sdnor8 uorlf,unJ palerauat flalruy;o flqrqels leruroJuof,-rs"nb : [ggl] 'rlk/d '8tI-tZI '(OOOr) 0gI pu" ('S).UIO:'y',{:ls,rofran Pu"'S'geN [pgf] 't86I'uolsog'rasn"wlrlg 'qyDyunururop

(saoedsralllurqrlal

'LZt-lLZ 'dd'11 '1on'('tp")'t'a1e; pue 'I\I'ullrv'fr4eutoeg puD crleurU$rv nr 'q 'prolurnytl 'sa,rrnl [t9I] 3:oaceds qnpour aqt Jo frlauoa8 a,rrlerarunue u" spr"MoJ : '.t6uog 'louralul ')ord .01,6I ,alIN 'gg7-Lgt'dd '1 '1on ''t11o141 'q 'proyutn1q 'sarlauea u"rlaqv pu" saarnf, aceds rlnpour aql Io ernlf,uls eqJ : [791] Jo

'rv|,-Lz?,'(rsot) 8r' ttlD w

efiyouy,p '1 'dnort r"pporu ra[]urqf,tall aql Jo sluallonbueuaqy : 'O 'Pro]urnl\i hgll '$t6 '1 'sar1a '09?-g0t'(ruOf) 6T'uoe1 -rr?A u"rqof,/ef pu" suorl"nba uolqos ur arnlf,nrls lecrtolouroqoC : 'W 'asepyrl [OSII 'surro;aoedscqoqrad,{q 'Iqnd ''0I-e9 '(gSOf S7HI ) ,t'qpn 'C 'g 'r,rolsoyq [69IJ 1o flrprtrr aql pu" a)"ds-u ur sturddeur Furoluof,-rssnb : 'tt10w 'Lt -62''(OOOr) Zn't nqoqgL'dnort uersqonJ" ro} les ?nuq aql Jo slasqnslu"rr"Aul : 'g 'tarresotonf[ggt] .99I_99I,(SSOI)guado| eos .qtvw .t 'V 'FoI 'sturdderu [ [lgl] ;o ftoaql aql ur lu"lsuof, alnlosq" u" uO : l"ruroluof,-rsenb '(asautdel) '(996l) pu" {lrpruro;uocrsen$ ug : 'y 'rro1A1 6g-9/ [99I] 2nqoing'{1rcr1f1tue-opnasd '3 'ua1p14Pl{ 'tltoN .uuy ,asuap .LVZ-Lrc,(tOOt) sdsng : gg1 are [SSt] 'rgt-g1r '(OOO1 relnurqf,ral uo uor?"rall : 'g'ualplqrll [lgt] ) 66'Utour'lua,nuJ'aceds ' LZr-96'(OgOr)L6'qtout ',(lqrqtuaury : 'g 'ua11n141cnf 'lueauJ'sdeur [SSt] leruroluocrsenbpu" sarraser?f,urod 'grZ-9ZZ'kni gZ'ttIDN 'yddy en4 'unrautog'acc1 -rns uu"ruarlTlreduroc t o1 paqddzs" "InruroJacerl s,traqlas : 'd 'H 'ueayc;rr1[ZSIJ 'sdnort 'I 'WDn aql ro1 deur srefl u"ruraly nWUgJ ur readde o1 Jo dlrarlcafrnspue {1qrqu1sprrroJuoJrssnb'ssaualruyf,Ir}aruoeC: 'y 'r:1ezns1e141 [tgt] 'ltDw 'ot8-628'(slor) e l n o E?'t 'puu1puocasaql sdnor8uersqf,nJroJruaroaqlssaulceduroc : 'd 'f 'qqale;A1 v [0gt] Jo 'tttow a{ncr 'sPr"rFq '?It-zot '(ggor) e9't o1 uorlecqdd" u? qll/r spltuaralrp rrlerpenb ro1sarrolcalcrlPasoIC: 'g 'rnse;1 [6]IJ

'(seor) '69r-9sr

V6'S

'N 'V ')ord 'dnort sselo turddeu

eql ]o suorlf,e rrpo8rg ; 'g 'rnseyq [g]I]

'002-69I'(zeor) gtl'q\vw 'g 'rnseyl 'uuy 'suorlegoJ a8uzqoxa pu" suor??uroJsu"rl parns€aur P^ratul : [l?I]

'06I-88I 'rnseyq '(ZgOf 'I 'WoW 'areds alng rallluqrreJ Jo sau"punoq o,$J : g ) 6? '/ 'af,"ds '0I-I '(I86I) 68'qow afi\ouy,p qnporu uo si oU crsapoat pue cqc,(roroq aql ]o selpadord f1r,rr1-rsu"rJ : 'H 'rnstltl 'WDN 'luauuroC ''8I-6/I '(616I) vg'aIaH 'g 'rnse11 'asuap are rapuqi(c auo qlr,$ sl"-rluaralrp laqarls-supluef eqJ : '\loN eqng 'eced'sralllurq)ral '9t9-sz9 '(szof) 59'7 eql ol f,rrlau uossralad-lra \ eql Jo uorsualxa arlJ : 'g 'rnsey'1

yo frcpunoq

[9?I] [g]I] []]I] [tll]

sef,uelalau

992

266

References

[168] Nelson, P. : Holomorphic coordinates for supermoduli space, commun. Math. Phys. t.L5 (1988), 167-175. [r0s] o'Byrne, B. : on Finsler geometry and applications to Teichmiller spaces, in [ A - 7 ] , p p . 3 1 7 - 3 2 8 ,1 9 ? 1 . [170] ohtake, H. : Lifts of extremal quasiconformal mappings of arbitrary Riemann surfaces,J. Math. Kyoto Uniu.22 (1982),191-200. [171] ohtake, H. : on the deformation of Fuchsian groups by quasiconformar mappings with partially vanishing Beltrami coefficients, J. Math. Kyoto unia.29 (19g9), 69-90. [172] Oikawa, K. : On moduli of Riemann surfaces, Sigaku 12 (1960/61) , 7g-I04 (Japanese). [123] okumura, Y. : on the global real analytic coordinates for Teichmiiller spaces, J. Math. Soc. Jap.n 42 (1990), 91-10r. [174] Patt, C. : Variations of Teichmriller and Torelli surfaces, J. d,Analyse Math. LL (1963),22r-247. Irz5]Penner, R. C. : The moduli spaceof punctured surfaces,in [A-112], pp. 313-340, 1987. [176j Polyakov, A. M. : Quantum geometry of bosonic strings, Phys. Lett. 10g B ( 1 9 8 1 ) ,2 0 7 - 2 1 0 . Rabin, J. M. and Freund, P. G. O. : Supertori are algebraic curves, Commun. lL77l M a t h . P h y s . 1 1 4 ( 1 9 8 8 ) ,1 3 1 - 1 4 5 . [178] Rauch, H. E. : The singularities of the modulus spaces, Bull. A. M. S. 68 (1962), 390-394. [17e] Rauch, H, E. : Transcendental view of the space of algebraic Riemann surfaces, B u l l . A . M . S . 7 L ( 1 9 6 5 ) ,1 - 3 9 . [180] Reich, E. and Strebel, K. : On the extremality of certain Teichmiiller mappings, Comm. Math. Hela.45 (1970), 35J-362. Riemann, B. : Grundlagen fiir eine allgemeine Theorie der Functionen einer [181] verinderlichen complexen Grosse, in [A-g2], pp. 3-48, 1851. [182] Riemann, B. : Theorie der Abel'schen Functionen, in [A-92], pp.88-144, 1857. [ 1 8 3 ]Riera, G. : Semi-direct products of Fuchsian groups and uniformization, Duke Math. J. 44 (1977),291-304. [184] Royden H. L. : Automorphisms and isometries of Teichmriller space, in [A-7], p p . 3 6 9 - 3 8 3 ,1 9 7 1 . [185] Royden, H. L. : Invariant metrics on Teichmtller space, in [A-8], pp. J93-399, t974. Saito, K. : Moduli space for Fuchsian groups, in Algebraic Analysis, Kashiwara, ll86l M. and Kawai, T. (eds.) Vol. II, pp.735-?86, Academic Press, London, 1988. [187] sakan, K. : Necessary and sufficient conditions for extremality in certain classes of quasiconformal mappings, J. Math. Kyoto Unia.26 (1986), 3f-37. 88] Sato, H. : Introduction to new coordinates to the Schottky space-The general [1 c a ^ s e , , IM . ath. Soc. Japn35 (1983),23-JE. [18e] Sato, H. : Limits of sequences of Riemann surfaces represented by Schottky groups, T6hoku Math. J.36 (1984), 521-539. [1eo] Schumacher, G. : On the geometry of moduli spaces, Manuscripta math. bO (1985),229-267. [re1]Segawa, S. : Martin boundaries of Denjoy domains and quasiconformal mappings, J. Math. Kgoto Uniu. g0 (1990), 297-976. f 1 o r ] Sekigawa, H. : The outradius of the Teichmiiller space, T6hoku Math. J.3O (r978), 607-6t2.

' o g z - w z ' ( g gsost r)

'qtDN opv '11scrrueu{ppur sursrqdroruoaruoq 'uu,rgng puroJuof,rsenb, 'd [gIZ1 ' 8 t t - I 0 t ':g (986I,t ZZI 'qtory 'uuv'1 'a 'C 'uea111ng scrureu,{p pur sursrqdrouoaruoq l"rrroJuorrsenb : [?IZJ 'I-86I '96?-99?'dd '[69-Y] ur 'suorlourrqoqradi(q;o dnor8 alarosrpi(rerlrqre ue yo ,(lrugur1c {roaql rrpo8raaql uO : '4 'q 'ue,rq1ns[SIZ] 'readde o1 "atu11 o7ofiy 'r1go1,t1 'f 'pupl puorasaq1 sdnort u"lsqrnJ rof arntllluol "t".9 "r{1,"O : '; 'eaaetng ;o [916] 'IZt-I0€ '(etor) 'tttow 89'apr 'lueuutoC 'saczyrnsuu?ruerlTuadoyo s8urddeur 'y 'laqarfg leuro]uo)rs"nb ug : [IIZJ

' r o z - r 6 r ' ( l l e rt)g

't11oygafi1ouy,p '1 's8urdderu 'X '1aqar1g Furroluof,rsrnb purarlxa Jo surl uO : [O1ZJ

'ggr-Lrr'(roor)

ITI'S

'W 'V '?ord 'sgnq pruroudlodpue suorlorucrqdrourololl: 'Z 'r{s/$o{pols [602] '9961'dueurrag'Sra,u.qxuncrg

'oqos 'dd'6'1on ''qle4;o sloadsy'('.p")'d'tuo1y1 T 8a,r,rarn'rpalrJ'96?,-197 pu"'V'pru.rao11 'nols'lA tnouoll u1 'ra1qouo11 xe1dutogloraaes ol suol1 to -nq.rpoC ur 'sse1curaqC lsry a,rr1e8au Jo splolru"ur uralsurg-ralqey lceduroe aql Jo arnl"^rnC : 'J 'g 'nrg [eOZl 3:oaceds r1nporuaql q )rrtaru uossralad-Ira6/r '(asauedel) 8t-tg'(686I) ,araqdsuuptuerU aql uo sorureu,{pxalduror uO : 'I{ '"In{qsHs 1g nsp'ng [r0Z] '62-I '(186I) 'uroN eP)g oz'dns 't?S 'uuv 'suorlcunl '141'ernryqsgs 1euorl"r;o fra8rns l?uroluorrspnbaq1ug : [g0Z]

' r r z- 8 8 r ' ( o o o r )

g tuotltsodrg nqoing 'uralqord ,{111oqegaqJ pu" uorlenba dX aqJ : '1 'e1ogg [S0Zl 'luaaul '288-888 '(SeOt) 88'Ulout 'J '"toHS 'suorlunba uolrlosJo surral ur sarlarr"A u"rqof,?f Jo uorl"zualf,"r"qC : [tgg] 'Il-tt '(696I) 'uuy 'sauuld t z'Won to 'H 'nzruqr{S -1pq raddn 1o lcnpord aql uo Burlcrado sdnort snonur?uof,srpuO : [t6Z] .8tt-ttt ,(feOt) .nyn oyofiy .r11o1tt1 LZ '1 'sdnort u?ruralx pu" saf,"Jrns uu"uarg uo arnlf,uls a,rrlcafor4 : 'H 'eBgS [ZOZ]

'929-6t9'(SeO s Zr )' a r n

'1 'suorleunolsu"rl uO : 'H 'earqg [1OZJ oqofiy 't11o1ty r"Inporu pue saceds raflSurqtrral '299-Ttq'(seor) 'tttoN tg'7 nqoqgl 'saceds ralllurqf,ratr pu" $lsrp-Issnb yo uorlezrral)"r"qC : 'H 'e8qg [967] 'atup olofiy T,gr-t1'(lAOf ) V7 llIDW 't'saceds ralllurq)ralJo sarlradord crrlauroat pue rrlflaue uO :'H'etqs [OOt] 'IIt-662 '(fSOf) 'S 'AI 'V tuoil 'auo snuat yo I0g saf,"Jrns uu"ruary uado ue Jo suorl"nurluoc lczduroc Jo {nporu aql : 'W '"qHS [86I] .(asauedel) g?,2-g0z,(leot) se nqoigg 'suorlrun3:urearls Iq saf,"Jrns uu"luary uado 1o uorl"zrleeg : 't{ teqgs [fOf] '6II-86 '(egOf 'Ulolr[ 'YeuuroC ) 8V'aPH 'C 'C 'sar"qlas 'sBurddrur ra[]urqf,reJ ulelraf,;o flradord lsuarlxa aqJ : [g6t] '929-109 '(sgot) 'utoufiO 'ttJ, 'po6tg 'arnle,rrnc g'fii a,rrletau puD 'g 'sagag lu"lsuof, Jo sa)"Jrns uo sersapoaB 1o turpoc ^o{r"I [ l€f,rrlauroeD : [96I] 'gg6t ''gZ-r9Z 'dd 'II 'IoA '[?Z-Vl ur 'suorlcun;: qfual : 'I 'r1v,rrog pu? 'W 'epddag [rcIJ lsapoat i(q saceds rallBurqrlel Jo uotl"firlaru"r"d '0lC-998 '(geOf) ge't 'tttory nqoqg;'pun1 puoces aq1;o sdnort aqlJo np"rlno : '11 'oloureur" pup 'g'e,taet5r1ag [C6t] u"rsqrnJJo sacudsralnurqrral

saf,uaraJau

L9Z

268

References

[216] Sullivan, D. : Quasiconformal homeomorphisms in dynamics, topology, and geometry, Proc. Internat. Congr. Math., pp. l2l6-L228, Berkeley, 1986. [217] Sullivan, D. P. and Thurston, W. P. : Extending holomorphic motions, .Acta Math. r57 (1986), 243-257. [218] Sunada, T. : Riemannian coverings and isospectral manifolds, Ann. Math. L2L ( 1 9 8 5 ) ,1 6 9 - 1 8 6 . Takhtadzhyan, L. A. : Uniformization,,local index theorem, and geometry of [219] the moduli spaces of Riemann surfaces and vector bundles, in Theto Functions, Bowdoin 1987, Ehrenpreis, L. and Gunning, R. C. (eds.), Proceedings of Symposia in Pure Math. Vol. 49, Part I, pp. 581-596, American Mathematical Society, Providence, Rhode Island, 1989. [220] Takhtadzhyan, L. A. and Zograf, P. G. : A local index theorem for families of d-operators on punctured Riemann surfaces and a new Kihler metric on their moduli spaces, Commun. Math. Phys.137 (1991), 399-426. [221] Tanigawa, H. : Orbits and their accumulation points of cyclic subgroups of modular groups, T6hoku Math J.43 (1991), 289-299. [222] Taniguchi, M. : Abelian differentials whose squares have closed trajectories on compact Riemann surfaces, Japan. J. Math.4 (1978), 417-443. [223] Taniguchi, M. : Variational formulas on arbitrary Riemann surfaces under pinching deformation, J. Math. Kyoto Univ.27 (1987),507-530; Supplements to my previous paper, dDid.28 (1988), 81-86. [224] Taniguchi, M. : Pinching deformation of arbitrary Riemann surfaces and variational formulas for abelian differentials, in Analytic Function Theory of One Complet Variable, Komatu, Y., Niino, K., and Yang C. (eds.) Pitman Research Notes in Math., Ser. 212, 330-345, Longman Scientific & Technical, 1989. [225] Taniguchi, M. : Abelian differentials with normal behavior and complex pinching deformation, J. Msth. Kgoto Unia.29 (1989), 45-56. [226] Taniguchi, M. : On the first variation of Green's functions under quasiconformal deformations, J. Math. Kyoto Univ.29 (1989), 591-600. [227] Taniguchi, M. : A note on the second variational formulas of functionals on Riemann surfaces, K6dai Math. "/. 12 (f989)' 283-295' [228] Taniguchi, M. : On rigidity of the ideal boundary of an infinite Riemann surface, Complea Variables14 (1990), r6l-167. O. : Extremale quasikonforme Abbildungen und quadratische DifTeichmffller, [229] ferentiale, in [A-106], pp. 335-531,1939. [230] Teichrniller, O. : Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenenorientierten Riemannschen Flichen, in [A-106], pp' 635-676, 1943. W. P. : Three dimensional manifolds, Kleinian groups and hyperbolic Thurston, [231] geometry, Bull. A. M. S. (New Series) 6 (1982), 357-381. [232] Thurston, W. P. : Zippers and univalent functions, in The Bieberbach Conjecture, Baernstein, A. et al. (eds.), pp. 185-197, American Mathematical Society, Providence, Rhode Island, 1986. [233] Thurston, W. P. : Earthquakes in twodimensional hyperbolic geometry, in [A2 5 1 ,p p . 9 l - 1 1 2 , 1 9 8 6 . [234] Thurston, W. P. : On the geometry and dynamics of diffeomorphisms of surfaces, Bull. A. M. S. (New Series)19 (1988), 417-431. A. J. : On a natural algebraic affne connection on the space of almost Tromba, [235] complex structures and the curvature of Teichmriller space with respect to the Weil-Petersson metric, Manuscripta math. 56 (1986)' 475-497.

'aarno

'ZLv-Ltv'(oOOr) rt,'uroaD'filO 't

'y .g ,1radyo7y1 lesra,rrun aql Jo frlauroa8 aql pue rularu )rloqrad,{q aq; : [OSZ]

'ZIV-rcV'(eeOr) EI'WDn 'l 'V 'uuef 'nS 'pwv 'uuf 'saf,"Jrns uu"ruargJo suorleruroJap alsed-1n3: .v .S ,1rad1o1q [SSZJ

'SIt-t8z '(fge 'qlory 'unutuop 'sare3:rnsuueruerg f ) Ztl'sfrUd 1o aeeds aql uo uorlf,unJ ulaz Sraqlag aql pu? runrlcads aql Jo sf,rlotdrudsy : .V .S ,fradloA\ [ZgZ] ' 9 6 2 - 9L z ' ( l e o t ) g z ' a r c ? D '#:O 't 'uralqord 'V .g ,1rad1o16 uaslarN eql pu" suorlf,unJ q13ua1 rrsapoag : [992J '9g6l '{ro^ rraN Pu? uqrag 'tepan-ratuudg 'Igt-Igt 'dd 'IIII '1on 'serlelaqlrlI q saloN arnlrarl ('tp") 'S 'ra1ng pue ''1 'raurra,uqcg''g 'qxrqazr\H'fg6I uuog iun|olqtaqrv ur 'sarr3lns uu"uarg ;o areds rlnporu aq1 ;o ,{tolodot eqJ : 'v 'S '1raa1o7y1[SSZJ 't LI-til'(ggOf ) gZ' ruoeC '{lO 't 'areds ralqnurqrlal ro} rrrlaru u"ruueruarq s.uolsrnql : 'y 'g 'pad1o7q [797] 'S?I-6II'(SgOt) gg'qlaur'lueauJ'sa,rrnf, 'V'S '1raa1o1q 3:oaceds qnpour aql ro; rosual uu?lualg aqt pu" surro; ureqC : [ISZJ '1,09I-98tI '(986I) '1 'Jauly 'ssslr LOI'WoW uossralad-Ia1\ aql tuorl alpunq auq a,rrlrsod e tururelqo uO : 'y 'g 'gad1o6 [g96] .166-696,(SeOf .I .reuv ) LOI-.WDN 'sa,rrnf, aceds rlnpour aqg frlauroat uossrala4-lla^\ eqt uO : 'y 'S 'tradlo^\ 1o ;o hgz] ' 9 9 2 - 9 t Z ' ( 8 8 6 I , 18 9 . ? , p a P H ' U I D W ' l u a u u o C 'rro1 parnlound aruo;o aceds rlnpour eqt 'y 'g '1rad1o6 Jo urroJ ralqSx aql uO : [696]

' tz9-r6,'(ssor) 8r| won

'uuy 'sarrn) .y .g ,yad1o6 elqels to Jo a*ds rlnpour aq1 ;o dtoloruoq eql uO : [6]Z] 'ttz-Ly?,'(eg6t) 'uuy 'ace;rns Llt'WDN oqoqraddq " Jo suorl"uroJap Jo i(rlauroa8 :rlcaldurds aql uO : 'v 'S '1raa1o16 [g]Zl '829 'uuy 'uorleurroJaP uaslalNlar{rued eqJ : 'v 'g '1rad1o1q [l,tz] -I09 '(286I) gII'ttlory

'Igt-tzt '(oror)60r'wvn 'uuv

'saleyrns .y .S ,yaA1o6 uu"uaru lczduroc roJ rlnpo.'' se erlrads {fua1 aq; : [g?ZJ ' LLg-tLg'(SfOt) Tg.rttory' 7 c$co7'aceds roJ rrrleru uossra]ad-lre6 aql Jo ssaualalduroc-uo1q: 'y 'g '1rad1o46 [976]

rafl]urq)ral

'6Lt-6r' '(OgOt) '#lO 't 'sdeurcruourreq;o,(roaq1ra[]ruqf,ral erII : 'W 'y1o16 6Z'rucaC [llZJ '(gS/lSOt) 'urag 'uueurarg ap sar"Jrns sap salnporu sal rns : 'V 'l-rad\ [gtz] Ol.tpqrnog .Zt-IZ ,(086I) .t6go1,,y .uuy 711 'sanbrrlaurosruou la salerlcadsosrsauueruu?ut[arrs?l?rre1 : 'J 'Il[ 'sergutrn [6pg] '8nS

'utoufig puv'tttr'po6tg ur readde o1'surolu a^eq lou paausuorlorucqoqrad,(q yo sdnort alrlelrasuof, {lruyur uor}r" aqJ : 'X r{pznqsl4l pue 'v '1.'tugan [1pg] l" Jo 90I-t0I '(ggot) 'S 'N 'V 'nng 'aceds ra11$ur{f,ral re^o scrureufq : 'y '1v\ 'qlea1 [otz] vl '0c9-Itt'(ggor) 'uuy 'noy rrsapoa8 rafllurqrral aqJ,: 'y'14 'qcaan [697] tzl'UIVW 't6I-t9I '(SeOf 'qIDn opy 'dnor8 snlqgru" qt-r^r ) tgl '4 'eqn; alqrledurors8urdderucrrlatuurrfsrsenb ;o uorsualxal"urro]uof,rscn$: [SSZJ 'i886I '{ro1 '3epan-ra8urrdg'98I-99I 'dd '9991 '1on ,traN pu" uqrag 'srrl"utrreqlpelur saloN ernlf,al '('p") 'n 'elurnberg 'uot1otto11 rn1ncyogut I to tctdol ur tfroaql rafl]urqlral o1 qceordde["uorl"rr"a pf,rss"If,v:'f 'v'equrorl [ggg] ' 092-6VZ'(tgOf ) 69' UlDutolducenuoyg'acedsrallnurqcra; uo f,rrlau uossraledlra1\ aql roJ uorlf,unJftraua u" uO : 'f 'v '"qurorJ [gt3] sef,uaraJau

692

)70

References

[260] Wolpert, S. A. : The Bers embedding and the Weil-Petersson metric, Duke Math. "/. 60 (1990),497-508. Yamada, A. : Precise variational formulas for abelian differentials, K6dai Math. [261] . / . 3 ( 1 9 8 0 ) ,1 1 4 - 1 4 3 . [262] Yamaguchi H.: Calcul des variations analytiques, Japan. J. Math. ? (198f), 319-377. Yoshida, M. : The Schwarz program, Sigaku 40 (1988), 36-46 (Japanese). [263] f264) Zograf, P. G. and Takhtadzhyan, L. A. : On Liouville's equation, accessory parameters, and the geometry of Teichmffller space for Riemann surfaces of genus 0, Math. USSR Sbornik60 (1988), 143-161. 12651 Zograf., P. G. and Takhtadzhyan, L. A. : On uniformization of Riemann surfaces and the Weil-Petersson metric on Teichmiiller and Schottky spaces, Math. USSR Sbornik60 (1988), 297-313. 12661Zograf., P. G. and Takhtadzhyan, L. A. : On the geometry of moduli spaces of vector bundles over a Riemann surface, Math. USSR Izuestiga 35 (f 990), 83-100.

vel vzl

vzr vzI z6 0sr LV 18, 8t I

'(u)e (u)g '(l'u)a (t'u)a t(o)a g

r=[{!d, !D} LD

,v '=[{(al'ltv]I

( t ' t t ) zv

68I 8AI

(c)t"v ( J' v) zv r(u)zv @) zv ( J' *H) zv

@)r"v (q)nv

tt tt 8t 88r 6Zr 86r 09r

(v)pv

88

sself, e)uele^rnba enF^ elnlosq" Surdderu flrluapr

za/le z8/I8

+ rela} z/{ne/ep - ralQ} z/{ne/et

{, - )*t ') = )sU qll^r ) alq€rre^ xelduroc n - zt;ll'a = z sU qtl^r z alqerre^ xalduroc a - oaurl'n = ffLeU qlr/rr o-l elqelrc^ xalduroa lred f.reur8erur lred lear eceds ueaprlcng xelduroc lpuorsueurp-u aceds ueaprlrng l"ar leuorsuaurp-u

V_Q {sIP }lun eueld-;1eq .ra,lao1 eueld-;pq reddn eraqds uueruerg eueld xalduoc 'slequrnu xelduroc sJaq(unu 1ea.rerlrle3auuou slaqunu leer ezlrlrsod

{oo}nu

slxe Ieer 'sJaqunu IBar sreqrunu leuor?er s.raBelur sraqurnu l€rnl€u

.(J'H)ev

tl t.l p?

z{

,t ze/e

z8/8 lrp*J=) f i , rl a. = z ax+n=m uII

eu oO ulI

*V V *H H

? )

+u +1r

u ?I

b

z

N

sloqul{s Jo tslT tL7,

sloqurfg Jo tslT

List of Symbols

272

a p2

lg"l.

C

clr

C .C'

(c,p) lc,pl Cf (D) XF D(R) Dif fo(R) Dif f +(R) d dtc ds2 dt', dt'o d"'* p F Fs Fs

li*(t,) f lf "l [/"]. [/t].

f,

148 188 31 29 .| 29 29 29 81

r97 247 19 19 t25 225 2 0 , 5 11 , 35 53 54

20r 40 48 48 35 3 0 ,1 2 1 162 162,247 127,159 13

frt

r02

j.l,l f'1")

i,

105 109 110

{Lr}

r49

[/: .9 * R] gwP f

rp fp f, 7 H

Htpl

247 201 7 , 2 8 ,4 0 t47 160 8 t97 r97 191 191

Hv

H B(H,r) h hv

\n(t) hwp i i* Jp K

Kt ,I{(f) I{n I{t I{a L = {Li}l=t LP(C) L*(D) LT(H, f ) L1 L'

!'[u)

t (* ls

t(c)

4(A A ) \u Mr Mt

Yo

Ms Mod(R) Mod(R). Mod(f) M(Q)

nr(j-) MT M(R) p(z)dz/ dz llJ Fq

193 191 199 200 201

20r 72 72 58 202 1 8 , 7 8 ,1 2 5 184 202 184 60 93 92 187 54 193 193 68 72 224 53 6 1 ,6 8 189 3 7 , 1 4 1 ,1 8 3 183 o 244 16 246 16, 162 247 t62 84 1 8 9 ,1 9 1 72 23 17, r24 1 7 ,1 8 , 8 8 ,1 2 5 153

'09I 'e6'8I z8I 86 6Zr gIU'1,8I'88I t8r 68r 8'I LNI 8'I Lil,VOI t7,I I9 IZr 98r

vzT,

-ll.'.ll oll.'.ll , l l '..l l u(.'.) (.'.) u(.'.) ['*) not

lanl dn

[.]

(ite lo lo c6

T (Q\Dot

0tr 6Zr 7,6r

c"e/e (i t" (r)'t

9ZZ I9 LT

68I 8?I I9I

zzr tT,l 6Zl IZT,VI.TI 67,I LEI,VI

zl 96 ZLT

zl VI,7,I 'rI '8I 0zI 'vl'tr'7,r 07,1 IL I9 ,e

ZI

+

((.r)z)o.,r Q)ot Q)A,I

Q)+r

("r).,r

pp(a),t'

(u),-r pir

uJ ta J

l"l Q)s oK

[/'s] (/'s) s d

LJ

l?)s'21

LZ 8I 7,LI VI'7,1 VI,ZI

0ur 87,2 7,t 8 7,LI LI ILI LVI 6z'17, t7,l t8 89 I ?,t'62'Lz'g 8V7, I6I

6Vr 8Ar rz4

,VI 6?,I 'tgl 89I LgI

09r EL Z,L ZL 09 99 86'99 297, 697,'20?,'l0z '69I z9I TI '69I 'gI ugl

7,gr'gr I6I

(a'u'U) I't''al Wl'"al log'al (og'u) WP'ul Q' )a JIU 'a oa {g

*a *U

a Q)cb

o

4 (a'a)t-u )L

ay

1,11"\ tl dt

"zp(z)dt cd.,

s6

l"l'g vlog o

JWd *7d *!d t--,},{ra}= d

d d

l"^rf dan

-(') *["] -[r] [']

ldl,t sloqur,{gJo ls-rT

t.17,

Index

Index

A

c

absolutely continuous on lines, 77 absolutely extremal selfmapping, 173 ACL,77 act properly discontinuously, 31 admissible, 174 Ahlfors'theorem, 202 Ahlfors-Weill theorem, 153 Ahlfors-Weill's section, 157 almost complex structure, 202 analytically finite type (g,n),75 attractive fixed point, 37 Aut(X)-conjugate, 36 axis, 38

Calder6n-Zygmund's theorem, 96 canonical base of the space of holomorphic Abelian differentials, 236 canonical form, 36 canonical homology base, 236 canonical lift, 121 canonical p-qc mapping of e , 102 canonical ;r-qc mapping of H, 104 canonical quasiconformal mapping of C with complex dilatation pr,102 canonical period matrix, 237 canonical system of generators, 5, 47 Carath6odry distance, 180 Cartan's theorem, 166 (C, e)-small, 248 closed geodesiccorresponding to C.r, 54 closed geodesiccorresponding to 1, 54 closed Riemann surface of genus g, 5 closed Riemann surface of genus g with nodes, 245 coboundary, 197 cocycle condition, 197 colla.rlemma, 174,249 complete, 168 complex dilatation, 18, 88 complex dynamics, 118 complex structure of a Riemann surface, 1 of C/f ,8 of D(R),249 o f .R , 2 9

B base point of Teichmiiller space, 120 Beltrami coefficient,16, 17,92,

r24,r25 Beltrami coefficient induced by a Riemannian metric, 22 Beltrami equation, 21 Beltrami differential, 124 Bergman projection, 188 Bers cohomology class, 199 Bers' Beltrami diferential. 153 B e r s ' e m b e d d i n g ,1 5 0 Bers' extremal problem, 172 Bers'fiber space, 180 B e r s ' p r o j e c t i o n ,1 5 0 Bers' simultaneous uniformization, L47 Bieberbach's area theorem, 152 biholomorphic mapping, 2, I59,246 biholomorphically equivalent, 2, 246 Brower's theorem on invariance of domains. 67

otfr,/r,sz of Mo,166

16I'6,(q pacnpur PrluereJrP rru"rllag ?ruol.ur€q 16I,d fq pacnpur FlluaraJrp rurerllag sruoureq '89I I6I ' prlua.ragrp rtu€rlleg cruorur"q gll'uorlrpuoc s(uo?lrueg t8z'uorlerr€^ s(Pr?r.ueP"H H

661

'elrnc crldqla t 'Lg ,cr1dq1a ZLI 'sse1cfSolotuoqoc ralqcrg

s 247,warcaq1 flqenp (uraroeql s(llneaqloq 96I gg1'fl.radord Sursearcapecuelsrp ' elartsrP tt

'6I'seceJrns ,lualearnba-.7 uueruerg go flgreg elqerluareJrp 69 ,gg ,1sr,u,1 gg1'uraloaql s.qrszlorC uqaq 6ZI tg7,'T,iT,'93'uorlcunu y eeJC uueruarll 3o uorle.reua3ap 1p6rsace;lns 62'raqurnu uorlf,asrelur crJleruoaE W 7 , 'M , ' 6 p 6 ' u o r l e r a u a 8 a p (uorlcunJ Ig {l3ua1 orsapoa3 p61 'arnlcnrls xaldtuoc Jo uorleuroJep pg 'rrsapoa3 C Ug'arnlelJnc u"rssneC

c ( I 0Z w toJ-Z l"luau€punJ 'g 'uorleler zt l"luauePunJ 6p'ureruop lelueurepunJ ,lapour uersqcnd 67 gp 'dnor3 u€rsqrnJ g7 'a?"ds a{clrd 29'Surppaquraurely-e{rrrJ ,saleurplooc e{?rrd 97 'Eurdderu rellnuqtrel purroJ 961 97,7,'PlpArolte^ NJ ,uorleuroJap 616 Nd 911'arnlcnrls xaldurocleururu-;f 'dnor3 ,(Solouoqoc relqtlg 961 tsrs 6lz' L9'uorl"turoJep uaslerNlarlcuad gv7,'glz ' gg '69 'saleurplooc uaslarNleq?ued

'1urod 621 leururur-X 696'rosual alnl€Arnc 96'dnor3 uorleruJoJsuerl Eurraloe 96'uorleurrogsue.rl Surrarr,oc (e?eJJns turrarroc lU 27'deru Surraaoc /U'3uua^oc 'elrnc pasolJ € relo? tg 'd punore pooq.roqq3rauel€urproot 6 6'pooqloqqSrau aleurprooc gt'(X)?nV ut ale3nfuoc 16' 6' lualuttrnba flpurro;uoc 'crrleul ueruu€ruerll e IZ dq pacnpur arn??nrls leruroJuoc 'Z reJnlrnrls I6 I€ruroJuoc '6 'Surddeur 16 leuroJuof,

g vz' 6 wJo wz'rw lo

uorlecgrlceduroc 'I 'ploJluetu xalduroc 69I .(J)J IgI Jo '(u)J I9I Jo ,6J IgI JO 'lgr 'dJ lgl Jo ,gJ T9I JO 097,'6w Jo

J

69

'adf1 69 leuorldacxe (^roaql crpoS.ra 0g '.;r rapun lualearnba

'LVZ'7,91 'gtl 'gal '0zl 'Gt '62'tz'?I'gI'zr' L'?,'lualearnba g'1el3e1ur cr1dr11e

xapul

Index

.!to

harmonic map, 218 Hartogs'theorem, 159 Hermitian inner product on ?|("(i-)), 200 Ilermitian inner product

on {,("(r)),200

holomorphic automorphic form (of weight -:4), L28 holomorphic family of Riemann surfaces, 180, 194 holomorphic function, 2, I59,249 holomorphic mapping, 2, I59 holomorphic quadratic differential, 73,128 holomorphic sectional curvature, 210 holomorphic tangent space of ?(f), 189,192 horizontal trajectory, 142 hyperbolic, 37, I72 hyperbolic complex manifold, 168 hyperbolic length, 53 hyperbolic,L@-norm, 150 hyperbolic metric, 54 I improved ,\-lemma, 118 infinitesimal deformation, 194 initial differential, 140 initial point, 28 irreducible. 174 isothermal coordinates. 20 J Jacobi's problem, 253

K K2ihlermetric,202 Klein-Maskit combination theorem, 250 Klein's combination theorem, 65 Kleinian group, 50, L79,217 Kobayashi distance, 168 Kobayashi pseudo-distance, 168

Kodaira-Spencer deformation theory, 181, 194 K-qc, 78, 120 L lattice group, 7 lie over a point p, 29 lift of a mapping, 30 lift of a path, 29, 30 limit set, 44 linear fractional transformation, 34 local coordinate, 2 local coordinate a.roundp, 2 local parameter, 2 lbcal parameter around p, 2 loxodromic, 37 -LP-smoothingsequence,84 ,\-lemma, 118 M mapping class group of iR, 16, 162 marked closed Riemann surface of genus g, 14 marked torus, 12 marking, 12, 14 Maskit coordinates, 179 maximal.60 maximal dilatation, 18, 78 matrix representation, 35 measurable automorphic form, 187 measured foliation, 73 Miibius transformation, 34 modular group, 9 moduli space of closed Riemann surfacesofgenus g, 16 moduli space of tori, 9 module, 84 Mori's theorem, 92 multiplier, 37 Mumford's compactnesstheorem, 175

'

791,e1nu.ro; Surcnpolda.r ,1urod paxg Burllada.r 29 gg'uo11caga.r 'elqrrnpar ?lI 'err-ds rellnuqcral pecnpar 771 '{"C' ' ' ''rC} dq pacnpar tLl 247, urc rcaq1 flrco.rdrcar I0Z'(J)Jyo acedslua3uel pa.r gg ,6 ae.t3ap;o dnor3 reeurl (uotl€ruroJsusrl lercads lear snrqol{ Isar ?g

vt

uorleur.rogsueJl leuorlc"{ r"eurl Iear .296'elnurro;Ieuor?€rr"^s(rlrneg

u 69

'g ,f3o1odo1 luarlonb ,g ,aeeds luarlonb 69

gg ,6 aar8ap;o dno.rE reaurl arrrlcalord lercads ,uorlce[o.rd LZ,g gz 'pueJ.lec .141.;otualqord 'u u7 la^al Jo dnor3qns acuanlSuoeledrcurrd '1er1ua1od 261 g6 (eln(uroJ s(ntadruod 916,1e.r3a1ur,ror1ef1o4 ggl 'serreseJ€curod 'gg ,19 ,cr.r1aurererurod tg ,a?uelslp arecurod Ig 'sar.rasuossreled 766 gg1 '1cnpo.ld lpl"?s uossreled 'porrad I 8z'qled gv?'yed' , ZLI LE,,cqoqe.red

gg,uorlrsodurocep slued gg ,J ,tq Jo ?r gg's1ued ef,eJrnsuueueru luerlonb 171'dnolE uersqeq4-rsenb d /gI'uorltegar letu.ro;uocrsenb (dnorEuerurely gp6 '1urod alqnop f.reurp.ro 116 Jo re)"Jrns uueurarg uado f lqrqe 1s 9 leruro;uocrsenb 'rl 91 luarcgaoorrrrerlleg TgZ,'gVZ'sapou Suruado q?lA{Surddeu xalduroc Isuorsuaurp-auo 1'a.rn1cnr1s l€ruroJuoarssnb 'gg 'Ig '62 1' plogueur xaldruoc l?uorsuerurp-euo OaI '81 'gI 'Surddetu purlo;uoarsenb o ,pralep.rpenb 79 8/'cb

b gg1 ,luau3es_d gg1 (qfua1-ol gg1 ,crsapoaE_g! 'ecuelsrp-al 991 gg1 ,luaru8es-ol gg1 'q13ua1-d ggl ,ggl ,aleurprooc-d ,cqoqradfq-opnasd 621 621'ursrqdrouroaJlp aosouy-opnasd (snonurluocsrp {1.redord Ig gg1,a.rn1cn.r1s aarlcaford

'lapotu uersqc\{ pezrlerurou 2p 'uorlnlos /6 Ierurou ,ale.reua3apuou 991

gvu'apou

6ZI'ureJoaql uo?srmlJ-ueslerN .uraloa{t s(ueslarN 71 'uelqord uorlezqeer ueslerN 08I gg .leure{ uaslarN gg 'uorsualxe ueslarN IgI'"ululel sn€rx-rreqeN ,e.rn1el.rnc 616 Ieuorlces crqdrouroloq erlrle3au gIU 'eJnle^rn? rf,crll a,rrle3au N

xaPuI

LLZ

278

reproducing kernel, 184 Ricci curvature, 209 Ricci curvature tensor, 209 Riemann sphere, 3 Riemann surface, 1 Riemann surface of type (g ,n), 75 R i e m a n n s u r f a c eo f t y p e ( g , n , m ) , 7 5 Riemann's mapping theorem, 25 Riemannian curvature tensor, 209 Riemannian metric, 20 Riemannian metric c o r r e s p o n d i n g t of , 2 3 Riesz-Thorin's convex theorem, 116 Royden's theorem, 168, 170

s same complex structure, 2 same conformal structure, 21 scalar curvature, 210 Schiffer's interior variation, 233 Schiffer-Spencer'svariat ion, 242 Schiffer-Spencer'svariation by attaching a handle, 242 Schifer-Spencer's variational formula, 242 Schottky group, 50 Schottky space, 50 Schwarzian derivative, 149 Schwaz-Pick's lemma, 51 Selberg zeta function, 253 Serre's duality theorem, 196 Shimizu's lemma, 45 Siegel upper half-space, 237 space of infinitesimal deformations, 194 special unitary group o f s i g n a t u r e( 1 , 1 ) , 3 5 straight line space, 176 straight line, 176 string theory,2I8,253 strong deformation, 247 strong deformation space, 247 strongly equivalent, 23 super Riemann surface, 253

Index

system of coordinate neighborhoods, 1 system of decomposing curves, 60 T Teichmiiller curve, 180 Teichmiiller distance, I25, 162 Teichmiiller mapping, 129 Teichmiiller modular group, 16, 162 Teichmiiller modular transformation, 16, 162, L72 Teichmiiller modular transformation group, 247 Teichmilller space ofgenus 1, 12,2I4 of genus g, 14, 127, of .r?,13, 14,120, 2I5 of a torus, 13 of l, I22, 123,148,151 Teichmiiller's existence theorem, 134 Teichmiiller's lemma, 138, 190 Teichmiiller's theorem. 59. 134 Teichmiiller's uniquenesstheorem, 132 terminal, 247 terminal differential, 140 terminal point, 28 3-manifold, 50 Thurston's boundary, 75 Thurston's compactifi cation, 75 topology of C/f ,8 of D(R),248 of R,29

otfr,/r,sz of Mo,166 of Mn,246 of Tn, 48 of ?(.R), 48,125 of T(f), 125 torus, 4 trace, 37 translation of the basepoint, 127, 159, twisting parameter, 62 d-horizontal line, 142

p21'eurural s,1red1o14 966'elnurro; s,1lad1o14 66't8'etuuray :,lfa6

LTZ'IOZ

'crrlalu ueruuetuarlf uossraladite6

166'crrlatu uossraledlre1\ gl1,' 196'ur.ro; uossraledjra1{ T,gT,' /Il.

6p6

(l"rel€lupenb e go xalrel t8 'alpueq e 3urqce11efq uorlerrel

/\

'??I 'aceds rallnuq?retr 0gI lesralrun 'dnor3 96 uorleuJoJsue.rl Eur.ra,rocl€srelrun 26'aceg.rnsSur.ralroclesrelrun 'SurJelor lZ lesreArun gg 'Sur.raaoclesre^run go sseuanbrun gZ (ueJoaq? uorlezftrrroJrun

n

6lZ

An Introduction to Teichmuller Spaces

Read more

An introduction to Teichmüller spaces

Read more

Univalent Functions and Teichmuller Spaces

Read more

An introduction to multicomplex spaces and functions

Read more

An introduction to Sobolev spaces and interpolation spaces

Read more

An Introduction to Sobolev Spaces and Interpolation Spaces

Read more

Introduction to Ramsey spaces

Read more

An Introduction to Sobolev Spaces and Interpolation Spaces

Read more

An Introduction to Sobolev Spaces and Interpolation Spaces

Read more

An Introduction to Sobolev Spaces and Interpolation Spaces

Read more

Introduction to Ramsey spaces

Read more

Introduction to Uniform Spaces

Read more

An introduction to noncommutative spaces and their geometry

Read more

An Introduction to Noncommutative Spaces and their Geometry

Read more

An introduction to Riemann surfaces, algebraic curves and moduli spaces

Read more

An Introduction to Confucianism (Introduction to Religion)

Read more

An Introduction to Judaism (Introduction to Religion)

Read more

An Introduction to Mormonism (Introduction to Religion)

Read more

An Introduction to Confucianism (Introduction to Religion)

Read more

An Introduction to Judaism (Introduction to Religion)

Read more

An Introduction to Confucianism (Introduction to Religion)

Read more

An Introduction to Magnetohydrodynamics

Read more

An Introduction to Logic

Read more

An Introduction to Judaism

Read more

An Introduction to Cryptography

Read more

An Introduction to Optimization

Read more

An Introduction to Reasoning

Read more

An Introduction to Pharmacovigilance

Read more

An Introduction to Cybercultures

Read more

An introduction to Hydrozoa

Read more

Recommend Documents

An Introduction to Teichmuller Spaces

An introduction to Teichmüller spaces

euolerrug Euo;1Euo11 slrsd uopusl IroA /(eN Eraqleprell ullreg o,(1oa Eepan-reEulrdS Springer-Verlag Tokyo Berlin Heide...

Univalent Functions and Teichmuller Spaces

Olli Lehto Univalent Functions and Teichmüller Spaces With 16 Illustrations * Springer- Verlag Norid Publishing Corp ...

An introduction to multicomplex spaces and functions

An introduction to Sobolev spaces and interpolation spaces

An Introduction to Sobolev Spaces and Interpolation Spaces

Lecture Notes of the Unione Matematica Italiana 3 Editorial Board Franco Brezzi (Editor in Chief) Dipartimento di Mat...

Introduction to Ramsey spaces

Annals of Mathematics Studies Number 174 This page intentionally left blank Introduction to Ramsey Spaces Stevo To...

An Introduction to Sobolev Spaces and Interpolation Spaces

Lecture Notes of the Unione Matematica Italiana 3 Editorial Board Franco Brezzi (Editor in Chief) Dipartimento di Mat...

An Introduction to Sobolev Spaces and Interpolation Spaces

Lecture Notes of the Unione Matematica Italiana 3 Editorial Board Franco Brezzi (Editor in Chief) Dipartimento di Mat...

An Introduction to Sobolev Spaces and Interpolation Spaces

Lecture Notes of the Unione Matematica Italiana 3 Editorial Board Franco Brezzi (Editor in Chief) Dipartimento di Mat...