Relative Index Theory, Determinants and Torsion for Open Manifolds
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Relative Index Theory, Determinants and Torsion for Open Manifolds
Jiirgen Eichhorn Universitat Greifswald, Germany
,~ World Scientific NEW JERSEY· LONDON· SINGAPORE· BEIJING· SHANGHAI· HONG KONG· TAIPEI· CHENNAI
Published by
World Scientific Publishing Co. Pte. Ltd. S Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
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RELATIVE INDEX THEORY, DETERMINANTS AND TORSION FOR OPEN MANIFOLDS
Copyright © 2009 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN-13 978-981-277-144-5 ISBN-IO 981-277-144-1
Printed in Singapore by 8 & JO Enterprise
Contents Introduction .....................................................
vii
I Absolute invariants for open manifolds and bundles ........................................................ 1 1 Absolute characteristic numbers ............................... 4 2 Index theorems for open manifolds ........................... 46 II Non-linear Sobolev structures ......................... 1 Clifford bundles, generalized Dirac operators and associated Sobolev spaces ................................... 2 Uniform structures of metric spaces .......................... 3 Completed manifolds of maps ............................... 4 Uniform structures of manifolds and Clifford bundles .. 5 The classification problem, new (co-)homologies and relative characteristic numbers ................................
62 63 89 121 124 141
III The heat kernel of generalized Dirac operators ......................................................... 1 Invariance properties of the spectrum and the heat kernel......................................................... 2 Duhamel's principle, scattering theory and trace class conditions ..........................................................
180
IV Trace class properties................................... 1 Variation of the Clifford connection........................ 2 Variation of the Clifford structure.......................... 3 Additional topological perturbations .......................
192 192 203 223
V Relative index theory .................................... 1 Relative index theorems, the spectral shift function and the scattering index ........................................
239
v
169 169
239
vi
Relative Index Theory, Determinants and Torsion
VI Relative (-functions, 1]-functions, determinants and torsion ..................................
252 1 Pairs of asymptotic expansions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 2 Relative (-functions ........................................... 256 3 Relative determinants and QFT ............................ 267 4 Relative analytic torsion ...................................... 269 5 Relative 1]-invariants .......................................... 272 6 Examples and applications ................................... 277
VII Scattering theory for manifolds with injectivity radius zero ....................................... 1 Uniform structures defined by decay functions ..........
299 299
2 The injectivity radius and weighted Sobolev spaces............................................................... 3 Mapping properties of e- t 6.. ......•.••.••..•••................ 4 Proof of the trace class property ............................
307 317 322
References .......................................................
331
List of notations ...............................................
338
Index ..............................................................
340
Introduction It is one of the main goals of modern mathematics to describe a mathematical subject, situation, result by a sequence of honest numbers. We remind e.g. in topology / global analysis of the rank of (co-) homology groups, homotopy groups, K -( co-) homology, characteristic numbers, topological and analytical index, Novikov-Shubin-invariants, analytical torsion, the eta invariant, all these numbers defined in the compact case. Including bordism and Wall groups, which are also of finite rank, one has an appropriate approach to the classification problem for compact manifolds. For open manifolds, all these numbers above are not defined in general. The ranks of the group of algebraic topology can be infinite, the integrals to define characteristic numbers can diverge, elliptic operators must no longer be Fredholm, the spectrum of self-adjoint operators must not be purely discrete, etc. We will prove at the beginning of chapter I that there are no non-trivial number valued invariants which are defined for all oriented (including open) manifolds and which behave additively under connected sum. Moreover, we prove, that for any n 2': 2 there are uncountably many homotopy types of open manifolds. Hence a classification, essentially relying on number valued invariants, probably should not exist. The main idea of our approach - brought to a point - is as follows. We consider pairs (P, PI), where P e.g. stands for a triple (manifold, bundle, differential operator), and we define relative invariants i(P, PI), where pI runs through a so called generalized component gen comp(P) which consists of all pI with finite distance from P. The distance comes from a metrizable uniform structure. To define the corresponding metrizable uniform structure is the content of chapter II and is one of the columns of our approach. Then the classification of the Ps amounts to the classification of the generalized components and the classification of the Ps inside gen comp(P). This treatise is organized as follows. In chapter I section 1 we vii
viii
Relative Index Theory, Determinants and Torsion
present classes of open manifolds for which the classical characteristic numbers via Chern-Wei! construction are defined, study their invariance and meaning. Here we include the important contributions of Cheeger/Gromov from [15], [16]. We call them absolute invariants since they are defined for single objects and not for pairs with one component fixed. Section two is devoted to some index theorems for certain classes of open manifolds and elliptic differential operators. It is visible that all these are very special classes and that the wish for a corresponding theory for all open manifolds requires a now, another approach. For us this is the relative index theory, applied to pairs. Then the main question is, what is an admissible pair of Riemannian manifolds or Clifford bundles with associated generalized Dirac operators? In a local classical language this would mean, what are the admissible perturbations of the coefficients in questions - and of the domains? We answer these questions in a very general and convenient language, the language of metrizable uniform structures. We define for a pair of objects under consideration a local distance, define by means of this local distance a neighbourhood basis of the diagonal and finally a metrizable uniform struture. In all our cases, the distance contains a certain Sobolev distance. For this reason, we give in chapter II 1 a brief outline of the needed facts and refer to [27] for more. II 2 is devoted to a very brief outline on uniform structures of proper metric spaces. Since diffeomorphisms enter into the definition of our local distances, we collect in II 3 some definitions and facts on completed diffeomorphism groups for open manifolds. In II 4, we introduce those uniform structures of manifolds and Clifford bundles and their generalized components, which are fundamental in the central chapters IV, V, VI. The final section II 5 contain the first steps of our approach to the classification problem for open manifolds. We introduce bordism for open manifolds, several bordism groups, reduce their calculation to that for generalized components, introduce relative characteristic numbers and establish generators for bordism groups of manifolds with non-expanding ends.
Introduction
ix
Chapter III is the immediate preparation for the chapters IV, V, VI. Section III 1 is devoted to the invariance of the essential spectrum under perturbation inside a generalized component, to heat kernel estimates, and we introduce in III 2 standard facts of scattering theory as wave operators, their completeness and the spectral shift function of Birman / Krein / Yafaev. As it is clear from our approach and the criteria in section III 2, the absolute central question is the trace class property of 2 -,2 e- tD - e- tD , E E gencomp(E). Moreover, the expression - 2 tD2 tD tr( e- e- ' ) enters into the integral for the relative index, relative zeta and eta functions. We establish this trace class property step by step in sections IV 1 - IV 3, admitting larger and larger perturbations. The proof of the trace class property is the heart of the treatise, really rather complicated and the technical basis for what follows. In section VI, we prove several relative index theorems and in V 2 properties of the scattering index. We remark that there are other well known relative index theorems e.g. in [8], [9] for exponentially decreasing perturbations. Both these theorems are very special cases of our more general result. In chapter VI, we apply our achievements until now to define relative zeta and eta functions, relative analytic torsion, relative eta invariants and relative determinants, which are in particular important in QFT. Section VI 6 presents numerous examples, special cases and applications of these notions. In particular, we present classes of open manifolds which satisfy the geometric and spectral assumptions which we assumed in the preceding sections. A particular simple case are manifolds with cylindrical ends for which we describe the scattering theory. Here we essentially rely on [3], [49]. An interpretation of our relative determinants in the case of cylindrical ends is given by theorem 6.17 in chapter VI. Until now, we always assumed bounded geometry, i.e. injectivity radius> 0 and bounded curvature together with a certain number of derivatives. Clearly, this restricts the classes of metrics under consideration. In [54], W. Muller and G. Salomonsen established a scattering theory without the assumption injectivity
x
Relative Index Theory, Determinants and Torsion
radius> 0 but they admit other perturbations. The difference 9 - h together with certain derivatives must be of so-called moderate decay. We reformulate their appraoch in our language of uniform structures and generalized components and extend their results to arbitrary vector bundles with bounded curvature. Part of this book has been written at the Wuhan University (China). The author thanks the DFG and Chinese NSF for support. The other part has been written at the MPI in Bonn. The author is particularly grateful to Gesina Wandt for her permanent engagement and patience in printing the manuscript.
Greifswald, December 2008
I Absolute invariants for open manifolds and bundles For closed manifolds, there is a highly elaborated theory of number valued invariants. Examples are the characteristic numbers like Stiefel-Whitney, Chern and Pontrjagin numbers, the Euler number, the dimension of rational (co-) homology and homotopy groups, the signature. Moreover, we have invariants coming from surgery theory etc. Taking into account a Riemannian metric, we obtain global spectral invariants like analytic torsion, the eta invariant. On general open manifolds, more or less all of this fails. Characteristic numbers are not defined, (co-) homology groups can have infinite rank etc. We have the following simple
Proposition 0.1 Let 9J1n be the set of all smooth oriented manifolds and V a vector space or an abelian group. There does not exist a nontrivial map c : 9J1 ----t V such that 1) Mn ~ Min by an orientation preserving diffeomorphism implies c( M) = c( M') and 2) c(M#M') = c(M) + c(M'). Proof. Assume at first Mn '1- I;n, fix two points at Mn, then Moo = Ml #M2#' .. , Mi = (M, i) ~ M has a well defined meaning. We can write Moo = Ml #Moo ,2 with Moo ,2 = M2#M3# . .. and get c(Moo) = c(M) + C(Moo ,2) = c(M) + c(Moo) hence c(M) = o. Assume Mn = I;n, and ord I;n = k > 1 which yields c(I;n# . .. #I;n) = k . c(I;n) = c(sn), c(I;n) = c(I;n#sn) = c(sn) = 2c(sn),
c(I;n) =
(1 + ~)
c(sn) = 0,
~c(sn),
c(sn),
(I;n) = O.
o 1
2
Relative Index Theory, Determinants and Torsion
The only real invariant, defined for all connected manifolds Mn known to the author is the dimension n. If one characterizes orientability / nonorientability by ±1, then there are two such invariants. That is all. Looking at the classification theory, we see a deep distinction between the case of closed or open manifolds, respectively. Denote by 9J1n ([cl]) the set of all diffeomorphism classes of closed n~manifolds. Then we have Proposition 0.2 #9J1 n ([cl]) = No. Proof. According to Cheeger, there are only finitely many diffeomorphism types for (Mn, g) with diam (Mn, g) < D, rinj (Mn, g) :; i, and bounded sectional curvature with bound K. Setting Dv = Kv = iu = l/ and considering l/ - ; 00, we count all diffeomorphism types of closed Riemannian n~manifolds, in particular all diffeomorphism types of closed manifolds. 0 On the other hand, for open manifolds there holds Proposition 0.3 The cardinality of 9J1([open]) is at least that of the continuum for n ~ 2. Proof. Assume n ~ 3, n odd, let 2 = PI < P2 < ... be the increasing sequence of prime numbers and let Ln(pv) = sn /Zpv be the corresponding lens space. Consider Mn := d l . L(pd#d2 . L(P2)# . .. , dv = 0,1. Then any 0, l~sequence (d l , d2, . .. ) defines a manifold and different sequences define non diffeomorphic manifolds. If n ~ 4 is even multiply by SI. For n = 2 the assertion follows from the classification theorem in [59]. 0 There are simple methods to construct from only one closed manifold Mn =1= En infinitely many nondiffeomorphic manifolds. This, proposition 0.3 and other considerations support the naive imagination, that "measure of 9J1([open]) : measure 9J1([cl]) =
Absolute Invariants
3
00 : 0". We understand this as an additional hint how difficult any classification of open manifolds would be. The deep distinction between the propositions 0.2 and 0.3 indicates that the chance to classify open manifolds (at least partially) by means of number valued invariants is very small. This is additionally supported by proposition 0.1. Concerning number valued invariants, there are two ways out from this situation, 1. to consider only those Riemannian manifolds for which absolute characteristic numbers and other invariants are defined, 2. to give up the concept of absolute characteristic numbers and invariants and to establish a theory of relative invariants for pairs of manifolds and bundles. In this chapter, we give an outline of absolute number valued invariants. In section 1 we introduce and discuss absolute characteristic numbers for open manifolds associated to a Riemmanian metric. These numbers are invariants of the component of the Sobolev manifold of metric connections. In the compact case, there is only one such a component and one gets back the well known independence of the metric. To define the Sobolev component we use the language of uniform structures. A comprehensive treatise of Sobolev uniform structures will be given in chapter II. We conclude section 1 with a short discussion of the Novikov conjecture for open manifolds. Many characteristic numbers appear as the topological index of certain differential operators. An outline of index theorems for open manifolds will be the content of section 2. To define relative number valued invariants for pairs of manifolds and bundles will be the topic of the chapters IV, V and VI.
Relative Index Theory, Determinants and Torsion
4
1 Absolute characteristic numbers for open manifolds Let (M4k, g) be closed, oriented, 9 an arbitrary Riemannian metric, Pi(M, g) the associated by Chern-Weil construction Pontrjagin classes, e(M, g) the Euler class, Lk the Hirzebruch polynomial. Then there are the well known equations
(}"(M4k)
=
J
Lk(M,g)
=
J
L k(P1(M,g), ... ,Pk(M,g))
=
(}"(M,g)
(1.1 ) and for (Mn, g) oriented
X(M n ) =
J
e(M, g) = X(M, g).
(1.2)
These equations express in particular that the r.h.s. are in fact independent of 9 and are homotopy invariants. We proved that the space of Riemannian metrics on a manifold splits w.r.t. a canonical uniform structure into "many" components and that on a compact manifold there is only one component (cf. e.g. [32]). The independence of 9 can be reformulated as the r.h.s. depend only on comp(g), since the space of Riemannian metrics on closed manifolds consists only of one component. We will extend the definitions of the l.h.s. and the r.h.s. to certain classes of open manifolds. In some cases there even holds equality. The main questions connected with such an extension are 1) the invariance properties, 2) b applications, the geometrical meaning. It is clear that the definition of characteristic numbers via ChernWeil construction can be extended to an open manifold if the Chern-Weil integrand is ELI, as a very special case if this integrand is bounded and (Mn, g) has finite volume. We present in chapter II a comprehensive discussion of Sobolev uniform structures. For our purpose here we briefly introduce the notion of a basis Q3 C I,p(X x X) for a uniform structure it on a set X. Q3 is a basis if it satisfies the following conditions.
5
Absolute Invariants
(B I )
(U{) (U~) (U~)
If Vi, V2 E Q3 then Vi n V2 contains an element of Q3. Each V E Q3 contains the diagonal ~ C X x X. For each V E Q3 there exists V' E Q3 S.t. V' ~ V-I. For each V E Q3 there exists W E Q3 s.t. WoW C V.
A uniform structure 11 is metrizable if and only if 11 has a countable basis. For a tensor field t on a Riemannian manifold (Mn, g) we denote m
b,mltl
:=
L sup 1\7lLt l
x,
IL=O
x
where Ilx == Ilg,x denotes the pointwise norm with respect to 9 and we set bltl == b,Oltl. By Itlp,r we denote the Sobolev norm
Itlp" '" Itlv,p"
~
Ut, IVitl~
1
dVOlx(9)) ,
and set lip == IIp,o. The same definitions hold for tensor fields t with values in a Riemannian vector bundle E. Let (Mn,g) be an open complete manifold, G a compact Lie group with Lie algebra (B, {2 : G ---t UN or (2 : G ---t SON a faithful representation, P = P(M, G) a principal fibre bundle and E = PXCEN the associated vector bundle which is endowed with a Hermitean or Riemannian metric. According to the faithfulness of (2, the connections on P and E are in a one-to-one relation, W +----+ \7w = \7. Denote by C(P, Bo, j,p) = C(E, Bo, j,p) the set of all connections w +----+ \7w = \7 with bounded curvature, i.e. satisfying (B o) : IRI :::; C, where R denotes the curvature and II the pointwise norm, and having finite p-action
J
IRV'wIPdvolx(g) <
00.
We fix P and E and write therefore simply C(Bo, j,p). Let J > 0 and set
Vi
+
{(\7, \7') E C(Bo, j,p)21 b,II\7 - \7'IV',p,l bl\7 - \7'1 + bl\7(\7 - \7')1 1\7 - \7'lp + 1\7(\7 - \7')lp < J}
6
Relative Index Theory, Determinants and Torsion
Lemma 1.1 ~ = {V8h>o is a basis for a metrizable uniform structure b,li1P ,l(C(Bo, J,p)).
Proof. We start with (U~): For each V V' E ~ such that V' ~ V- 1 .
E ~
there exists
Hence we have to estimate only
bl\7'(\7' - \7)1 ::; bl(\7' - \7)(\7' - \7)1 +b 1\7(\7' - \7)1 ::; C b,ll\7' _ \712 +b,l 1\7' - \71 and
1\7'(\7' - \7)lp < 1(\7' - \7)(\7' - \7)lp + 1\7(\7' - \7)lp < c2 bl\7' - \711\7' - \7lp + 1\7(\7' - \7)lp, l.e.
b,ll\7' - \71V",p,l ::; P1(b,11\7' - \71V',p,l), where P1 is a polynomial without constant term. (U~) is done. For (U~) : For each V E ~ there exists W E ~ such that WoW ~ V we have to estimate in
b,11\71 - \721V'1,p,1 ::;b,l 1\7 1 - \71V'l,p,l +b,l 1\7 - \721V'1,p,1
(1.3)
only the term b,ll\7 - \721V'1,p,1. But
b,ll\7 - \721V'1,p,1
bl\7 - \7 21+b 1\71(\7 - \72)1 +1\7 - \7 21p + 1\7 1(\7 - \72)lp < bl\7 - \7 21+b 1(\71 - \7)(\7 - \72)1 +bl\7(\7 - \72)1 + 1\7 - \7 21p + +1(\7 1 - \7)(\7 - \72)lp + 1\7(\7 - \72)lp < b,ll\7 - \7 21V',p,1 +2b,11\71 - \71V'l,p,l .b,l 1\7 - \721V',p,1,
together with (1.3)
b,11\71 - \721V'1,p,1 ::; P2 (b,11\71 - \71V'l,p,l, 1\7 - \721V',p,1),
7
Absolute Invariants
(U~)
where P2 is a polynomial without constant term.
is done. D
Denote by b,mnq(®E) or nq,p,r(®E) or b,mnq,p,r(®E) the completion of m
~nq(®E) = {ry E nq(®E) Ib,mlryl:= Lsupl\7lLrylx < oo} IL=O
x
or n~,p(®E)
{ry
E
nq(®E) Ilrylp,r 1
.~n~,p(®E)
(J~I'Vi~l~dVOl'(9)r < Do) ~nq(®E)
n n~,p(®E)
with respect to b,m II or IIp,r or b,m IIp,r = b,m II + IIp,n respectively. We obtain nq,p,d etc. by replacing \7 ------t d and n···(E, D) by replacing \7 ------t D. Here n* (® E) are the forms with values in ®E = P Xc ®. Denote by b,l(P,l(Bo, f,p) the completion w.r.t. b,lUp,l.
Theorem 1.2 a) b,l(P,l(Bo, f,p) is locally arcwise connected. b) In b,l(P,l(Bo, f,p) coincide components and arc components. c) b,lCP,l(Bo, f,p) has a decomposition as a topological sum b,l
b,lCP,l(Bo, f,p) =
L compp,l(\7i). iEI
d)
b,lcompP,l(\7) = {\7' E b,lCP,l(Bo, f, p) I b,ll\7 - \7'IV',P,l < oo}
+ (completion of ~nl(®E' \7) n n~,p(®E' \7) w.r.t.b,lllV',P,l) = \7 + b,ln1,p,1(®E, \7). = \7
Proof. The only fact to prove is a). b) and c) are consequences of a) and d) follows from \7' = \7 + (\7' - \7). Let E > 0 be so
Relative Index Theory, Determinants and Torsion
8
small that in Uc:C'v) b,ll' - '1V',p,l and the metric of b,lcP,I(Bo, j, p) are equivalent. Put for V' E Uc:('V') , b,IIV - V'IV',p,1 < E, V t := (l-t)V+tV' = V+t(V'-V). IfV v E ~O(QjE' V)nO~'P(QjE' V) and b,IIV v - VIV',p,1 ---+ 0 then V v,t = V + t(V v - V) ---+ v-+oo
V + t(V' - V) = V t , i. e. V t E b,lcP,I(Bo, j,p). Moreover, b,IIV t1 - V t2 1V',p,1 = It 1 - t 2 1' b,IIV' - VIV',p,1 t---+t O. 0 1-+ 2
Lemma 1.3 The elements V or,lcP,I(Bo, j,p) satisfy (Bo) and
JIRV'I~
dvolx(g) <
00.
By the definition of b,IC p ,1 its elements are C 1 (since they arise by uniform convergence of O~th and lrst derivatives) hence RV' is defined. If Vv ---+ V, Vv E C(Bo,J,p), V = V v + (V - V v), then, for fixed 1/,
Proof.
RV'
= RV'v+(V'-V'v) = RV'v + dV'(V - V v) + ~[V - V v, V - V v]. 2
(1.4)
Each term of the r. h. s. of (1.4) is bounded, hence RV'. Moreover IRV'vl E L p, dV'(V - Vv)1 E Lp and [V - V v, V - V v] < C .b IV - V vi . IV - V vi E Lp. 0 Now let w f-----t VW = V be given. After choice of a bundle chart with local base SI, . .. , SN : U ---+ E Iu the curvature 0 will be described as OSi = L Oij ® Sj, where (Oij) is a matrix j
of 2~forms on U, Oij(Sk, Sl) = Oij,kl = Rij,kl. An invariant polynomial P : Mat N ---+ C defines in well known manner a closed graded differential form P = P(O) = Po + PI + ... , where Pv is a homogeneous polynomial, Pr(O) = 0 for 2r > n. The determinant is an example for P. If w is not smooth then P(O) is closed in the distributional sense. Let oAO) be the 2r-homogeneous part (in the sense of forms) of det (1 + Oij).
Lemma 1.4 Each invariant polynomial is a polynomial zn al,'" ,aN'
Absolute Invariants
9
Lemma 1.5 Ifw E b,lcP,l(Bo,f,p) andr ~ 1 then
(1.5) Proof. ~~~
For the pointwise norm Ilx there holds IDI; = IDij,kll;, where Dij,kl = Dij(ek,e/) and el,' .. en is an or-
i,j k
thogonal base of TxM. According to our assumption we have IDI~ = J IDI~ dvol < 00 and IDlx ::; b for all x E M. The proof is done if we could estimate 100r(D)lx from above by IDlx. By definition (1.6) where summation runs over all 1 ::; i l < ... < ir < Nand all permutations (i l ... ir) -----+ (jl ... jr). E denotes the sign of this permutation. We perform induction. For r = 1 follows O"l(D) = ~ Dii . The inequality
(1.7) s,t
implies in particular 10"1 (D) I; ::; 2NIDI;. For arbitrary forms
Corollary 1.6 Let P be an invariant polynomial, w E b,lcP,l(Bo, f,p), r > 1. Then each form Pr(D) is an element of b,lD 2r,p,1.
10
Proof.
Relative Index Theory, Determinants and Torsion
o
This follows form 1.4, 1.5 and (1.8).
Denote by H*,p,{d} = Z*,p,{d} j B*,p,{d} or bH* = bZ* jb B* the Lpor bounded cohomology, respectively, where {d} refers to the closure of d as coboundary operator. Corollary 1. 7 Under the assumptions of 1.6, P and w define
well defined classes [PQ(n W )] E H 2 Q,p,{d}(M), [PQ(n W )] Eb H2(J(M).
o Now arises the natural question, how does [PQ(n W )] depend on w? We denote I = [0,1]' it : M - - t IxM the imbedding it('r,) =
(t, x) and furnish I x M with the product metric we write Hq,p,{d}
(~ ~).
Here
== Hq,p etc ..
Lemma 1.8 For every q ;:::: 0 there exists a linear bounded mapping K : nq+I,p,d(I x M) --+ nq,p,d(M) resp. K :b nq+I,d(I x M) --+b nq,d(M) such that dK + Kd = ii - i* - O.
P
f roo.
S·
mce gIxM =
(1 0). .
., b
IS an Isometnc 1m edd'mg
0 g
and i; is bounded. i; maps into nq,p,d(M) because Idi*'Plx = li*d'Plx :s; C 'ld'Plx' Denote Xo = and for 'P E n q+l ,p,d(I X M) 'Po(X I , ... , Xq) := 'P(Xo, Xl"'" Xq). Then 'Po E nq,p,d(I x M), l'Pol(t,x) :s; 1'PI(t,x), and we define
-it
I
K 'P(X1 , ... , Xn)
:=
J
i;'Po(XI, ... , Xq)dt.
o
Thus K is bounded too. The equation dK + K d = ii - t~ is standard. Replacing nq,p,d by bnq,d gives the same conclusions.
o
Absolute Invariants
11
Lemma 1.9 Let f, 9 : M --+ N be smooth mappings, F : I x M --+ N a smooth homotopy, f*, g* : O,q,p,d (N) --+ O,q,p,d (M) , F* : O,q,p,d(N) --+ O,q,pd(I x M) resp. f*,g* : bO,q,d(N) --+ bO,q,d(M), F* : bO,q,d(N) --+ bO,q,d(I x M) bounded and rp E O,q,p,d(N) resp. rp E bO,q,d(N) closed, i. e. rp E Zq,P(N) resp. rp E bZq(N). Then there holds (g* - f*)rp E Bq,P(M) resp. (g* - f*)rp E bBq(M). Proof. We consider the case O,q,p,d. According to our assumption, we have Krp := KF*rp E O,q-l,p,d(M) and (g* - f*)rp = ((F 0 tl)* - (F 0 io)*)rp = (iiF* - i'OF*)rp = (ii - i'O)F*rp = (dK + K d) F* rp = dK F* rp = dK rp. The case of bounded forms will be treated by the same equation. D Now we are able to prove one of our main theorems.
Theorem 1.10 Let Q : MN(C) --+ C be an invariant polynomial, r 2: 1, P = 1 or 2. Then each component U of b,lcP,l(Bo, f,p) determines uniquely a cohomology class [Qr(o'U)] E HP,2r(M) resp. [Qr(o'U)] E bH2r(M). Proof. Assume wo, Wl E U. Then, according to theorem 1.2, d) "7:= Wl -Wo E b,lO,l'P,l(~E'WO) and Wt = wo+t"7, -8 < t < 1+8, is contained in U. We have to show [Qr(o'wo)] = [Qr(O,Wl)]. Consider (1.10)
J
For all t E]- 8,8+ 1[ is 100tl P dvol < 00 and Io'tlx is bounded at M. This follows from (1.10) and the assumption Wo, Wl E U. If P :] - 8,1 + 8[xM --+ M denotes the projection (t, x) --+ x, pi = p* P resp. E' = p* E the liftings of the bundles to ] - 8, 1 + 8[xM, p (which covers p) the associated mapping of the bundle spaces, then p*wo, P*Wl are connections for the lifted bundles. tp*Wl +(I-t)p*wo = p*wo+tp*"7 is again a connection Wi. According to (1.10), there holds O,W' = p*o'o+tap·wop*"7+~[P*"7,P*"7]. p*
12
Relative Index Theory, Determinants and Torsion
is bounded. Thus !lw' is surely p--integrable and bounded if this holds for dP*wop*"l. But this follows from the equation (P*WO)ij = P*(WO,ij) for the connection matrix, "l E b,l!ll,p,d(IB E , wo) and from the boundedness of p*. w', !lw' define well determined p-integrable resp. bounded co cycles at ]-8, 1+8[xM. Let it again be the mapping x -----t (t, x). Then io(E', w') resp. ii(E', w') can be identified with (E,wo) resp. (E,W1). it, 0 :s; t :s; 1, is a smooth bounded homotopy between io and i l . According to 1.9, io!lr(!l') and ii!lr(!l') are in H 2r,p resp. bH 2r cohomologous, i. e. Qr(!lO) and Qr(!ld are in H 2r,p resp. bH 2r cohomologous. 0
Definition. For a component U of b,lCP,l(Bo, f,p) we define the r-th Chern class Cr (PU, p) by
cr(E, U,p) = cr(P, U,p) := -1) ( [o"r(!lU)]. 27f r Then we have C r E H 2 r,p, Cr E bH 2 r . Remark 1.11 For WO,W1 E b,lcP,I(Bo,f,p) the co cycles O'r(!lwo /(27fiY, O'r(!lwl/(27fiY are contained in the Chern class Cr (E) and therefore they are cohomologous, but they do not need to be cohomologous in H 2r,p. Take for example an w E !lp,I(Bo, f,p) and apply a gauge transformation 9 with w-g*w tJb,l!lp,l,d(IBE,w). There holds l!lwlx = l!lg*wlx' An explicit example is given by M = R2, P = M X UN, w the canonical flat connection, the gauge transformation 9 at the point (x, y) given by ei (x2+y2) . id, where id denotes the unit matrix. Then Iw - g*wi<x,y) = Ig- 1d91(x,y) = li(x dx - y dy) . idl(x,y) = IN(x 2 + y2) I! is neither bounded nor p--integrable. For this reason our above approach seems to be adequate to the general situation on noncom pact Riemannian manifolds. 0 Definition. For (! : G -----t ON, E = P X G lR N denote by EC or pc the complexification of E or P, respectively. Any connection
13
Absolute Invariants
w on E resp. P extends in a canonical manner to a connection
on EC resp.Pc and we have an inclusion of the components U of b,lcP,l(P, B o, f,p) into the components UC or,lCp,l(PC , B o, f,p). Then we define the k-th Pontrjagin class Pk(P, U, p) by
Let P f be the Pfaff polynomial for skew symmetric 2N -matrices, f2 : G - - t S02N, E = P XG IR2N. Then we call for a component U of b,lcP,l(pC ,Bo,f,p) e(E, U,p) :=
(2~)NPf(DU)
the Euler class of U. There holds e E H 2N,p(M), e E bH2N (M).
Now come in characteristic numbers. Consider f2 : G - - t UN, let be dim M = 2k and Q an invariant polynomial, w E b,lCP,l(pC , B o, f,p), Q(DW) = a + Q1(D) + ... + Qk(D). Then Q 'l. ... tk. := Q.tl 1\ ... 1\ Q.tk with i1 + ... + ik = k defines a characteristic 2k-form and a characteristic number f Qil ... ik = Qil ... ik(P,w)(M) if the latter integral exists. In particular we consider classes CiJ ... ik := Cil 1\ ... 1\ Cik and have to ensure the existence of the corresponding integral.
Lemma 1.12 a) If k
= 1 and w
E
b,lCl,l(Bo, f, 1), then
f
Cl
M
converges. b) If k > 1 and wE b,lC 1,1(Bo, f, 1) or w E b,lC 2,1(Bo, f, 2) then converges. Proof. a) is clear. We have to prove b). At each x E M, CiJ ... ik is a sum of monomials a . Didl 1\ ... 1\ Dikjk . There holds according to lemma 1.5 ID iJjl 1\ ... 1\ Dikjk Ix < Dl . IDlx resp. ~ D2 . IDI; if p = 1 resp. p = 2. D
14
Relative Index Theory, Determinants and Torsion
Corollary 1.13 Under the assumption 1.12, for any invariant polynomial Q converges QiI ...ik·
J
M
o This follows from 1.4 and the proof of 1.12. Lemma 1.12 b) is also valid in the case {! : G ----t ON, dim M = 4k for Pil ...ik' i l + .. '+ik = k, k 21, resp. in the case (!: G----t S02N, dimM = N for the Euler form e(E,w, 1,g) 1.12 a) for N = 1, 1.12 b) for N > 1). The above characteristic numbers are defined until now only for a chosen connection w. One would like that the charakteristic numbers are constant at least at the components of b,lcP,l(Bo, j,p). This is in fact the case for p = 1. Theorem 1.14 The characteristic numbers are constant at the components of b,lCl,l(Bo, j, 1). Proof. If w, w' are contained in the same component U, then according to 1.10, Qil ...ik(W) and Qil ... ik(W') define the same cohomology clas in H 2k ,l (M) resp. H 4k,l (M), i. e. there exists an absolutely integrable cp with dcp = Qil ...ik(W) - Qil ...ik(W'). A fundamental result of Gaffney then says dcp = 0 for (M, g)
J
M
complete and dcp itself absolutely integrable.
0
Thus one gets characteristic numbers Qil ... ik (P, U) (M). Remark 1.15 For w, w' E b,lC 2,l(Bo, f, 2) and deg(QiI ... i k ) 2 4 the characteristic numbers QiI ... ik(W)(M), Qil ... ik(W')(M) are defined. If Qil ... ik(W), Qil ...ik(W') define the same cohomology class in H 2k,1(M 2k ) resp. H 4k,l resp. H 2N ,\ then the characteristic numbers coincide. 0
A very special but interesting case in our considerations is the case vol(M) < 00. Consider b,lC(Bo). It is defined by means of ~ = {v,,},,>o, where V" = {(\7, \7') E C(Bo)21 b,ll\7 - \7' IV' < 6}. Theorem 1.16 Ifvol(M) < 00 then characteristic numbers are constant on each component of b,lC(Bo).
Absolute
Invariants
15
Proof. According to 1.10 each component U of b,IC(Bo) determines uniquely a cohomology class [QiI ... ikl bH2k(M2k) or bH4k or b H2N respectively. Taking two cocycles of this class, there exists a bounded CI-form cp such that there difference equals to dcp. cp, dcp are bounded, vol(M) < 00, thus cp, dcp are absolutely integrable and the theorem of Gaffney gives the desired result.
o Remark 1.17 vol(M) < 00 implies b,IC(Bo) = b,ICp,I(Bo, f,p). Thus the conclusion of 1.16 also holds for the components of b,ICP,I(Bo, f,p). 0
We call the quasi isometry class of 9 the uniform structure US (g) generated by g. For all metrics of US (g) the cohomology spaces H*'P (Mn , g') coincide. The same holds for bH* (Mn , g'). This leads immediately to Theorem 1.18 The cohomology classes Qil ... ik(W) resp. characteristic numbers Qil ... ik(W)(M) in 1.10 respectively 1.14, 1.15 are the same for all metrics g' E US(g). 0
The situation completely changes if w itself depends on g. Then it will be wrong in general that for g' E US (g) c(w (g)) rv c(W (g') ). The case w = w (g) is essentially the case P = bundle of orthogonal frames of (Mn, g), \7 = Levi-Civita connection \7 9 . Therefore we briefly describe the metrics which come into question and describe their admitted variation (for fixed M). Let
M(Bo,p, f) = {g I9 complete, satisfies (Bo) and
JIR91~
dvolAg) <
oo},
b,21 9 - 9 'I 9,p,2
= big -
+
-- b,21 9 - 9 'I 9 + Ig - g'l 9,p,2-g'I9 + bl\7 9 - \79'19 +b 1\79(\7 9 - \79')1 +
(J (Lq - q'I~,. + t,
1
I(V');(V' - V'') 1;,.)
dVOlx(q))'
16
Relative Index Theory, Determinants and Torsion
and set
M(Bo,p, f)21 C(n, i5)-lg ~ g' ~ C(n, i5)g and b,2lg - g'19,P,2 < i5}.
Va = {(g, g') Here C(n, 15)
E
= 1 + 15 + i5y'2n(n -
Lemma 1.19 s:B
1).
= {V,,}8>O is a basis for a metrizable uniform
structure.
0
Denote by b,2 MP,2(B o, p, f) its completion. Proposition 1.20 a) b,2 MP,2(B o, p, f) is locally arcwise con-
nected. b) In b,2 MP,2(B o, p, f) coincide components with arccomponents. c) b,2MP,2(Bo,p,f) has a representation as a topological sum b,2MP,2(Bo,p, f)
= L:b,2compP,2(gi). iEI
d) comp(g) = {g'
E
b,2 MP,2(B o,p, f) 1 b,2lg - g'lg,p,2 < oo}.
0
Proposition 1.21 If g' E comp(g) then \7g' E comp(\7 g) is the
sense of theorem 1.2, d).
0
Hence we obtain well defined characteristic classes C(\7 g) C(g) and characteristic numbers C ... (\7 g)(M) = C ... (g)(M) as above. The main important cases are the Euler form e =
E(g), X(M n , g)
:=
J
E(g)
M
and the signature case
CJ(Mn, g)
:=
J
L(g),
M
where L(g) is the Hirzebruch genus.
Absolute Invariants
17
There arise the following natural questions. 1) How does E(g) depend on g? 2) What is the topological meaning of X(Mn, g)? 3) Under which conditions does there hold X(Mn,g) = X(Mn), i.e. the Gauf3-Bonnet formula? The same questions should be put for dMn, g), O'(Mn). To the first question we have a partial answer.
Proposition 1.22 If g' E b,2 comp 1,2(g) then
and D
In the case g' tj:. b,2comp1,2(g) we can't say anything. The examples in [3] for X(Mn,g) -I X(Mn,g'), O'(Mn,g) -I O'(Mn,g') are of this kind, i.e. g' does not lie in the component of g. Concerning the second question, we start with a simple case in dimension two which has been discussed by Cohn-Vossen [21] and Huber [43] and has been endowed with particular short proofs by Rosenberg [64], which we present below for completion. Theorem 1.23 Let (Mn, g) be a finitely connected complete noncompact Riemannian surface with curvature Kg. a) If K E L1 then X(M) ~ K dvolx(g).
J
M
b) If vol( M2 , g) <
00
and K E L1 then
X(M) =
J
K dvolx(g) = X(M,g).
M
Proof. M2 is diffeomorphic to a compact surface with p points deleted. A neighbourhood of each point is diffeomorphic to Sl x R+ and the metric can be put in the form
Relative Index Theory, Determinants and Torsion
18
gl1(e,t)de 2
p
+ dt 2 .
Set Mk = M \ USlx]k,oo[.
The GauB-
1
Bonnet theorem for surfaces with boundary yields X(Mk ) = K dvolx(g) + W12, where W12 is the connection I-form as-
J
J
Mk
8Mk
sociated to an orthonormal frame on M. X(M) = X(Mk ), hence one has to show limk->oo W12 2:: 0 for a) and limk->oo W12 =
o for
J
J
8Mk
8Mk
b). W.r.t. the orthonormal frame e 1 = y'gUde and e = dt the first structure equation del = W12 n e2 gives W12 = 1ft (y'gU)de and the second one gives K dvol x (g) = 0 12 = dw 12 = 22 (y'gU)dedt. J Kdvolx(g) < 00 implies lim J !~y'gUde = 2
:t
M
oor
k->oo8Mk
J :t~ y'gUde = canst = C. k-> oo8Mk lim
In the case b), vol(M, g) <
J y'gUdedt < 00 which implies lim J y'gUde = 0, k->oo8Mk hence lim J W12 = lim 1ft J y'gUde = C, C = O. In the case k->oo 8Mk k->oo 00, i.e.
M
a),
J
J
y'gUde
I"V
C· k
+D
as k
---t
00. C < 0 would imply
8Mk
y'gUde = 0 for k sufficiently large. But this is impossible
8Mk
for a positive integrand.
0
In the case for arbitrary n, there are many approaches to study the equation X(M, g) = X(M). To have X(M) defined, one must require that each homology group over lR is finitely generated. Sufficient for this is that M has finite topological type, i.e. it has only finitely many ends El, . .. ,E s , each of them collared, U(Ei) 9:! 8Ui x [0,00[. Then M can be given a boundary 8M to get a compact manifold M. The case n odd is absolutely trivial. Proposition 1.24 Assume (M 2k +1, g) is of finite topological type, 9 arbitrary. Then
X(M) =
J
E(g) = X(M, g) if and only if x(8M) =
o.
M
Proof. For n
=
2k + 1, the Euler form E(g) vanishes since the
Absolute Invariants
19
Pfaffian of an odd dimensional skew symmetric matrix is zero, J E(g) = X(M, g) = O. On the other hand, 0 = X(M U M) = aM
2x(M) - X(aM) = 2X(M) - X(aM).
0
The more interesting case are even dimensional manifolds. We recall some definitions. For a local orthonormal frame el , ... , en the connection I-forms Wij satisfy the equations de
i
=
L
Wij 1\ e
j
and
Wji
=
-Wij'
j
They are related with the curvature 2-forms DiJ by Dij =
dWij -
L
Wik 1\ Wkj'
k
Denote by S(M) the tangent sphere bundle which is a (2n -1)dimensional manifold. For a point (x,~) E S(M) let el , ... ,en be a frame such that el is dual to~. We put I I (g)
L
:=
Ck
O:::;k
L sign (a )D
a (2)a(3)
1\ ... 1\ D a (2k)a(2k+l)
a
I\W a (2k+2)1
(1.11)
1\ ... 1\ Wa(n)l'
where we will not specify the
Ck
:z=
and
means the sum over
a
all permutations a of {2, ... , n}. II (g) can be understand as pull back on an (n - I)-form on M to S(M) by means of pr: S(M) ---r M. If X E TS(M) at (x,~), X = Xl + X 2 with Xl tangent to M and X 2 tangent to S~-l then Dij(X) = Dij(Xl ) and similarly for Wil(X), If M is compact with boundary aM and e is the section of S(M) over aM given by the outward normal vector, then e*Dij(X) = Dij(Xl ), the same for Wil. Then, according to Chern,
X(M) =
J
E(g)
M
+
J
aM
e* II(g)
=
J
E(g)
M
+
J
aM
II(g).
(1.12)
Relative Index Theory, Determinants and Torsion
20
Assume now that (Mn, g) = (M2m, g) is even-dimensional and of finite topological type. By gradient flow of an appropriate Morse function we can introduce coordinates (Xl, . .. , Xn-l, Xn = r) at each end such that 0 ~ r < 00, gin = 0, 1 ~ i ~ n - 1, gnn = 1. Let as above Mk be characterized by Xn = r ~ k. Then
X(M) = X(Mk ) =
J
E(g)
+
Mk
J
II(g).
(1.13)
8Mk
At each end E T Mlc: splits as T M = WEB JR. Suppose additionally that W splits as
and that with respect to this splitting 9 has the form (1.15) Then S. Rosenberg calculated in [65] the expression (1.11) at each end can could show if fj(r) ---+ 0, fj(r) ---+ 0, then
J E(g)
---+
Mk
J E(g) M
J
and
T-400
r~oo
II(g)
---+
O. We will not repeat
8Mk
the really simple calculations but state Rosenberg's
Theorem 1.25 Let (Mn, g) be open, complete and of finite topological type. Assume that in an open neighbourhood of each end E M splits as a product manifold N2 x ... x N r• x JR with the metric fl(r)g2 EB· .. EB l;. (r)gr. + dr2, where gj is a metric on N j . If fj(r) ---+ 0 and fj(r) ---+ 0, then X(M) = E(g) == X(M, g). T---+CX)
T---+OO
J
M
In particular, any evendimensional manifold of finite topological type admits complete warped product metrics satisfying GaufJ0 Bonnet (setting N2 = 8M). Corollary 1.26 Assume the hypothesis of 1.25 and additionally 9 E b,2 M I ,2(Bo, f, 1). If g/ E b,2 comp l,2(g) then X(M) E(g') == X(M, g/). 0
J
M
Absolute Invariants
21
Remark 1.27 We see in 1.26 a considerable improvement of 1.25 since now the admitted class of metrics is much larger. D If one gives up the integrability of the Ws in (1.14), i.e. the product structure of the cs, then one must strengthen the conditions to the fj. This has been done by Rosenberg too.
Theorem 1.28 Let (Mn,g) be open, completeandoffinitetopological type. Assume that in an open neighbourhood of each end c, TMlc = W 2 EB ... EB Wr< EB IR and the metric is of the form fi(r)g2 EB ... EB l;Jr)gr< + dr2 with gi a metric on Wi' If fi (r) ----+ 0, ff( r) ----+ 0 and fj fi- I and (fj fi- I )' are bounded r---+oo
r-too
for all r, i, j then
X(M) =
J
E(g). D
Example. Let M\G/ K be an arithmetic quotient of an evendimensional split rank-one symmetric space. Then at each component 8Mi of 8M, 8M is the total space of a fibration over a torus TI with a torus T2 as fiber. We have T MlvxlR = WI EB W 2 EB IR for open V c 8M where the fibration restricted to V is trivial. Wi is the tangent space to the torus T i . But in general the G-invariant metric 9 does not respect this splitting. Donnelly has shown in [24] that each end c has the structure N x IR, N at most two-step nilpotent. The Lie algebra n of N splits as n = V2 EB V3 of root spaces, V3 = Z(n), and the invariant metric at the identity of N has the form (1.16) where g2 is a metric on V2, g3 a metric on V3. [n, n] C Z(n) and the G-invariant distribution V2 is not integrable. Hence theorem 1.25 is not applicable in general. In the hyperbolic case G/ K = SO(n, 1)/ SO(n), one has V2 = n, which yields GauJ3Bonnet. D
Relative Index Theory, Determinants and Torsion
22
Corollary 1.29 Assume the hyptheses of theorem 1.28 and additionally 9 E b,2 M 1 ,2(Bo, f, 1). If g' E b,2 comp l,2(g) then X(M) = E(g') == X(M, g'). 0
J
There is another GauB-Bonnet case which does not fall under 1.2 - 1.29. Proposition 1.30 Let (M2m, g) be open, complete, oriented, of finite topological type and the metric at 00 constant with respect to r, i.e. there exists an ro ~ 0 such that g(rI, x) = g(r2, x) for all x E 8M and rI, r2 > roo Then
X(M) =
J
E(g)
== X(M, g).
M
Proof. Let k> ro manifold. Hence
+ 5.
Then Mk U Mk yields a smooth closed
J
J
E(gMkUMk) = 2 E(gMk), MkUMk Mk 2X(Mk ) - X(8Mk ) = 2X(M)
J
(1.17)
E(gMk)'
Mk Forming lim in (1.17) gives the desired result.
o
k-+oo
A special case of 1.28 would be a metric cylinder at infinity, glu(oo) = geM ® +dr2. This is simultaneously a warped product with warping function f(r) = 1. f(r) = 1 does not satisfy f(r) ---+ 0, 1.25 is not applicable. Clearly, such T-+OO
J IRIPdvolx(g) = 0 or u(oo) IRIP dvolAg) = 00, similarly either IE(g)1 dvolx(g) = 0 u(oo) u(oo) or J IE(g)1 dvolx(g) = 00. In the second case J E(g) exists u(oo) but IE(g)1 ~ Lp, p ~ 1. an (M2m,g) satifies (Bo) but either
J
J
Absolute Invariants
23
Another class of examples which submits very useful insights are surfaces of revolution. We state from [65] without proof :]0,00[---+ lR be smooth, f(O) f'(O) = 0 and (M2 = {z = f(x 2 + y2)}, induced metric from lR3 ) be the associated surface of revolution. Then
Proposition 1.31 Let f
X(M) =
2~
J
K dvolx(g) = X(M, g)
(1.18)
M
if and only if
r~ f'(r) ---+
r-oo
±oo. o
Hence, if f is for all r > 0 strongly convex or concave, (1.18) holds. In both cases M has for r > 0 positive curvature and infinite volume. On the other hand, we have 1.15 in the case of 1.23 b) in the finite volume case, i.e. one can have X(M) = X(M, g) as in the finite volume case. For this reason we should find additional conditions which assure in the finite volume case or the infinite volume case, respectively, that 1) X(M, g) is a (proper) homotopy invariant, 2) X(M, g) = X(M) if M has finite topological type. We start with vol( Mn , g) < 00 and IK I ::; 1 where the letter (after rescaling) is equivalent to (Bo). Then
X(M,g) =
J
E(g)
M
is well defined and for g' E b,2 comp l,2(g)
X(M, g) = X(M, g').
(1.19)
Lemma 1.32 Let (Mn,g) be complete, vol(M,g) < 00 and IKI ::; 1. Then Mn admits an exhaustion by compact manifolds with smooth boundary, Mf C M7) C ... , U Mr = M, such that k
vol(8Mr) ---+ 0 and for which the second fundamental forms II(8Mr) are uniformly bounded.
Relative Index Theory, Determinants and Torsion
24
o
This is just a corollary of theorem 1.33 below.
If we take such an exhaustion as just described then
X(Mk ) = X(Mk,g)
+
J
(1.20)
II(8Mk)'
8M;;
J
II(8M, g)
8M n
-----t
k--->oo
k
0, X(Mk ) E lL, hence for k sufficiently
large X( Mk , g) E lL, but we are far from a certain convergence of (X(Mr,g))k and don't know anything about the topological properties of such a limit if it exists. To obtain more insight and definite results we follow [3] and consider the following additional hypothesis. For some neighbourhood U(oo) c M, some profinite or normal covering space U(00) has the injectivity radius at least (say) 1 for the pull back metric, (1.21) Together with JKJ
:s:
1 on U(oo) we write geo~(M)
U = M then we denote geo( M) this hypothesis that
:s: 1.
:s:
1. If
00
In any case we assume in
U or Mare profinite or normal coverings.
M M is profinite if there exists a decreasing sequence {r j h of subgroups of finite index, r j C 7rl (M), such that nr j =
Here
-----t
7rl(M). The key for everything is the following very general theorem which assures the existence of sufficiently" smooth" exhaustions and which yields 1.32 in the case of vol(M, g) < 00.
Theorem 1.33 (Neighborhoods of bounded geometry). Let (Mn, g) be complete, X C Mn a closed subset and < r 1. Then there is a submanifold with smooth boundary such that for some constant c( n) depending only on n
un
a) b) c)
°8un :s:
X cUe Tr(x) = r - tubular neighbourhood of X, vol(8U) :s: c(n) . vol(Tr(X) \ X) . r- 1 , (1.22) 1 JII(8U)J :s: c(n) . r- . (1.23)
Absolute Invariants
25
D We refer to [17] for the proof. Now we will discuss X(M, g) in the profinite or normal case, geo(M) ::; 1. Here we follow [16]. Put for j : Al C A2 and real coefficients ,6i(AI' A2) = dim{j*(H i (A 2)) C Hi(A I )} and ,6i(A) = dim{j*(Hi(A, oA)) C Hi(A)}. bi shall denote the usual Betti number. Then for Al C A2 C A3 C A4 and A C Y a finite closed and f : Y --+ Z, 9 : Z --+ Y simplicial, determining a homotopy equivalence,
(1.24) and (1.25) Put for p : yn --+ yn profinite with ind(rj) = dj and corresponding covering spaces Pj : Yjn --+ yn
and define inf X(yn) similarly. A --+ 00 is defined by partial ordering of finite sub complexes induced by inclusion. Using (1.24) and a diagonal argument, there are subsequences S = YJ(e) S.t.
exists. From (1.25) we infer immediately that Si(yn, S) is a homotopy invariant. Suppose Si(yn, S) < 00, i = 0, ... ,n and sup X(yn) = inf X(yn), then the latter number is also a homotopy invariant. Theorem 1.34 Suppose (Mn,g) complete, vol(Mn, g) < either profinite or normal and geo(M) ::; 1. a) Then X( Mn, g) is a proper homotopy invariant,
00,
M
26
Relative Index Theory, Determinants and Torsion
b) in the case
M
profinite
X(M,g)
=
supX(M)
=
infx(M),
c) if additionally M has finite topological type,
X(M,g) = X(M). Proof.
UMk =
Assume
M
----t
M profinite, let Ml C M2 C ... ,
M be an exhaustion of M by compact submanifolds
k
with boundary and denote Mk - R = {x E M k /dist(x,8Mk ) = R}. For j sufficiently large, theorem 1.33 is applicable and we apply it to pjl(Mk_1 ), Pjl(Mk) with c = ~. This yields submanifolds Ajk C Pjl(Mk) C B jk . Given c > 0 arbitrary, there exist k o, N(k) such that for k > k o, j > N(k)
IX(Mn,9)- :jX(Bjk )
I ::;
X(Mn,g)- :j
J
E(g)
Bjk
< c.
+
(1.28) We see this immediately from (1.12) and (1.22), (1.23): X(Mn,g) = X(MJ:,g) + X(M n \ MJ:,g), here /X(Mn \ MJ:,g)/ becomes arbitrarily small for k sufficiently large.
X(Mn,g) - :j
J
E(g)
< /X(Mn,g) - X(MJ:,g)/
Bjk
+
X(MJ:,g) - :j
J
E(g) ,
Bjk
27
Absolute Invariants
X(Mr, g) -
~j
J
E(g)
<
X(Mr,g) -
~j
Bjk
J
E(g)
Pjl(MJ:)
+ 1 dj
J
E(g)
B j k\ Pj l(MJ:)
but this becomes arbitrarily small for j and k sufficiently large. Finally
according to (1.22). (1.28) is proven. We obtain from (1.24) f3i(A jk ) :S f3i( pj l(Mk )) :S f3i( pj l(Mk ), !VIj )
:s bi(Ejk)
(1.29)
and from the exact cohomology sequence of the pair (E jk , Ejk \ A jk ) together with the excision property
---+
Hi-l (Bjk \ Ajk)
---+
Hi(Bjk)
---+
---+
Hi(Bjk, Bjk \ Ajk) ~ Hi(A jk , &A jk )
Hi(Bjk \ A jk ) ---+
...
The manifold Ejk \ Ajk satisfies (Eo), (1) for j > N(k) and for k sufficiently large, (1.30)
Relative Index Theory, Determinants and Torsion
28
According to a theorem of Gromov, (1.31) We infer from (1.29) - (1.32) that we can replace in (1.28) X(B jk ) -1 by X(Pj AIk,AIj), hence
becomes arbitrarily small, any proper homotopy equivalence preserves a subsequence of
(,t X(pj1(AIk), Mj)j,k'
X(AIn,g) is a
proper homotopy invariant. By the same argument we conclude in the profinite case assertion b). If AI has finite topological type then for k sufficiently large (Ji( pj 1(AIk), M j ) = (Ji(Mj ) and 1
-
1
-
-
X(pj (AIk), AIj) = X(AIj) . d:x(AIj) = X(AIj) = X(AI) J
o
yields assertion c).
The case of a normal covering M ----t AI will be discussed in theorem 1.38. The second characteristic number of particular importance is given by O'(AI, g) = L(AI, g), where L(AI, g) is the Hirzebruch
J
M
genus. For closed AI it is the topological index of the signature operator, i.e. it coincides with the topological signature. For simple open manifolds this equality does not longer hold in general. Nevertheless, we could ask for O'(AI, g) the same questions as for X(AI, g), the question for the invariance properties and the topological significance of 0'( AI, g). Concerning the invariance, a first answer is given by proposition 1.22. But we consider also other variations of g. A key role plays again the formula for the compact case with boundary, oAI = N,
O'(AI, g)
+ ry(N, g) +
J
IIa(N, g) = O'(AI),
N
(1.32)
Absolute Invariants
29
where IIa(N, g) essentially involves the second fundamental form and TJ(N, g) is the eta invariant. If Mn is open and Ml C M2 C "', U Mk = M, an appropriate exhaustion such that k
J IIa( 8Mk)
0 and TJ(8Mk ) ---t 0 then we would have in fact <J(Mk,g) ---t <J(M). Hence we should ask for conditions which assure TJ(8Mk ) ---t O. There is a clear (and for our case complete) answer. ---t
Theorem 1.35 Let (N 4l- 1 , g) be compact satisfying geo(N) ::; 1. Then there is a constant c = c( 41 - 1) such that (1.33) D We refer to [16], [27] for the proof. Now we define sup o-(M) , inf o-(M) quite analogous to the Euler characteristic as follows. Let M4l be complete, M4l ---t M profinite and Mtl C M4l a compact submanifold with boundary. Put
sup o-(Mk ) supo-(M) and similarly inf o-(Mk), inf o-(M). Here as always <J(Mk) is defined as the signature of the cup product pairing on j* H2l(Mtl, 8Mtl) C H2l(Mtl). Theorem 1.36 Let (M4l, g) be complete, vol( M, g) < 00 and suppose M either projinite or normal and geo(M) ::; 1. Then there holds a) Assume M normal. Then <J(M, g) is a proper homotopy invariant of M. b) In the case M ---t M projinite, for any exhaustion Ml C M2 C "', UMk = M, by compact manifolds, k
<J(M, g) = sup o-(M) = inf o-(M).
30
Relative Index Theory, Determinants and Torsion
c) If, additionally, M has finite topological type, a(M, g) =
1
~im
-
-d a(Mj).
)--+00
j
Proof. In the normal case M ----t M below a) follows from theorem 1.38. The proof of b) is quite analogous to that of theorem 1.34 b), using a chopping of M according to theorem 1.33, (1.32) and theorem 1.35. c) then follows from b) and the fact that for sufficiently large k, 1 a (p)-:-l(Mk )) = 1.a(Mj). D J
J
We now turn to the normal case M ----t M, being even more explicit than in the profinite case. The first key here is the extension of Atiyah's L 2 -index theorem for normal coverings M ----t M of closed M to normal coverings M ----t M, M = Mjr, rinj(M) ~ 1, (Mn,g) complete, vol(Mn, g) < 00, IKI ::; 1. We denote by 1{q,2(M) the space of L 2-harmonic q-forms, by PHQ ,2 : L 2(NT* A) = nq,2 ----t 1{q,2 the orthogonal projection. PH has Schwartz kernelliq (x, y) which is a symmetric Coo double form whose pointwise norm satisfies
(1.34) (1.34) comes from geo(M) ::; 1 and the elliptic estimate for the Laplacian. liq(x, y) is invariant under the isomtries r, hence the pointwise trace trliq(x, x) can be understood as function on M and we put as usual
bq,2(M)
:= trrPHQ,2(M) =
J
trliq(x, x) dvolx(g) <
00.
M
q
bq,2(M) is just the von Neumann dimension dimr H ,2 (M) of the r -module -q2 H ' (M). We define the L 2-Euler characteristic and L 2-signature by n
X(2)(M)
:=
L(-1)q{jq,2(M) q=O
31
Absolute Invariants
and
if(2)(M)
:= trr(*P1t2k,2(M4k)).
Now we state the Lrindex theorem for open manifolds with finite volume and bounded curvature. Theorem 1.37 Suppose (M, g) complete with vol(Mn, g) < IKI :::; 1 and M - + M normal with geo(M) :::; 1. Then
00,
X(M,g) = X(2)(M)
(1.35)
(}"(M, g) = if(2) (M).
(1.36)
and
o We refer to [15], [27] for the proof. We recall the existence of good chopping sequences Ml C M2 C 00 _
.. " UMk =
M, vol(8Mk )
-+
0, III(8Mk) I
:::;
c, Ih%(x, y) :::;
1
c(n)1, where h% denotes the kernel corresponding to projection on the harmonic q-forms for p-l(Mk ) C M. Then we obtain
bq ,2(8Mk ) = 0
(1.37)
bQ,2(M \ Mk , 8(M \ M k )) = O.
(1.38)
lim
k-.oo
and lim
k-.oo
Define ~Q,2(B) by
Q
~Q,2(B) := dimrim (HQ,2(p-l(B),p-l(8B)) C H ,2(p-l(B))) (1.39) and for A C B
It follows from the properties of dimr that (1.41 )
and
Relative Index Theory, Determinants and Torsion
32
We remark that (1.41) and (1.42) are the adequate reformulation of (1.24), (1.29) in the language of dimr. We established in theorem 1.37 the equations X(M,g) = X(2)(M), CJ(M,g) = CJ2(M). Now we discuss the invariance properties of the right hand sides. This is the content of
IKI ::; 1, vol(M,g) < and assume for some normal covering geo( M) ::; 1. a) If Ml C M2 C "', UMk = M is an exhaustion then Theorem 1.38 Let (Mn,g) be complete, 00
lim ~q,2(Mk) = lim lim ~q,2(Mk' M[) = 'b q,2(M).
k---->oo
(1.43)
k---->oo [---->00
This implies the homotopy invariance of the 'bq,2(M). b) X(M,g) resp. (J(M,g) is a homotopy invariant resp. proper homotopy invariant of M. c) If M has the topological type of some Mk C M, then
and (1.45) Proof. b) follows immediately from theorem 1.37 and a). For c) suppose that M has finite topological type. Then there exists an exhaustion Ml C M2 C ... S.t. each inclusion Mk -----t M is a homotopy equivalence. This implies
and we obtain (1.44) from (1.43) and moreover X(M, g) = X(Mk ). Hence there remains to show a). For this we must refer to [15]. D
We apply these results on characteristic numbers to 4-manifolds. Let (M4, g) be open, complete and oriented, * : A2 M -----t A2 M the Hodge operator, *2 = 1, A2 = A~ EB A:". The special orthogonal group acts on the space of algebraic curvature tensors Cl
33
Absolute Invariants
C;
(cf. [57]). Let = U +S + W be the corresponding (fiberwise) decomposition into irreducible subspaces. Then this induces for the curvature tensor R = R9 a decomposition R = U + 8 + W. For R = R9 = R+ + R_, we denote by Ric = Ric 9 the Ricci tensor, by 7 = 7 9 the scalar curvature, by K = K9 the sectional curvature and by W = W9 = W+ + W_ the Weyl tensor. There are decompositions for the pointwise norms Ilx as follows 2 2 2 IRI2 IR+12 + IR_12 = IUI + 181 + IWI 41W+12 + IW_12 + 21Ric 12 2 2 61UI + 2181 , 2 241U1 .
IRic 12 72
~72,
(1.46) (1.47)
(l.48)
We obtain still other decompositions if we consider the curvature operator R as acting from A2 = A! EB A=- to A! EB A=-, for an orthonormal basis e I , e2, e3, e4 1 R(ei /\ ej) = "2 Rijklek /\ el = nij,
z=
n = (nij ) = matrix of curvature forms, nij(ek, el) = R ijkl . We can write R with respect to the orthogonal basis el/\ e2 + e3/\ e4, el/\e4 +e2/\e3, el/\e3+e2/\e4 in A!, el/\e3+e2/\e4, el/\e2-e3/\e4, el /\ e4 - e2 /\ e3 in A=-, as
R= with A and
(~ ~)
= A*, C = B*, D = D*, trA = trD =
(~ ~)
-
I~
= W, W+ = A -
~, B
I~' W-
= Ric -
= D - ;2' We
obtain for the first Pontrjagin form PI 1 1 PI - 87r 2tr(R /\ R) = - 87r 2 tr(A /\ A) + tr(D /\ D)
-~2 (-2)(IW+12 87r
IW_12) dvol
~2 (IW+1 2 -IW_12) dvol 47r
~2 (IR+1 2 -IR_12) dvol 127r
~7g
34
Relative Index Theory, Determinants and Torsion
!Pl
and for 0"(M4, g) = J L(g) = J = 12~2 J(IW+12_IW_12) dvol. Assuming g E b,2 M 1,2(Bo, 1, f), 0"(M4, g) is well defined. The Euler form E(g) has the representation
E(g)
1 87r 2tr( *R)2 dvol
~2 (IUI 2 87r
181 2 + IWI2) dvol
1 + D2) dvol 87r 1 327r 2 (IRI2 - 41Ric 12 + T2) dvol.
-2tr(A2 - 2BB*
For g E b,2 M 1 ,2(Bo, 1, f), Hence we obtain
J E(g)
= x(M, g) is well defined.
Proposition 1.39 Let (M4, g) be open, complete, oriented and E b,2M 1 ,2(Bo, 1,1). Then O"(M, g) and X(M, g) are well deD fined and an invariant of comp(g).
g
J
Remark 1.40 According to (1.46) - (1.48), IR g l 2 dvol < 00 would be sufficient for the existence of 0"( M, g) and X( M, g). But this condition would not establish a uniform strucutre, we would not have components and invariance properties (where we used in particular Gaffney's theorem). Moreover, we need the bounded curvature property for the connection with the theorems 1.37, 1.38. D We obtain from proposition 1.39 and its proof the simple
Corollary 1.41 If (M4, g) is additionally Einstein then X(M,g) ~ 0 and lo-(M4,g)1 :S ~X(M4,g). Moreover, X(M4, g) = 0 if and only if (M4, g) is fiat. If (M4, g) is Einstein then 8 == 0, B == 0 and 12~2(IW+12 - IW_12) :S ~8!2(IUI2 + IW+12 - IW_12). Hence 0"(M4,g) :S ~X(M,g). Changing the orientation replaces
Proof.
35
Absolute Invariants
(J(M 4,g) by -(J(M4,g) and we get altogether
IdM4 ,g)1 <
~X(M4,g).
D
The same estimate holds for ~-pinched Ricci curvature.
Proposition 1.42 Suppose the hypotheses of 1.39 and additionally that the Ricci curvature of (M4, g) is negative and ~ pinched, i. e. there exists A > 0 s. t.
-Ag < - Ric < -
-~Ag. 3
(1.49)
Then there holds for all g' E comp(g) C b,2 M 1,2(Bo, 1, J)
1(J(M4,g')1 :::;
~X(M4,g')'
(1.50)
Proof. We have
and
X(M4, g) =
J
E(g) =
J
8~2 (1U1 2- ISl 2+ IWI2) dvol.
Sufficient for (1.50) would be ISI 2:::; is (1.49) as pointed out by [57].
1U12
and sufficient for this D
Examples 1.43 1) Examples for 1.39 with infinite volume are e.g. manifolds M4 of the smooth type M4 = Mt u 8Mt x [0, oo[ where the curvature at the cylinder 8Mt x [0, oo[ is bounded and asymptotically fiat in the sense IRI dvol < 00. This can
J
8M~x[O,oo[
be easily realized by warped product metrics.
36
Relative Index Theory, Determinants and Torsion
2) Examples for 1.39, 1.41, 1.42 with finite volume are given by hyperbolic 4-manifolds of finite volume. 3) Generalizations of these examples are given by variation of 9 inside comp(g). 0
Theorem 1.44 Let (M4, g) be open, complete, vol ( M4, g) < 00, IK I ::; 1 and suppose that (M4, g) admits a normal covering (!Ii, g) satisfying geo( M) ::; 1. a) If x( M4, g) < 0 then M4 does not admit a complete Einstein metricg' satisfyingvol(M4 ,g') < 00, IKg!l::; 1, geo(M4,g')::; 1 for some normal covering. b) If x(M4 ,g) > 0 and 10'(M4)1 > ~X(M4,g) then M4 does not admit a compl!;te Einstein metric g', s.t. vol(M4, g') < 00, IKg!1 ::; 1, geo(M4,g') ::; 1. Moreover, there does not exist a complete metric g' satisfying - Ag' ::; Ric (g') ::; and IKg! I ::; 1, vol(M 4,g') < normal covering.
(X)
~ Ag'
and geo(M4,g') ::; 1 for some
Proof. a) Suppose the existence of an Einstein metric g' with the required properties. Then x( M4, g), x( M4, g') are well defined. X(M4,g) = X(M 4,g'), according to theorem 1.38 b). But this contradicts X(M 4 , g') = J(IUI 2 + IWI2) dvol ~ O. b) and c): Quite analogously we derive by means of theorem 1.38 b), corollary 1.41 and proposition 1.42 a contradiction. 0
8;2
Until now we defined characteristic numbers in the following cases 1) R E L1 and bounded, vol(M) arbitrary, 2) R bounded, vol(M) < 00. There remains the case R bounded, vol(M) = 00. It is clear that in this case we will not get characteristic numbers by integration. (Mn, g) is called closed at infinity if for any 'P E C(M), 0 <
Absolute Invariants
37
A-I < cp < A, A > 0 some constant, the form cpo dvol generates a nontrivial cohomology class in bHn (Mn, g). A fundamental class for M is a positive continuous linear function m : bO,n(M) ---+ IR such that (m, dvol) # 0 and (m, d'lj;) = O. Proposition 1.45 M has a fundamental class if and only if M is closed at infinity. Proof. Denote £( dvol) for the linear hull of dvol, let 0 tt[dvol] E bHn(M) and set (m, dvol) = 1, mlb"Bn == O. Then we obtain by linear extension m on £( dvol) EB b Bn as positive continuous linear functional. The Hahn-Banach theorem for the extension of such functionals yields the desired m. The other direction is absolutely trivial. 0 Define the penumbra for K eM.
xEK
CL(M \ Pen+(M \ K, r)). We call an exhaustion MI C M2 C ... ,
U Mi = M,
by compact
i
submanifolds a regular exhaustion if for each r
~
lim vol(Pen+(Mi , r))/vol(Pen-(Mi , r))
0
=
1.
Z->OO
It is clear that then automatically
lim vol(Pen+(Mi , r))/vol(Mi)
1,
lim vol(Mi)/vol(Pen-(Mi, r))
1.
z->oo
z->oo
Examples 1.46 1) (Mn, g) = (IRn, gstandard) admits a regular exhaustion. 2) Any (Mn,g) with sub exponential growth admits a regular exhaustion. 3) The hyperbolic space admits no regular exhaustion. 0
Relative Index Theory, Determinants and Torsion
38
Let {MJf~l be a regular exhaustion and set for w E
Then l(mi,w)1 ~ sup Iwl x x
= blwl, i.e. Imil
bnn
~ 1, the mi belong to
the unit ball in (bnn)*. This unit ball is compact in the weak star topology, according to the Banach-Alaoglu theorem, hence the sequence {mih has a weak star limit point m. m is then called associated to the regular exhaustion {Mi h. Proposition 1.47 Let m be associated to a regular exhaustion { Mi k Then m is a fundamental class for M. Proof.
There remains only to show (m, d'l/;)
= O. Let
COO(M) such that 0 ~
vf w-
OS;
=
E
0 outside
(vol(Pen+(M" 1)) - vol(M,))'lwL
hence
Therefore we would be done if we could show . 1 hm vo I(M) i
2-->00
f
= O.
M
Integration by parts yields
-f
d
If d
~ 2(vol(Pen+(Mi , 1)) -
vol(Mi))bl'l/;l,
Absolute Invariants
39
o
which implies the assertion.
Define for W E b,lCp(Bo), [Qil ...ik(W)] E bHn(M) a (bounded) characteristic class and a regular exhaustion {Mih with associated fundamental class m the characteristic number
Then, according to proposition 1.47, Qil ... ik(P, comp(w))[m] is well defined. In particular we obtain in this case avarage Euler numbers, avarage signatures, which are special cases of Roe's (avarage) topological index (cf. [60]). Average characteristic numbers are also considered in [44], [45], [42]. Some simple geometric examples are calculated in [44]. In all cases discussed until now, we restricted to the case of connections (or metrics) with finite p-action or bounded curvature or both. The next proposition shows that this is in fact a restriction. Proposition 1.48 Let (Mn , g) be open, complete, satisfying (1), G a compact Lie group, P = P(M, G) a G-principal fibre bundle, (! : G - - t U(N) resp. O(N) a faithful representation, E the associated vector bundle, p ::; 1. Then there exist G-connections W such that their p-action is infinite or the curvature is unbounded or both, respectively. Proof. Consider the closed unit ball B1(0) c lRn and set up in B1(0) constant I-forms Wij, Wij = -Wji or Wij = -Wji' 1 ::; i, j ::; N, respectively, such that some nij = dWij - 2: Wik 1\ Wkj k
are =1= O. Now consider an infinite sequence Uv = Uc:,,(x v ) of closed geodesic balls with pairwise distance ~ d > 0, introduce in each geodesic ball normal coordinates u 1, ... , un, 2: (Ui) 2 ::; i
choose over Uv orthonormal bases el,v,'" eN,v and define with respect to these bases local connection matrices W~j,v by W~j,v(Ul"'" un) := Wij' If In~j,lIl~ dvolx(g) = all =1= 0, set Ev ,
J
u"
Relative Index Theory, Determinants and Torsion
40
W:J'. ,v = (a v + 1.... ) !PW:J. v. all,
This connection over
U Uv
is smoothly
v
extendable over the whole of M and gives a connection with J In"l~ dvolx(g) 2:: L, J In"l~ dvolx(g) 2:: L, 1 = 00. Setting
M
w:J'. ,v
v
v~
=
l/ .
o
(a v
+ 1.... )!p . w:J'. v av,
yields examples for the other cases.
The conditions of finite p-action or boundedness can be reformulated in the language of classifying spaces and classifying mappings. We start with G = U(N). Let VN,k ~ GN,k be the Stiefel bundle over the complex Grassmann manifold G N,k of all ksubspaces c eN and S the matrix valued function on VN,k defined by S(Vl, ... ,Vk) = aij:= (bij)t, where Vl, ... ,Vk is a unitrary k-frame, el, ... , eN the standard base in eN and Vi = N
L, bijej. j=l
Proposition 1.49 a) rU = S*dS is a U(N)-invariant connection form at VN,k. b) Let be m = (n + 1)(2n + 1)k3 . If P is a U(k)-principal fibre bundle over a manifold of dimension ::; nand W a connection form for P, then there exists a smooth bundle morphism fp P --> Vm,k = Pn,U(k) such that fJ,r = w.
o We refer to [55], p. 564, 568 for the proof. rO is called a n-universal connection for U(k). In a similar man.
ner one defines on the real StIefel bundle V~ k n-universal O(k)-connection
roo
'
O(k)
--> G~ r
,
an
For an arbitrary compact Lie group G one constructs by means of a faithful representation G --> O(k) an n-universal connection rG on the n-universal bundle Pn,G --> Bn,G (cf. [20], p. 570). According to proposition 1.49, we refine the bundle concept and consider instead of a bundle P pairs (P, fp), fp : P --> Pn,G a
Absolute Invariants
41
C 1-classifying bundle map. (P, fp) is called a (p,J)-bundle if f;I'o E C 1Cp(J,p) = {w a C 1 connectionIJlnwl~dvolx(g) < oo}, i.e. Jlnf;'YGI~dvolx(g) < 00. In the same manner we define (P, fp) to be a lr-bundle if f;I'o E C 1Cp(Bo), i.e. blnf;-w I < 00. The for the applications most interesting case is the case assuming (Bo) and finite p--action. Hence we assume (Bo) for (Mn,g). (P,fp) is a (b,p, f)-bundle, if f;I'o E b,lq,l(Bo, f,p)· Two (b,p, f)-bundles (P,Jp) , (P, f~) are called equivalent if fj,I'o, f'~I'0 are contained in the same component of b,lC~l(Bo, f,p). Assume G to be a subgroup of U(N), dim M n = 2k. At the level of base spaces we consider classifying maps f M : M ---t Bn,o. A pair (M, fM) is called a (p, c)-bundle if all classes fivrch ...ik' i 1 + .. ·i k = k, are elements of H 2k ,p(M). (M,fM) is called a (b,c)bundle if all classes fivrcil ...ik are elements of bH 2k (M). (M, fM) is called a (b, p, c)-bundle if all classes fivrcil ...ik are elements of bH 2k,p(M). It is clear that a given fp : P ---t Pn,o uniquely determines f M : M ---t Bn,o. The case G ~ O(N), dimM = 4k, is quite parallel. Then we consider the Ph ...ik' i 1 + ... ik = k and define (M, fM) to be a (p, po )-bundle if all classes fivrPh ...ik' i 1 + ... ik = k are elements of H 4k ,p(M). Analogously for (b,po)- and (b,p,po)bundles (M, fM)' If we replace Ph ...ik by the class of Hirzebruch genuss Lk then we get the notion of a (p, L k )-, (b, Lk)- or (b, p, Lk)-bundle (M, fM), respectively. Theorem 1.50 a) Suppose G c U(N), dim M = 2k. (M, g) satisfying (Bo), p 2 1. A (b,p, f)-bundle (P,Jp) defines a unique (b,p)-bundle (M, fM)' If (P, fp), (P, f~) are equivalent then fivrch ...ik = f'~Cil ...ik for all Cil ...ik' i 1 + ... + i k = k. If additionally p = 1 and (M, g) is complete then even the corresponding characteristic numbers coincide. b) Suppose G ~ O(N), dimM = 4k, (M,g) satisfying (Bo), p 2 1. A (b,p,f)-bundle (P,fp) defines a unique (b,p,po)-
42
Relative Index Theory, Determinants and Torsion
bundle (M,fM) which is simultaneously a (b,p, Lk)-bundle. If
(P, fp), (P, f'p) are equivalent then fMPil ... ik = f'~Pil ...ik and fMLk = f'~Lk' If additionally P = 1 and (M, g) is complete then the corresponding characteristic numbers coincide.
The proof follows immediately from the definitions and theorem 0 1.14. Example 1.51 It it possible that b,lC~,l(Bo, 1, f) = 0. Let (M2, g) be an infinitely connected open complete Riemannian manifold with bounded sectional curvature K, K = K+ - K_. K, K 2 0 { - K, K:==; 0 K+ = { 0, K < 0 ,K_ = 0, K > 0 ,Then there holds
J K_ dvol = 00 (cf. [43], theorem 13). In particular J IKI dvol = 00 which implies J I[2W(g)1 dvol = 00. The proof essentially relies on the GauB-Bonnet theorem (as one would expect) for compact surfaces. But this theorem holds for any metrizable connection in the orthogonal 2-frame bundle P(M2, 0(2)) over M2 ([47], p. 305/306). The sectional curvature K is defined by [21,2 = K dvol. As conclusion we obtain b,lCp (B o, 1,1) = 0. 0 We conclude this section with some remarks concerning the Novikov conjecture for open manifolds. As very well known, the Novikov conjecture for closed manifolds stimulated many outstanding topologists to prove this and on this road deep results in C* algebraic topology, C* K -theory and geometric group theory have been achieved. Hence, the Novikov conjecture has not only its own meaning but even more meaning as a stimulating question. If Mn is open and we consider the classifying diagram
M 1 M
---+
f
Bn:
43
Absolute Invariants
and a E H*(B1r) then
(L(M) . j*a, [M]) will not be defined in general. For this reason, Gromov proposes to consider fJa(M) = (L(M) . j*a, [M]) for a E H;(B1r)' Then the NC for open manifolds would mean the "invariance of fJa(M) under proper homotopy equivalences". Probably much more appropriate would be an approach in the sense of our " open category", i.e.
1) everything is uniformly metrized, we have (I), (B k ), uniform triangulations etc., 2) maps are bounded and uniformly proper, in particular this holds for homotopy equivalences, 3) one works within functional algebraic topology. Hence one should consider
(L(M) . j*a, [M])
with
L(M)
E
L p , j*a
E
Lq .
Of particular meaning would be the cases (1.51)
or (1.52)
respectively. If we suppose (M, g) satisfying (Bo) then automatically L(M) E bH*(M). (Bo) does not restrict to topological type since any open manifold admits a metric g satisfying even (Boo) and (I). In the second case one should additionally assume (1.53)
i.e. there is a spectral gap of .6.* above zero. In this case H*,2 = H*,2 = L 2 -harmonic forms, C*,2, C*,2 are L2-complexes and
44
Relative Index Theory, Determinants and Torsion
form an L 2 -Poincare complex. Every L 2-( co-) homology class can be represented by an L 2 -harmonic (co-) cycle. Bordism of L 2 -Poincare complexes can be defined easily. We proved in [34] that (1.53) is invariant under bounded uniformly proper homotopy equivalences. W.l.o.g., classifying maps can be assumed to be bounded and uniformly proper,
Mn
---t
Bn = M n U cells.
We present now 3 versions of NC (for open manifolds).
1. Version. In the class of open oriented manifolds (Mn, g), 9 E b,2 M 2,2(Bo, 2, f) with inf O'e(~*(g)l(ker~*).L) > 0 is
(L(M)f*a, [MD,
a E H*,2(B n ),
f
bounded and
uniformly proper classifying map, invariant under bounded and uniformly proper homotopy equivalences.
(NCOl)
Criticism. This version should hold only in very restricted cases. Starting point in the compact case is the equality (1.54) where the l.h.s. is a priori a homotopy invariant and the r.h.s. is a certain characteristic number. The Lrversion of (1.54) is already wrong in simple open cases. Let (M4k, g) be an open manifold with cylindrical ends, i.e. (M4k, g) = (M,4k U 8M,4k X [0,00[, g) with glaMl4kx[O,oo[ ~ giaMI + dt 2. Then it is well known that
0'(M4k)
=
O'L2(M 4k ) =
J
Lk(M) - ry(8M,4k) ,
i.e. already the starting point which guarantees the invariance of L(M) in the simplest case is wrong. Hence the first version of NC for open manifolds makes sense only for that classes of open manifolds for which
Absolute
in the case n
Invariants
45
= 4k holds.
2. Version of NC, relative version. Fix (Mn,g) and suppose M 1 , M2 E genbcompL,iSO,rel(M, g)
Ml \Kl ~ M\K
M2 \K2 ~ M\K with a Riemannian collar at oK!, oK2, oK. Then we define
J
J
Ki
K
cr(Mi' M) :=
L(Mi) -
L(M)
cr(M!, M 2) := cr(Ml' M) - cr(M2' M)
=
J
J
L(Ml) -
Kl
L(M2)
K2
= cr(Kl U K 2)
~ JL(M,J - ~(aK,) Kl
(
J
L(M2) -
~(aK2)))
\)2
= cr(Kl) - cr(K2)' The relative NC becomes
J
L(Ml)f{a =
Kl
JL(M2)f~a
(NC02)
K2
if there exist ~12 : Ml ----t M 2, ~21 : M2 ----t M 1 , bounded, uniformly proper, ~21 ~12 rv idM1 , ~12~21 rv idM2 bounded and ~.p. and ~21 ~12 = id outside [(1 c MI, ~12~21 = id outside K2 C M2 and fi : Mi ----t Bn are bounded and u.p. classifying maps, a E H*(Bn). This relative version has the advantage that we require no conditions for (Mn, g) and NC splits to NC for the generalized Lipschitz components (cf. [27], [33]). 3. Version of NC. Consider (Mn,g) open, oriented with (Bo), > 0, embeddings N4k '---t Mn x IR,1 with trivial normal bundle
rinj
46
Relative Index Theory, Determinants and Torsion
and bounded second fundamental form such that PD[N] = f*a, f : Mn ---t B7r bounded and uniformly proper classifying map and such that (1L 2(N4k) is defined (i.e. dim 1{2k,2(N) < 00).
a E Hn-4k,1 (B7r ),
Then the number (1a(M) := (1L2(N 4k ) is invariant under bounded and uniformly proper homotopy invariants.
(NC03) How to attack these conjectures will be the content of a forthcoming investigation.
2 Index theorems for open manifolds Let (Mn, g) be closed, oriented, (E, hE), (F, hF ) ---t Mn smooth vector bundles, D : Coo(E) ---t Coo(F) an elliptic differential operator. Then L2(E) ::::l V]5 ~ L2(F) is Fredholm, i.e. there exists P : L2(F) ---t L 2(E) s.t. PD - id = Kl, DP - id = K 2, Ki integral operators with Coo kernel Ki and hence compact. It follows dim ker D, dim coker D < 00, indaD = dim ker D dim coker D is well defined and there arises the question to calculate indaD. The answer is given by the seminal Atiyah-Singer index theorem
Theorem 2.1 where
indtD = (ch (1(D)T(M), [MJ).
o Assume now (Mn,g) open, E,F,D as above. Kl,K2 still exist as operators with a smooth kernel where in good cases one can achieve that the support of Ki is located near the diagonal. But there arise several troubles. 1) If Ki bounded is achieved then Ki must not be compact.
Absolute Invariants
47
2) If Ki would be compact then indaD would be defined. 3) If indaD would be defined then indtD must not be defined. 4) If indaD, indtd (as above) would be defined then they must not coincide. There are definite counterexamples. There are 3 ways out from this difficult situation. 1) One could ask for special conditions in the open case under which an elliptic D is still Fredholm, then try to establish an index formula and finally present applications. These conditions could be conditions on D, on M and E or a combination of both. In [2] the author formulates an abstract (and very natural) condition for the Fredholmness of D and assumes nothing on the geometry. But in all substantial applications this condition can be assured by conditions on the geometry. The other extreme case is that discussed in [22], [50], [48], where the authors consider the L 2 -index theorem for locally symmetric spaces. Under relatively restricting conditions concerning the geometry and topology at infinity the Fredholmness and an index theorem are proved in [11] and [12]. 2) One could generalize the notion of Fredholmness (using other operator algebras) and then establish a meaningful index theory with applications. The discussion of these both approaches will be the content of this section. 3) Another approach will be relative index theory which is less restrictive concerning the geometrical situation (compared with the absolute case) but its outcome are only statements on the relative index, i.e. how much the analytical properties of D differ from those of D'. This approach will be discussed in detail in chapter V. 4) For open coverings (.tV!, g) of closed manifolds (Mn, g) and lifted D there is an approach which goes back to Atiyah, (cf. [4]). This has been further elaborated by Cheeger, Gromovand others. The main point is that all considered (Hilbert-) modules are modules over a von Neumann algebra and one replaces the usual trace by a von Neumann trace. We will not dwell on this approach since there is a well established highly elaborated theory. Moreover special features of openess come not
48
Relative Index Theory, Determinants and Torsion
into. The openess is reflected by the fact that all modules under consideration are modules over the von Neumann algebra N (7r), 7r = Deck( if - t M). We refer to the very comprehensive representation [46]. This section is a brief review of absolute index theorems under additional strong assumptions. It shows that these approaches are successful only in special situations. In chapter V we will establish very general relative index theorems. We start with the first approach and with the question which elliptic operators over open manifolds are Fredholm in the classical sense above. Let (Mn, g) be open, oriented, complete, (E, h) - t (Mn, g) be a Hermitean vector bundle with involution T E End (E), E = E+ EEl E-, D : COO(E) - t COO(E) an essentially self-adjoint first order elliptic operator satisfying DT + TD = O. We denote D± = Dlcoo(E±)' Then we can write as usual
(2.1) The index indaD is defined as indaD .- indaD+:= dim ker D+ - dim coker D+ dim ker D+ - dim ker D-
(2.2)
if these numbers would be defined. Denote by 02,i(E, D) the Sobolev space of order i of sections of E with D as generating differential operator. We essentially follow [2].
Proposition 2.2 The following statements are equivalent a) D is Fredholm. b) dim ker D < 00 and there is a constant c > 0 such that
where (ker D).L ker D in L2(E).
==
1{.L
is the orthogonal complement of
1{
=
Absolute Invariants
49
c)
There exists a bounded non-negative operator P 0,2,2(E, D) ---+ L 2(E) and bundle morphism R E COO (End E), R positive at infinity (i. e. there exists a compact K C M and a k> 0 s. t. pointwise on EIM\K, R ~ k), such that on 0,2,2(E, D) D2
= P+R.
(2.4)
d) There exist a constant c > 0 and compact K C M such that
o The main task now is to establish a meaningful index theorem. This has been performed in [2].
Theorem 2.3 Let (Mn , g) be open, complete, oriented, (E,h,r) = (E+ EEl E-,h) ---+ (Mn,g) a Z2-graded Hermitean vector bundle and D : c;:(E) ---+ c;:(E) first order elliptic, essentially self-adjoint, compatible with the Z2 -grading (i. e. supersymmetric), Dr + r D = o. Let K C M be a compact subset such that 2.2 a) for K is satisfied, and let f E cOO(M,lR) be such that f = 0 on U (K) and f = 1 outside a compact subset. Then there exists a volume density wand a contribution Iw such that indaD+ = (w(l - f(x)) dvolx(g) + I w , (2.6)
J
M
where w has an expression locally depending on D and Iw depends on D and f restricted to 0, = M \ K. 0
Until now the differential form w dvolx(g) is mystery. One would like to express it by well known canonical terms coming e.g. from the Atiyah-Singer index form ch (J(D+) U T(M), where T(M) denotes the Todd genus of M. In fact this can be done.
Index Theorem 2.4 Let (Mn, g) be open, oriented, complete, (E, h, r) ---+ (Mn, g) a Z2-graded Hermitean vector bundle, D :
50
Relative Index Theory, Determinants and Torsion
C':(E)
------t C':(E) a first order elliptic essentially self-adjoint supersymmetric differential operator, DT + TD = 0, which shall be assumed to be Fredholm. Let K C M compact such that 2.2 d) is satisfied. Then
indaD+ =
J
ch (J(D+) U T(M)
+ In,
(2.7)
K
where ch dD+) uT(M) is the Atiyah-Singer index form and In is a bounded contribution depending only on Din, n = M \ K.
o Remarks 2.5 a) As we already mentioned, Zrgraded Clifford bundles and associated generalized Dirac operators D such that in D2 = 6,E + R, R ~ c· id, c > 0, outside some compact K c M, yield examples for theorem 2.3. A special case is the Dirac operator over a Riemannian spin manifold with scalar curvature ~ c > 0 outside K eM. b) Much more general perturbations than compact ones will be considered in section V 1. 0 The other case of a very special class of open manifolds are coverings (M,g) of a closed manifold (Mn,g). Let E,F ------t (Mn,g) be Hermitean vector bundles over the closed manifold (Mn, g). D : COO (E) ------t COO(F) be an elliptic operator, (M, g) ------t (M, g) a Riemannian covering, D : C':(E) ------t C':(F) the corresponding lifting and f = Deck (Mn, g) ------t (Mn, g). The actions of f and D commute. If P : L2 (M, E) ------t 1t is the orthogonal projection onto a closed subspace 1t c L 2 (M, E) then one defines the f -dimension dimr 1t of 1t as dimr 1t := trrP, where trr denotes the von Neumann trace and trrP can be any real number ~ 0 or = 00. If one takes 1t = 1t(D) = ker D C L 2 (E), 1t* = 1t(D*) ker(D*) C L2(1') then one defines the f-index indrD as indrD := dimr 1t(D) - dimr 1t(D*).
Absolute Invariants
51
Atiyah proves in [4] the following main Theorem 2.6 Under the assumptions above there holds
o It was this theorem which was the orign of the von Neumann analysis as a fastly growing area in geometry, topology and analysis. Moreover, the proof of theorem 2.3 is strongly modeled by that of 2.6. Another very important special case which is related to the case above of coverings are locally symmetric spaces of finite volume. There is a vast number of profound contributions, e. g. [7], [22], [48], [50], [51]. We do not intend here to give a complete overview for reasons of space. But we will sketch the main features and main achievements of these approaches. Let G be semisimple, noncompact, with finite center, KeG maximal compact, X = G / K a symmetric space of noncompact type, f c G discrete, torsion free and vol(f\G) < 00. Then X = f\X = f\ G / K is a locally symmetric space of finite volume. If VE , Vp are unitary K -modules then we obtain homogeneous vector bundles E = G / K x K VE ----t G / K = X, F = G / K x K Vp ----t G / K = X, over X and corresponding bundles E, F ----t X over X. A G-invariant elliptic differential operator D : COO(E) ----t COO(F) descends to an elliptic operator D : COO(E) ----t COO(F). There arise the following natural questions: to describe the D in question, to establish a formula for the analytical index, to calculate the index via a topological index and an index theorem. We indicate (partial) answers given by Barbasch, Connes, Moscovici and Muller. Denote by R(k) the right regular representation R(k)f(g) = f(gk), TE : K ----t U(VE)' Then k ----t R(k) 0 TE(k) acts on COO(G) ® VE. We identify COO(E) with (COO(G) 0 VE)K, similarly L2(E) with (L2(G) ® VE)K. If ® is the Lie algebra of G, ®c its complexification, U(®) the universal enveloping algebra of ®, TE : K ----t U(VE), Tp : K ----t U(Vp) are unitary
Relative Index Theory, Determinants and Torsion
52
representations then (U( ®) ® Hom (VE, VF))K shall denote the subspace of all elements in U(®) ®Hom (VE , VF ) which are fixed under k ------+ AdG(k) ® TE(k-1)t ® TF(k). Let d = L: Xi ® Ai E i
(~(®) ® Hom (VE, VF))K. Then
D = L:R(Xi ) ® A
defines a
differential operator D : Coo(E) ------+ Coo(F) commuting with the action of G. We state without proof the simple
Lemma 2.7 a) Any G-invariant differential operator D ------+ Coo(F) is of the form
:
Coo(E)
(2.8) above. b) The formal adjoint
d*
=
D*
corresponds to
L xt ® A7 E (U(®) ® Hom (E, F))K, i
where x ------+ x* denotes the conjugate-linear anti-automorphisms of U(®) such that x* = -x, x E ®e. 0 For a unitary representation 7f : G
A
------+
U(H(7f)) and d =
L: Xi® i
(U(®) ® Hom (VEl VF))K define 7f(d) : H(7f)oo ® VE H(7f)oo ® VF by E
------+
Here H(7f)oo denotes the space of COO-vectors of 7f. 7f(d) induces an operator drr : (H(7f) ® VE)K ------+ (H(7f) ® VF)K.
Proposition 2.8 Suppose that d is elliptic. Then ker drr = {u E (Hom (7f)oo ® VE)K I drru = O}
coincides with the orthogonal complement of im d; = {d;v I v E (H(7f)oo ® VF)K}
in (H(7f) ® VE)K.
o
53
Absolute Invariants
Corollary 2.9 a) kerd 7r is closed in (1i(n-) ® E)K.
b) The closure of d; coincides with the Hilbert space adjoint of d7r •
D
Corollary 2.10 Suppose that d is elliptic and
e
e
1f
=
J
1f>..
d>',
1i(1f) =
A
J
1i(1f>..) d>'
A
is an integral decomposition of 1f. Then
e
ker d7r =
J
(2.9)
ker d7r ), d>..
A
D
Now we come to the main part of our present discussions, the locally symmetric case. Identifying L2(E) with (L2(f\G) ® VE)K, and taking into consideration the decompositions
Rr
= R~ EB R~,
L 2(f\G)
=
L 2,d(f\G) EB L 2,c(f\G)
of the right quasi-regular representation R r of G on L2 (f\ G), we obtain the decomposition
L2(E) L 2 ,d(E) L 2,c(E)
L 2,d(E) EB L 2,c(E), (L 2,d(f\G) ® VE)K, (L 2,c(f\G) ® VE)K,
similarly for F = f\F. Consider now the operators D = dRr and Dd = dRrd : C':'(E)
-----+
C':'(F).
Theorem 2.11 Under the assumptions above (on G, K, f), ker D
= ker Dd
(2.10)
and dim ker D <
00.
(2.11)
54
Relative Index Theory, Determinants and Torsion
Denote by G~ the set of all equivalence classes of irreducible unitary representation 7f of G whose multiplicity mr(7f) in R~ is nonzero. In particular L 2 ,d(r\G) = L mr(7f)H(7f). 1fEG~
Theorem 2.12 Let KeG be maximal compact, rEG discrete and torsion free, TE : K ~ VE , TF : K ~ VF unitary representations, E = GIK XK VE , F = GIK XK VF , E = r\E, F = r\F and D = dRr a corresponding locally invariant elliptic differential operator acting between L2 (E) and L2 (F). Then indaD = dim ker D - dim ker D* is well defined and
(2.12)
o Corollary 2.13 Let X = r\GIK be a locally symmetric space of negative curvature with finite volume and L2(E) => 'DD ~ L2 (F) a locally symmetric elliptic differential operator then ind D is defined and depends only on the K -modules K ~ U(VE ), U(VF ) which define E, F, E = r\E, F = r\F. 0
The value of the formula in theorem 2.12 is very limited since in general the mr(7f) are not known. Hence there arises the task to find a meaningful expression for it. This has been done with great success e. g. in [22] and [51], [52] where they essentially restrict to generalized Dirac operators. To be more precise, we must briefly recall what is a manifold with cusps. Here we densely follow [50]. Let G be a semis imp Ie Lie group with finite center, KeG a maximal compact subgroup. Pa split rank one parabolic subgroup of G with split component A, p = U AM the corresponding Langlands decomposition, where U is the unipotent radical of P, A a IR-split torus of dimension
Absolute Invariants
55
one and M centralizes A. Set S = U M and let f be a discrete uniform torsion free subgroup of S. Then Y = f\ Y = f\ G I K is called a complete cusp of rank one. Put K M = M n K, KM is a maximal compact subgroup_ of M. If X M = MI Ky there is a canonical diffeomorphism ~: IR+ x U x X M ----+ Y. Set for t 2: 0 ~ = €([t, oo[ xU x X M ) and call yt = f\~ a cusp of rank one. Another, even more explicit description is given as follows. Let fM = M n (Uf), Z = SIS n K. Then there is a canonical fibration P : f\Z ----+ f M \XM with fibre f n U\ U a compact nilmanifold and a canonical diffeomorphism ~ : [t,oo[xf\Z ~ yt. The induced metric on [t,00[Xf2\Z looks locally as ds 2 = dr 2 + dx 2 + e-brdui{x) + e-4brdu~A(x), where Ibl = \ dx 2 is the invariant metric on X M induced by restriction of the Killing form. Now a complete Riemannian manifold is called a manifold with cusps of rank one if X has a decomposition X = XOUX1 U· . ,UXs such that Xo is a compact manifold with boundary, for i, j 2: 1, i =1= j holds Xi n Xj = 0 and each Xj, j 2: 1, is a cusp of rank one. The first general statement for generalized Dirac operators on rank one cusps manifold is Theorem 2.14 Let X be a rank one cusp manifold, (E, h, \7,.) ----+ (X, gx) a Clifford bundle and D its correspond-
ing generalized Dirac operator. Then D is essentially self-adjoint and (2.13) dim(ker D) < 00. The spectrum of H = D2 consists of a point spectrum and an absolutely continuous spectrum. If L2(E) = L 2,d(E) EEl L 2,c(E) is the corresponding decomposition of L 2(E) and Hd = HIL 2,d(E) then for t > 0 e- zHd is of trace class. (2.14) D
As we mentioned after corollary 2.13, the main task, main objective consists in the case of a Z2-grading to get an expression for indaD. For the sake of simlicity we restrict to spaces
56
Relative Index Theory, Determinants and Torsion
x = Xo U Y1 as above with one cusp Yi, Yo U Y1 = Y = f\G/ K. Let (E = E+ EB E-, h, V', .) ~ (Y, g) be a Z2-graded Clifford bundle such that E± IYl = f\E±, where E± are homogeneous vector bundles over G / K and let D+ : Coo (Y, Et) ~ Coo(y, E-) the corresponding generalized Dirac operator. We recall KM = MnK, X M = M/KM . D+ induces an elliptic differential operator Dt : Coo(IR+ x fM\X M , Et) ~ Coo(IR+ x fM\X M , EM)' where E! are locally homogenous vector bundles over f M \XM . From this come a self-adjoint differential operator DM : Coo(fM\XM,Et) ~ Coo(fM\XM,EM ) and a bundle isomorphism (3 : Et ~ EM such that Dt = (3 (r! + DM). We set DM = DM + ~id, m = dim U.>.IAI + 2 dim u2.>.IAI, A the unique simple root of the pair (P, A). W. Muller then established in [50] the following general index theorem for a locally symmetric graded Dirac operator. Theorem 2.15 Assume ker DM = {O}, let 7](0) be the eta invariant of DM and WD+ the index form of D+. Then
indaD+ =
J
WD+
+ U + ~7](0),
(2.15)
x where the term U is essentially given by the value of an L-series at zero and an expression in the scattering matrix at zero. 0 Finally, application of an elaborated version of theorem 2.15 allows to prove the famous Hirzebruch conjecture for Hilbert modular varieties. This has been done by W. Muller in [51]. There is another approach to Fredholmness by Gilles Carron, which relies on an inequality quite similar to 2.2 d). Let (E, h, V', .) ~ (Mn, g) be a Clifford bundle over the complete Riemannian manifold (Mn, g) and D : Coo(E) ~ Coo(E) the associated generalized Dirac operator. D is called nonparabolic at infinity if there exists a compact set K c M such that for any open and relative compact U c M \ K there exists a constant C(U) > 0 such that C(U)I
(2.16)
Absolute Invariants
57
To exhibit the consequences of this inequality, we establish another characterization of it.
Proposition 2.16 Let (E, h, \7,.) - - t (Mn, g) and D as above and let W(E) be a Hilbert space of sections such that a) C~(E) is dense in W(E) and b) the injection C~(E) <-t n~~!(E, D) extends continuously to W(E) - - t n~~~(E, D). Then D : W(E) - - t L 2(E) is Fredholm if and only if there exist a compact K C M and a constant C(K) > 0 such that
o Remark 2.17 The norm
--t
ID
Corollary 2.18 D : COO(E) - - t COO(E) is non-parabolic at infinity if and only if there exists a compact K C M such that the completion of C~(E) w. r. t. N K (·),
NK(
1
(2.19)
yields a space W(E) such that the injection C~(E) n~~!(E, D) continuously extends to W(E).
--t
0
The point now is that we know if D is non-parabolic at infinity then D : W(E) - - t L 2(E) is Fredholm. We emphasize, this does not mean L 2(E) ::) DD - - t L2(E) is Fredholm. We get a weaker Fredholmness, not the desired one. But in certain cases this can be helpful too.
D-)
0 Suppose again a Z2-grading of E and D, D = ( D+ 0 ' L 2(E) = L2(E+) EB L 2(E-), W(E) = W(E+) EB W(E-). Following Gilles Carron, we now define the extended index indeD+
58
Relative Index Theory, Determinants and Torsion
as indeD+ .- dim ker wD+ - dim ker L2Ddim{'P E W(E+) I D+'P = O}- dim{ 'P E L 2 (E-) I D-'P = O}.
(2.20)
If we denote hoo(D+) := dim(kerw D+ / kerL2 D+) then we can (2.20) rewrite as
hoo(D+) hoo(D+)
+ indL2 D+ + dim ker L2D+ -
dim ker L2D-. (2.21)
The most interesting question now are applications and examples. For D = GauE-Bonnet operator, there are in fact good examples (cf. [12]). For the general case it is not definitely clear, is non-parabolicity really a practical sufficient criterion for Fredholmness since in concrete cases it will be very difficult it to establish. In some well known standard cases which have been presented by Carron and which we will discuss now it is of great use. Proposition 2.19 Let D : COO(E) ---+ COO(E) be a generalized Dirac operator and assume that outside a compact K c M the smallest eigenvalue Amin(X) of Rx in D2 = \7*\7 + R is ~ O. Then D is non-parabolic at infinity. D
We obtain from proposition 2.18 Corollary 2.20 Assume the hypothesis of 2.18. ---+ L 2 (E) is Fredholm.
Wo(E)
Then D : D
Under certain additional assumptions the pointwise condition on Amin (x) of Rx can be replaced by a (weaker) integral condition. Denote R_(x) = max{O, -Amin(X)}, where Amin(X) is the smallest eigenvalue of Rx.
Absolute Invariants
59
Theorem 2.21 Suppose that for a p > 2 (Mn, g) satisfies the Sobolev inequality
(j
~
lui"'" (x) dVOl.(9))
,
IduI 2 (x) dvolx(g) for all u
E
cp(M)
:; J
C':(M)
(2.22)
M
and
JIn_I~(x)dvolx(g)
<
00.
M
Then D : Wo(E)
----7
o
L 2 (E) is Fredholm.
Another important example are manifolds with a cylindriccal end which we already mentioned. In this case, there is a compact sub manifold with boundary K C M such that (M \ K, g) is isometric to (]O, oo[xoK, dr 2 + g8K). One assumes that (E, h)lJo,oo[x8K also has product structure and DIM\K = /I . (tr + A), where /I. is the Clifford multiplication with the exterior normal at h} x oK and A is first order elliptic and self-adjoint on EI8K. Proposition 2.22 D is non-parabolic at infinity. Proof. There are two proofs. The first one refers to [5]. According to proposition 2.5 of [5], there exists on M \ K a parametrix Q : L 2 (EIM\K) ----7 n~~!EIM\K' D) such that QDcp = cp for all cp E C~(EIM\K)' Hence for C~(EIM\K)' U ~ M \ K bounded,
The other proof is really elementary calculus.
C~(EIM\K)'
Icp(r,y)1 =
II~drl
::;
For cp E
vr '1~~IL2'
Hence
Relative Index Theory, Determinants and Torsion
60
The authors of [5] define extended L 2-sections of EI]0,oo[x8K as sections
.}.x.EU(A) be a complete orthonormal system in L 2 (EI8K) consisting of the eigensections of A. Then we can a solution
L
c>.e->.r
.(y)
(2.23)
>'Eu(A) and
.
<po(r, y)
=
L
= 0 for A <
c>.e->.r
.(y) ,
O. In this case
L
=
CO,i
>'Ea(A)
>'Ecr{A)
>'>0
o This proposition can also be reformulated as Proposition 2.24 Denote by p~o or P
D
= 0 on K
and P
= 0 on 8K.
b)
Absolute Invariants
61
and P-:;o'P
=0
on oK.
o There is a very general approach to index theory as established by Connes, Roe and others. The initial data are as follows: D an elliptic differential operator as above, Q3 an operator algebra, the K -theory Ki (Q3) of Q3, the cyclic cohomology H C* (Q3) of Q3. Then one constructs the diagram
D
1 ID
----+
1 (I D, m) = indtD
?
~ indaD = (Ind D, ()
Here ID is of cohomological nature, m a fundamental class, (ID , m) a pairing, Ind D comes from ellipticity and the 6 term exact sequence of K -theory, ( E H C* (Q3) and (Ind D, () is the Connes' pairing. Choice of Q3, i, (, m, Ind D yields a concrete index theory. We refer to [60], [61], [62], [74] for details. The classical index theory on closed manifolds is given by the choice i = 0, Q3 = ideal K of compact operators, Ind D E Ko(K) = projectors - projectors, HC o :7 ( = trace, trInd D = indaD, ID = classical index from, m = [M]. The lack of all these (absolute) index theories for open manifolds is that they either refer to very special cases or there are not enough serious applications. This was one of the motivations for us to establish a general relative index theory as in chapter V.
II Non-linear Sobolev structures Relative invariants i(P1 , P2 ) are invariants defined for pairs (PI, P2 ), where P2 has to be considered as a perturbation of PI, where P e.g. stands for a triple (manifold, bundle, differential operator). Our approach consists in defining a metrizable Sobolev uniform structure for the set of Ps and defining generalized components gencomp(P) = {Pi Idistance(P, Pi) < oo}. The admitted perturbations of P are just the pi E gencomp(P). In special cases, the generalized components coincide with path components which was the motivation for the notation "generalized component" . To perform the indicated program, we have to collect results on Sobolev spaces for open manifolds, to introduce uniform spaces and to connect both concepts in the theory of non-linear Sobolev structures of manifolds and Clifford bundles. Since Sobolev maps enter into the definitions, we enclose an outline of manifolds of maps between open manifolds. The notion of generalized components of manifolds offers a certain approach to the classification problem. In a first step one "counts or classifies" the generalized compoenents gen comp( Mn, 9). In a second step one "counts or classifies" the manifolds (Min, 9' ) inside a generalized component gen comp (Mn, 9). Here we understand under "count or classify" not a complete classification but steps toward this goal. A "complete classification" is nowaday even for closed manifolds not available. This chapter is organized as follows. We start in section 1 with Sobolev spaces, embedding theorems, module structure theorems for open manifolds and establish some theorems which will be essential in chapter IV. The general approach to uniform structures of proper metric spaces will be presented in section 2. Completed manifolds of maps, in particular completed diffeomorphism groups, are the content of the third section. The heart of chapter II is section 4, where we introduce uniform structures 62
Non-linear Sobolev Structures
63
of manifolds and Clifford bundles. As a first application, we give an outline of the classification approach for open manifolds. In particular, we define new (co-) homologies, define relative characteristic numbers and apply this to bordism theory for open manifolds.
1 Clifford bundles, generalized Dirac operators and associated Sobolev spaces In the first part of this section, we consider the Sobolev spaces associated to Clifford bundles, the W - and H -spaces. In the second part we introduce Sobolev spaces for arbitrary Riemannian vector bundles, the O-spaces which we essentially need in section 3. We recall for completeness very briefly the basic properties of generalized Dirac operators on open manifolds. Let (Mn, g) be a Riemannian manifold, m E M, Cl(TmM, gm) the corresponding Clifford algebra at m. Cl(Tm, gm) shall be complexified or not, depending on the other bundles and structure under consideration. A Hermitean vector bundle E - t M is called a bundle of Clifford modules if each fibre Em is a Clifford module over Cl(Tm, gm) with skew symmetric Clifford multiplication. We assume E to be endowed with a compatible connection \IE, i.e. \I E is metric and \I~(Y .
= (\I~Y) .
+ Y . (\I~
X, Y E r(T M), E r(E). Then we call the pair (E, \IE) a Clifford bundle. The composition
r(E) ~r(T*M®E) ~r(TM®E) ~r(E) shall be called the generalized Dirac operator D. We have D = D(g, E, \I). If Xl, ... Xn is an orthonormal basis in TmM then n
D =
LXi' \I~i' i=l
64
Relative Index Theory, Determinants and Torsion
D is of first order elliptic, formally self-adjoint and
D2 =!:::,.E +R, where !:::,.E = (V'E)*V'E and R E r(End(E)) is the bundle endomorphism 1 n Rip = "2 XiXj RE (Xi , Xj)ip.
L
i,j=l
Next we recall some associated functional spaces and their properties if we assume bounded geometry. These facts are contained in [8], [36] and partially in [27]. Let E --t M be a Clifford bundle, V' = V'E, D the generalized Dirac operator. Then we define for ip E f( E), p 2:: 1, r E Z, r 2:: 0, liplwp,T
liplHp,T
-
.-
(J ~ IV;I~ (J ~ ID;I~
1
dVOI.(9)) , 1
dVOIA9)) ,
{ ip E f(E) Iliplwp,T <
Wf(E) WP,T(E) .-
oo} ,
completion of Wf w. r. t .
H~(E)
'-
HP,r(E)
'-
{ip E r(E) IlipIHP,T <
completion of
H~
I
IWP,T,
I
IHP,T.
oo} ,
w. r. t.
In a big part of our consideration we restrict to p = 1,2. In the case p = 2 we write w 2 ,r == wr, H 2 ,T == Hr etc.. If r < then we set
°
WT(E) .- (W-T(E)r, HT(E) .- (H-r(E)
r.
Assume (Mn, g) complete. Then ego (E) is a dense subspace of Wp,l (E) and HP,l (E). If we use this density and the fact IDip(m)1 :S
e . lV'ip(m) I,
Non-linear Sobolev Structures
65
we obtain 1IHp,l ~ Gf • 1lwp,l and a continious embedding
Wp,l (E)
<-t
HP,1 (E).
For r > 1 this cannot be established, and we need further assumptions. Consider as in the introduction the following conditions
rinj(M, g) = infxEM rinj(X) > 0, I(V'9)iR91 ~ Gi , 0 ~ i ~ k, 1(V'9)iREI ~ Gi , 0 ~ i ~ k.
(I) (Bk(M, g)) (Bk(E, V'E))
It is a well known fact that for any open manifold and given k, 0 ~ k ~ 00, there exists a metric 9 satisfying (I) and (Bk(M, g)). Moreover, (1) implies completeness of g.
Lemma 1.1 Assume (Mn,g) with (1) and (Bk)' Then G;;o(E) is a dense subset of wp,r(E) and HP,r(E) for 0 ~ r ~ k + 2. See [27], proposition 1.6 for a proof.
Lemma 1.2 Assume (Mn,g) with (1) and (Bk)' exists a continuous embedding
D
Then there
Proof. According to 1.1, we are done if we can prove
for 0 ~ r ~ k + 1 and E G;;o(E). Perform induction. For r = 0 IIHP,O = 1lwp,o. Assume 1IHP,r ~ G· 1lwp,r. Then
1IHP,r+l
< G· (1IHP,r + IDr DIHP,r) < G· (1lwp,r + IDlwp,r).
Let a~i' i = 1, ... , n coordinate vectors fields which are orthonormal in m E M. Then with V'i = V' ~ 8x'
IV'S DI~
~ G·
L iI, .. o,i.s,j
lV'il'" V'i s8~j . V'jIP.
66
Relative Index Theory, Determinants and Torsion
Now we apply the Leibniz rule and use the fact that in an atlas of normal charts the Christoffel symbols have bounded euclidean derivatives up to order k - 1. This yields
I~r DI~ ~ C· ~ I~il'" ~ir+1 I~ for r ~ k, it ,... ,i r +l
i. e.
altogether
o Remark 1.3 For p = 2 this proof is contained in [8]. Theorem 1.4 Assume (Mn,g) with (1) and (B k ) and with (B k ) and p = 2. Then for r ~ k
H 2 ,r(E) == Hr(E)
~
Wr(E) ==
0 (E,~)
w 2 ,r(E)
as equivalent Hilbert spaces.
Proof. According to 1.2, Wr(E) ~ Hr(E) continuously. Hence we have to show Hr (E) ~ W r(E) continuously. The latter follows from the local elliptic inequality, a uniformly locally finite cover by normal charts of fixed radius, uniform trivializations and the existence of uniform elliptic constants. The proof is performed in [8]. Another proof is contained in [66]. 0
Remark 1.5 1.4 holds for 1 ~ p <
00
(cf. [27]).
o
As it is clear from the definition that the spaces WP,k(E) can be defined for any Riemannian vector bundle (E, hE, ~E). We assume this more general case and define additionally
Non-linear Sobolev Structures
67
and in the case of a Clifford bundle b,. H(E)
:~ { U E C'(E) I b",Dlul :~ ~ ~~p, IDiUI. <
00 }
,
b,SW(E) is a Banach space and coincides with the completion of the space of all Q E r (E) with b,s IQI < 00 with respect to b,s II. Theorem 1.6 Let (E, h, \7 E ) be a Riemannian vector bundle satisfying (I), (Bk(Mn, g», Bk(E; \7». a) Assume k ~ r, k ~ 1, r - ~ ~ s - ~,r ~ s, q ~ p, then (1.1)
continuously. b) If k ~ r > ~
+s
then WP,T(E)
<-t
b,SW(E)
(1.2)
continuously. We refer to [36] for the proof.
D
Corollary 1.7 Let E -+ M be a Clifford bundle satisfying (I), (Bk(M», (Bk(E», k ~ r, r > ~ + s. Then
HT(E)
<-t
b,s H(E)
(1.3)
continuously. Proof. We apply 1.4, (1.1) and obtain
HT(E)
<-t
b,SW(E).
(1.4)
Quite similar as in the proof of 1.2, (1.5) continuously.
D
A key role for everything in the sequel plays the module structure theorem for Sobolev spaces.
Relative Index Theory, Determinants and Torsion
68
Theorem 1.8 Let (Ei' hi, \7 i ) --t (Mn,g) be Clifford bundles with (I), (Bk(Mn,g)), (Bk(Ei , \7i)), i = 1,2. Assume 0::::: r::::: rl, r2 ::::: k. If r = 0 assume
r-'2:. < rl P r-'2:. < r2 P r-'2:. < rl -
r-~
I
~
~ P2 ~
PI
< r,- PI ~ } or < r21 - ~ P2 < PI
0
{
~ P2
I
or
< rl -
r-'2:. < r2 ~
P
<
1
P2
~
PI
~ P2
}
(1.6)
1.. + 1.. and PI P2
r-'2:. < rl P r-'2:. < r2 P r-'2:. < rl -
{ or
+ r2 -
< 1..+1.. PI P2
P
If r > 0 assume 1.P -<
~
PI
P
~
PI
~ P2
~
PI
+ r2 -
< rl r-- < r2 P r-'2:. < rl P
r-~
~
P2
~
PI
~ P2 ~
PI
+ r2 -
} }
(1.7)
~ P2
Then the tensor product of sections defines a continuous bilinear map
o
We refer to [36] for the proof. Define for U E CO(M), c > 0
uc(x) :=
VOl~c(X)
J
u(y) dvoly(g).
Bc(x)
Proposition 1. 9 Let (Mn, g) be complete, Ric(g) ;::: k, k E JR. Then there exists a positive constant C = C(n, k, R), depending
69
Non-linear Sobolev Structures
only on k, R such that for any c E]O, R[ and any u E W1,1(M) C1(M)
J
lu - ucldvolx(g) :::; C·
c·
M
Jl\lul
dvolx(g).
n
(1.8)
M
Proof. For u
E C':(M) the proof is performed in [40], p. 31-
33. But what is only needed in the proof is 00 (even only l\lul dx < 00).
J
J luldx, J l\lul dx <
The key is a the lemma of Buser (cf. [10]),
J lu -
uc(x)1 dy :::; C·
Bc(x)
J l\lul
c·
dy
(1.9)
Bc(x)
for all x E M, x E]O, 2R[ and u E C1(Bc(x)).
D
But for completeness, we will give a complete proof of the proposition since it plays in chapter IV a crucial role. For this, we recall two well known lemmas which are contained e.g. in [40],
[41]. Lemma 1.10 let (Mn, g) be open, complete Ric (g) ~ k, {2 > O. Then there exists a sequence (Xi)i of points Xi E M, such that for any c > (2 there holds a) The family (Bc(Xi))i is a uniformly locally finite cover of M and an upper bound N for the number of non-empty local intersections is given by N = N (n, (2, c, k) .
b) For any i
-I j,
Bg(Xi) 2
n Bg(x J·) 2
=
0.
D
Lemma 1.11 Let (Mn , g) be open, complete, Ric (g) ~ k. Then, for any 0 < r < R and any x E M vol(BR(x)) :::;
VOlt~R; vol(Br(x)), vo k r
(1.10)
where volk(t) denotes the volume of a ball of radius t in the complete simply connected Riemannian n-manifold of constant curvature k. In particular, for any r > 0 and any x E M
(1.11)
70
Relative Index Theory, Determinants and Torsion
Corollary 1.12 For any x E M, 0 < r < R,
Proof. This follows from (1.10), (1.11) and the standard estimate for YOlk (t). 0
Now we proceed with the proof of proposition 1.9. Let now, according to lemma 1.10, c EjO, R[, (Xi)iEI such that M = U B 2c (Xi) , Bc(Xi) n Bc(xj) = 0 for i i= j and i
#{i
E fix E
B 2c (Xi)} :S N = N(n, k, R) :S 16 n eS J(n-l)l k l·R.
Then, for U E W1,1(M)nC1(M), dx = dvolx(g) , dy = dvoly(g) ,
J J Iu J J Iu
L
Iu-ucldy
2 Bc(Xi)
<
L
uc(xi)ldy
2 Bc(Xi)
+
L
IUc(Xi) - U2c(Xi)ldy
2 Bc(Xi)
+
L
c -
U2c(Xi)ldy.
(1.13)
2 Bc(Xi)
According to (1.9)
L
J
2 Bc(Xi)
IU-Uc(Xi)ldy:S C·c
L 2
J Bc(Xi)
l\7uldy:S N·C·c
J
l\7uldy.
M
(1.14)
71
Non-linear Sobolev Structures
Moreover, we obtain
J
L t
IUc(Xi) - U2c(Xi) Idy
Bc(Xi)
= Lvol(Bc(xi))IUc(xi) -U2c(Xi)1 i
: ; ~ J lu ::; L J lu J t
Bc(Xi)
t
B2c(Xi)
u2c(xdldy U2c(Xi) Idy
IV'Uldy
::; 2N . C· c
(1.15)
M
and
J IU
L t
c -
U2c(Xi) Idy
Bc(Xi)
::;;; J {VOl(~c(X)) J Bc(Xi)
IU(Y) - U2C(Xi)ldY } dx
yEBc(Xi)
::;;; J {VOl(~c(X)) J L J J VOl~c(X)
lu(y) - U2C(Xi)ld Y } dx
XEBc(Xi)
=
yEB2c(Xi)
lu(y) - U2c(Xi)ldy·
t
B2c(Xi)
dx.
(1.16)
Bc(Xi)
Using (1.19), we estimate
J B2c(Xi)
lu(y) - U2c(Xi)ldy ::; 2C . c
J
B2c(Xi)
lV'uldy.
(1.17)
72
Relative Index Theory, Determinants and Torsion
We infer from lemma 1.11
(1.18)
(1.19)
(1.16), (1.17), (1.19) yield
L ~
J
IUe
-
U2e(Xi)ldy
~ K· C· N· c
Bc(Xi)
J
l\7uldy,
(1.20)
M
and (1.13) - (1.15), (1.20) imply
Jlu - uel
dvol(g)
J
~ 3(1 + K)N . C . c l\7ul dvol(g).
M
M
o Corollary 1.13 Let (E,h, \7) ---+ (Mn,g) be a Riemannian vector bundle, (Mn, g) with (1), (Eo), r > n + I, 0 < c < rinj and TJ E W 1,r(E). Then E W 1,O(M) == L1(M), where
GTe
Proof. Set u(x) = ITJ(x)l. Then u(x) = GTe(x) and, according to Kato's inequality,
J
l\7uldx =
Hence we obtain from
J1\7ITJlldx ~ J
I\7TJldx <
lui = ITJI
E L1 and 1.9,
00.
ITJI - ITJle E L 1, (1.21)
73
Non-linear Sobolev Structures
o In sections 3,4 and 5, we consider not only Clifford bundles, Clifford connections but arbitrary Riemannian vector bundles. For this reason, we introduce very briefly general Sobolev spaces and the corresponding main theorems which are completely parallel to the preceding ones for Clifford bundles. Details and proofs can be found e.g. in [27], chapter I 3. Let (E,h,\7 h ) ----+ (Mn,g) be a Riemannian vector bundle. Then the Levi-Civita connection \79 and \7 h define metric connections \7 in all tensor bundles T;: @ E. Denote smooth sections as above by COO(T;: @ E), by Cgo(T;: @ E) those with compact support. In the sequel we shall write E instead of T;:@E, keeping in mind that E can be an arbitrary vector bundle. Now we define for p E JR, 1 ~ p < 00 and T' a non-negative integer
D,~'P(E) == D,f(E) nO,P,T (E) == np,T (E)
COO(E)I'Plp,T < oo}, completion of D,f(E) with respect to 1 . Ip,T'
no,p,T (E) == np,T (E)
completion of C:;"(E) with respect to 1 . Ip,T and {'P 1 'P measurable distributional
D,0,p,T (E) == D,p,T (E)
{'P
E
section with
1'Plp,T < oo}.
Here we use the standard identification of sections of a vector bundle E with E-valued zero-forms. D,q,p,T (E) stands for a Sobolev space of q-forms with values in E. For p = 2, we often use the notations 112,0 =
IIL2 = 112.
Fur-
74
Relative Index Theory, Determinants and Torsion
thermore, we define m
b,ml
L
IV'i
i=O
b,mO(E)
{
b,mn(E)
completion of
1
c,: (E)
b,m 1
with respect to b,m 1. I.
b,mO(E) equals the completion of ~O(E)
=
{
COO(E)
1
b,ml
with respect to b,m 1. I. Denote by b,OOO(E) the locally convex space of smooth sections
DP,r(E),
op,r(E),
b,mn( E), b,mo( E) are Banach spaces and there are inclusions np,r(E) ~ op,r(E) ~ op,r(E), b,mn(E) ~ b,mO(E). If p = 2, then
n (E), 02,r (E), 02,r (E) are Hilbert spaces. 2 ,r
0
np,r (E), op,r (E), op,r (E) are different from one another in general. Proposition 1.15 If (Mn, g) satisfies (1) and (B k ), then
o Embedding theorems are of great importance in non-linear global analysis and even more the module structure theorem which we present now.
75
Non-linear Sobolev Structures
Theorem 1.16 Let (E, h, 'VE) --+ (Mn, g) be a Riemannian vector bundle satisfying (1), (Bk(Mn, g)), (Bk(E, 'V)), k ~ 1.
a) Assume k
~ r, r - ~ ~ s - ~, r ~ s, q ~ p. Then
(1.22)
continuously. b) If r - !!:P > s, then (1.23)
continuously. Now we come to the general module structure theorem.
Theorem 1.17 Let (Ei,hi,'V i ) --+ (Mn,g) be vector bundles with (1), (Bk(Mn,g)), (Bk(Ei , 'Vi)), i = 1,2. Assume 0:::; r:::; rl, r2 :::; k. If r = 0 assume
(
P 1 - PI r-!!:
<
1
P
r-}
< rl < r2 < PI1
PI
+ r2 _
-.l.. +-.l.. PI
.!l
P2
.!l
P2
}or{
< rl r-!!: < r2 ) < P21 P 0
If r > 0, assume 1P < -.l.. + -.l.. and - PI P2 r -!!:
r { r
P
-!!:
P
-!!:
P
< rl < r2 < rl -
r-!!: P r-!!: P r-!!: P
or
P2
.!l
PI
I
.!l
PI
.!l
P2
.!l
PI
+ r2 -
.!l
P2
.!l
PI
.!l
P2
}
76
Relative Index Theory, Determinants and T01'sion
Then the tensor product of sections defines a continuous bilinear map D,Pl,T1(E1 , \71) x D,P2,1'2 (E2' \72)
----+
D,p,1'(E1 ® E 2, \7 1 ® \7 2). (1.24)
o Corollary 1.18 Assume r = rl = r2, P = PI = P2, r > ~. (a) If El = M x IR, E2 = E, then D,p,1'(E) is a D,p,T(M x IR)-
module. (b) If El = M x IR = E 2, then D,p,1'(M x IR) is a commutative, associative Banach algebra. (c) If El = E = E 2, then the tensor product of sections defines a continuous map
o The invariance properties of Sobolev spaces will be the topic of our next considerations. Given (E,h, \7 E ) ----+ (Mn,g), for fixed E ----+ M, r 2: 0, p 2: 1, the Sobolev space D,p,1'(E) = D,p,1'(E, h, \7 E , g, \79, dvolx(g)) depends on h, \7 = \7 E and g. Moreover, if we choose another sequence of differential operators with injective symbol, e. g. D, D2, . .. in case of a Clifford bundle, we should get other Sobolev spaces. Hence two questions arise, namely 1) the dependence on the choice of h, \7 E, g, 2) the dependence on the sequence of differential operators. For later applications we pose a third question, that is 3) must h, \7 E , 9 be smooth? We start with the first issue and investigate the dependence upon the metric connection \7 = \7 E of (E, h). If \7' = \7,E is another metric connection then 'TJ = \7' - \7 is a I-form with values in lBE' \7' - \7 E D,1(lB) = D,(T* M ® lBE). Here lB is the bundle of the skew-symmetric endomorphisms. \7 = \7 E induces a connection \7 = \7 IBE in lB E and hence a Sobolev norm 1\7' - \71V',p,T = 1\7' - \7lh,V',9,V'9,p,r.
Non-linear Sobolev Structures
Theorem 1.19 Assume (E, h, "VE) (Bk(M)), (Bk(E, "V E)), k ~ l' > ~
77
(M n , g) with (1), 1. Let "V' = "V,E be a
--t
+
second metric connection with (Bk(E, "V'E)) and suppose
I"V' - "VIV,p,r-l <
(1.25)
00.
Then
np,r](E, h, "V, g) = nM(E, h, "V', g),
0 ~ e~
(1.26)
l'
as Sobolev spaces. Proof. First we remark that our assumptions imply (Bk(
np'(!(E, "V)
c np,(!(E, "V')
(1.27)
and show its continuity. We are done if we prove (1.27) for smooth elements
l
e=
1,
We apply the module structure theorem to ("V' - "V)
1'-1- -Pn l' -
1-
n
> 0--,
p n 1- ~ > 0- - , p P n > o - -p since
~ + 1 - ~ = (1' - ~) - ~
1
-
p
1
1
p
p
< - +-.
n
l'
> -, P
78
Relative Index Theory, Determinants and Torsion
Hence
1(\7' - \7)'Plp ::; C1 ·1\7' - \71V',p,r-1 ·1'P1V',p,l, 1\7''Plp ::; C1 . 1\7' - \71V',p,r-1 . l'PIV',p,l + 1\7'Plp, 1\7''Plp ::; C2 . l'PIV',p,l. l'PIV'/,p,1, = 1'Plp + 1\7''Plp ::; C3 . l'PIV',p,l, (1.28) C 3 = C3 (1\7' - \71V',p,r-1), D,p,l(E, \7) ~ D,p,l(E, \7') continuously. We conclude similarly for
(!
= 2,
\7,2'P = (\7' - \7)(\7' - \7)'P + (\7' - \7)\7'P + \7(\7' - \7)'P + \7 2'P
(1.29) and apply the module structure theorem to the single terms of (1.29). In (\7' - \7)(\7' - \7) the left hand term \7' - \7 E D,(T*2 ® 0 - ~, 2-!!:>0-!!: p p' r-1-!!:+2-!!:>0-!!: p p p 1(\7' - \7)(\7' - \7)'Plp ::; C4 ·1\7' - \71~,p,r-1 ·1'P1V',p,2.
(1.31)
1(\7' - \7)(\7' - \7)\7'Plp < Cs · 1\7' - \71V',p,r-1 . 1\7'PIV',p,l < Cs · 1\7' - \71V',p,r-1 . l'PIV',p,2. (1.32)
79
Non-linear Sobolev Structures
1(\7(\7' - \7))'Plp
< <
C6 • 1\7(\7' - \7)IV',p,r-2 'l'PIV',p,2 C6 • 1\7' - \71V',p,r-1 . l'PIV',p,2'
(1.33)
So (\7' - \7) \7 'P is done. We obtain from (1.29)-(1.33)
< C7 · l'PIV',p,2, l'PIV'I,p,2 < C8 • l'PIV',p,2, 1\7,2'Plp
C8
= C8 (1\7' -
\71V',p,r-I).
Now we turn to the general case. Assume the result holds for n' v 'P, ... , n'I?-I. v 'P, 1. e.
Then I?
\7'I?'P =
L \7i-I(\7' -
\7)\7'I?- i 'P
+ \71?'P.
(1.34)
i=1 By assumption, 1\71?'Plp < 00. There remains to consider the terms \7i-I(\7' - \7)\7'I?- i 'P. Iterating the procedure, i. e. applying it to \7,I?-i and so on, we have to estimate expressions of the kind
with il + .. ·+il? = Q, il? < [!. Applying the corresponding version of the Leibniz rule, we finally have to estimate
such that nl + 1 + n2 + 2 + ... + ns-I + 1 + ns = Q < r, ns < [!, nl,' .. ,ns-I > O. The module structure theorem can now be
Relative Index Theory, Determinants and Torsion
80
applied in virtue of the sequence of inequalities
n
r - 1- nl - - > r - 1- (nl p
n
r - 1 - n2 - - > r - 1 - (nl p n
r - 1 - n2 - -
p
n =r - 1- p
+r -
+r -
n
+ n2) -
-,
+ n2) -
n
p
-, p
n
1 - n2 - -
p
1 - (nl
+ n2) -
n
-
p
n
> r - 1 - (nl + n2) - -; p
r - 1 - (nl
n
+ n2) -
- > r - 1 - (nl p
n
r - 1 - n3 - - > r - 1 - (nl p
r - 1- (nl
n
+ n2) -
n =r - 1- p
+r -
-
p
+r -
1 - (nl
n
+ n2 + n3) -
+ n2 + n3) -
-, p
n
-, p
n
1- n3 - -
p
+ n2 + n3) -
n
-
p
n
> r - 1 - (nl + n2 + n3) - -; p
r - 1-
(nl
n
+ ... + n s -2) - - > r p
- 1-
(nl
+ ... + ns-l) -
n r - 1 - ns-l - - > r - 1 - (nl + ... + ns-l) p n r - 1 - (nl + ... + n s-2) - - + r - 1 - ns-l p
n =r - 1- p
+r -
1 - (nl
+ ... + ns-l) n
> r - 1 - (nl + ... + ns-l) - -; p
r - 1- (nl
+ ... + ns-l) -
n
n
p
p
() - ns - - > 0 - -,
n
n
p
p
- > 0 - -,
n
-
p
n -, p n -
p
n p
Non-linear Sobolev Structures
r - 1 - (n1
+ ... + n s -1) -
n
-
p
+ (] -
81
n
ns - -
p
n
n
p
p
= r - 1 - - + (] - (n1 + ... + ns) - n > 0--. p
This yields
I(Vm1 (V" - V')) ... (V'n s- 1 (V" - V')) (V'ns 'P) Ip 1+.+n 1) . lV'ns I < C9 . IV" - V'1(s-1)(r-1)-(n V',p,r-l 'P V',p,(J-n S-
s
< C . IV" - V'1(s-1)(r-l)-(n1+·+ns-t}·1 I 9 V',p,r-1 'P V',P,(J' 1V"(J'Plp ~ C lD · l'PIV',p,(J l'PIV'I,p,(J ~ C11 I'PIV',p,(J, C11 = C11 (IV" - V'1V',p,r-l), np,g(E, V') ~ np,g(E, V") continuously, 0 ~ (] ~ r. The other continuous inclusion would follow from Lemma 1.20 Under the hypotheses of 1.19,
IV" - V'1V',p,r-l <
00
implies IV" - V'1V",p,r-l <
00.
Proof. This is only a special case of what we just have proven. Set rJ = V" - V', assume IV" - V'1V',p,r-l = IrJlV',p,r-l < 00, then IV"- V'1V",p,r-l = IrJlV",p,r-l ~ P(IV"- V'1V',p,r-l) ·lrJlV',p,r-l < 00, where P is a polynomial in the indicated variable. 0
Now let IV" - V'1V",p,r-l < 00 and repeat the first part of the proof exchanging V' and V" in all formulas and arguments, to obtain
np,g(E, V")
~
np,(J(E, V') continuously, 0
~
(]
~
o
This finishes the proof of 1.19.
Corollary 1.21 The conditions IV" - V'1V',p,r-l < V'1V",p,r-l < 00 are equivalent.
r.
00
and IV" -
o
82
Relative Index Theory, Determinants and Torsion
The next step is to understand changes of the metric g. Theorem 1.22 Assume (E, h, \7) - - - t (Mn, g) with (1), (Bk(Mn,g)), (Bk(E, \7)), k ~ r > ~ + 1, and g' with C . g1 9 ::; g' ::; D . g, (Bk(Mn,g)) and /\7 - \7g/g,p,r-1 = (J /g 1 r-1 g g'/~,x dvolx(g))p + I: /(\7g )t(\7 - \7g)/p < 00. Then I.
1
i=O
Op,g(E,h, \7,g)
= OP,{!(E,h, \7,g'),
0::; (2::; r,
as euivalent Sobolev spaces. Proof. The assumption C· 9 ::; g' ::; D· 9 implies C 1 dvol x (g) ::; dvolx(g') ::; D1 dvolx(g) and C2 /t/ g,x ::; /t/gl,x ::; D 2 /t/ g,x for all tensors t of degree ::; r. Hence
o ::; {2gl::; r, andg1we must check only the result when replacing \7g
by \7 . \7g, \7 come first into the calculations with the second vector bundle derivatives:
g1 Now use the estimates of the proof of 1.19, but applied to \7 g1 \7g. \7 is no longer a metric connection with respect to 9 but g1 for \7g - \7 as an element of 0 1 (End T) all estimates remain D valid. g1 Since the estimate of (\7 0 \7)i involves the j-th derivatives g1 g1 of \7 - \7g, 0 ::; j ::; i-I, only /\7 - \7g /V'9,p,r-2 seemingly g1 enters the estimates. But we need in fact /\7 - \7g/V'9,p,r-1 to apply the module structure theorem. One sees this e. g. that r > ~ + 1,
n
r - 2 - n1 - -
>
n r - 2 - n2 - -
>
p
p
83
Non-linear Sobolev Structures
do not imply
n
r - 2- p
+r -
2 - (nl
+ n2) -
n - > r - 2 - (nl p
+ n2) -
n p
and so on, since not necessary r - 2 - !!p > O. Sufficient for this and the whole module structure procedure would be r > !!p + 2. Thus
Theorem 1.23 Assume (E, h, \7) -----7 (Mn, g) with (I), (B k (Mn , g) ), (B k (E, \7)), k ~ r > ~ + 2 and g' with C . 9 :::; g' :::; D· g, (Bk(Mn,g')), (1) and l\7g' - \7glg,p,r-l < 00. Then OM(E, h, \7, g) = OM(E, h, \7, g),
0:::;
{J :::;
r,
as equivalent Sobolev spaces.
0
Finally we study what happens by replacing (h, \7 h ) (h', \7h').
-----7
Theorem 1.24 Let (E,h,\7 h) -----7 (Mn,g) be a Riemannian vector bundle with (I), (Bk(Mn, g)), (Bk(E, \7 h)), k ~ r > ~+1, h' a second fibre metric with metric connection \7h' and with (Bk(E, \7h')), C· h :::; h' :::; D· hand l\7h' - \7hlh,V'h,g,p,r_l < 00. Then
as equivalent Sobolev spaces. Proof. \7h' is not necessarily metric with respect to h but the Sobolev space OM(E, h, \7' , g) is nevertheless well defined. Then C . h :::; h' :::; D . h implies OM(E, h, \7h, g) = OM(E, h', \7 h, g) and OM(E h' \7h' g) = OP,Q(E h \7h' g)
'"
'"
.
\7h' - \7 h is no longer a section of T* ® IB E but still a section of T* ® End E, by our assumption \7h' - \7 h E
Relative Index Theory, Determinants and Torsion
84
nP,r-1(End E, h, V h, g).
We decompose Vhf cp = (Vhf - Vh)cp vhcp etc. and conclude as in the proof of 1.19
nM(E,h,Vh,g)
+
c nM(E,h,Vhf,g) nM(E, h', Vhf, g),
o:s f2 :S r,
continuously. The same inclusion then holds for the Sobolev spaces of E* and EndE = E*®E, all endowed with the induced metrics and connections. In particular
nP ,r-1(End E, h, V h, g) c nP ,r-1(End E, h, Vhf) nP,r-1(End E " h' Vhf , g) , which implies IV
nM
(
hf
- Vhlhf,'Vhf,p,r_1 <
E, h', Vhf, g) ~
nM
(
00.
From this we get
E, h, V h, g) ,
O:S f2 :S r,
continuously.
D
Combining theorems 1.19 - 1.24, we obtain as main result:
Theorem 1.25 Let (E, h, V) ----t (Mn, g) be a Riemannian vector bundle with (I), (Bk(Mn, g)), (Bk(E, V)), k ~ r > ~+1. Suppose h' is a fibre metric on E with metric connection V' and g' a metric on Mn with (I), (Bk(Mn, g')), (Bk(E, V')) satisfying c· fh :S h' :S D . h, C1 . 9 :S g' :S C2 . g, IV' - Vlh,'V,9,p,r-1 < 00, IV9 - V 919,p,r-1 < 00. Then
nM(E, h, V, g) = nPd!(E, h', V', g'), as equivalent Sobolev spaces.
O:S f2 :S r, D
We are left with the dependence on the sequence of differential operators. The most important and frequently used cases are sequences based on Vi, (V*V)i, ~i, Di respectively. We will present far reaching results. First we compare the Sobolev spaces based on V (up to now the main point of interest) and on the Bochner
Non-linear Sobolev Structures
85
Laplacian ~ = ~B = \7*\7. Concerning this issue, there is a remarkable contribution from Gorm Salomonsen [66] which we fit into our discussions. Let again (E, h, \7) ----t (Mn, g) be a Riemannian or Hermitean vector bundle, and \7* the formal adjoint to \7 with respect to the canonical Hilbert scalar product, (\7*cp,1jJ) = (cp, \71jJ), cp E Cgo(T* M ®E), 1jJ E Cgo(E), (\7*cp,1jJ) = h(\7*cp, 1jJ)x dvolx(g). Locally this can be written as \7* = _gi j \7i. We do not assume (1) or completeness for (Mn, g) . Until otherwise said, we denote by ~ = \7*\7 the Bochner Laplacian, i.e. the Friedrichs' extension of ~. For our purpose we must indicate the differential operators defining the Sobolev spaces. Therefore we write D,p,r(E, h, \7, g) = D,p,r(E, \7) or D,p,2S(E, ~), respectively. We restrict to the case p = 2 because here and there we apply Hilbert space methods. Since we do not assume (1), (Bk) for (Mn, g), the Sobolev spaces o.2,r ~ 2 ,r ~ D,2,r do not automatically coincide. Explicitly
J
n
{cplcp distributional section of E s. t. \7 i cp E L 2 (T*i ® E), 0 ~ i ~ r}, 1
Icpl'V,2,r D,2,2S(E, ~)
(~IV\OIL)' ~ ~ IV'1'1
14'
{cplcp distributional section of E, ~ i cp E L2 (E) , 0 ~ i ~ s},
r
(t, I""I'IL
1
~ ~ I""l'iL,
D,~s(E, ~)
n
{cp E COO(E)I Icpl~,2,2s < oo},
2,2s(E, ~)
n~s(E, ~),
o.2,2S(E, ~)
Cgo(E)II~'2'28 .
We consider ~ as a self-adjoint operator on the domain given by Friedrichs' extension.
86
Relative Index Theory, Determinants and Torsion
Theorem 1.26 Assume for (E, h, \7) and (Boo(E, \7E)). Then
rp,2k(E, \7) = D,2,2k(E, ~),
(Mn, g) (Boo(M, g))
------t
k = 0, 1,2, . ..
as equivalent Sobolev spaces.
o
We refer to [66] for the proof. Since ~ = ~F is self-adjoint, ~! and k
Icplt.,2,k =
L 1~!cpIL2'
cp
E
C~(E)
j=O
and
D,2,k(E,~) = C~(E)IILl,2'k
are well defined. One can now easily prove an analogous result to 1.26. Theorem 1.27 Assume (E, h, \7) and (Boo(E, \7)). Then n2,k(E,~)
------t
(Mn, g) with (Boo (M, g))
= n 2,k(E, \7)
o
as equivalence of Sobolev spaces, also for k odd.
By means of spectral calculus we can define D,2,S(E,~) as Coo(E)112'(1+Ll)~ even for s E JR+ , where IInl2 = C r (1+t.)2" ((1 + ~)~cp, (1 + ~)~cp) = ((1 + ~)scp, cp). n2,-S(E,~) will be 8
defined as the dual of D,2,S(E, ~). Refering to [66], we state without proof Theorem 1.28 With the assumptions of 1.27 and S E JR, \7 maps n2,S(E,~) into n 2,S-1(T* 0 E, ~). If W E b,oon(Hom (E, F)), (E,h F , \7 F ) ------t (Mn,g), then W maps n2,s (E) into n2,s (F) .
o
87
Non-linear Sobolev Structures
Other canonical differentiable operators in geometric analysis used for the definition of Sobolev spaces are the generalized Dirac operators D and the full Laplacian ~. As (the graded) ~ is a special case of D2, we concentrate on D. Let (E, h, \7, .) ----t (Mn, g) be a Clifford bundle, D its generalized Dirac operator. Then we define as above
02,T(E, D) := {'P I'P a distributional section and Di'P E L 2(D), 0::; i ::; r}, T
T
L IDi'PIL L
l'PID,2,T
rv
i=O
IDi'PIL2'
i=O
O;(E, D) .- {'P E COO(E) 11'PID,2,T < oo} == HT(E), 02,T (E, D) .- O;(E, D)IID,2,r == H 2,T(E), 2,T(E, D) '- C~(E)IID'2,r == H2,T(E).
n
There is a sequence of closed subspaces
For p = 2, lemma 1.2 can be sharpened as Lemma 1.29 Let (E, h, \7, .) ----t (Mn, g) be a Clifford bundle. There are continuous inclusions (1.35)
n2,T(E, \7) n2,T(E, D), '-+
r = 0,1,2, ....
(1.36)
If (Mn , g) is complete then
We refer to [66] for the proof. Theorem 1.4 can be modified and sharpened as follows.
o
88
Relative Index Theory, Determinants and Torsion
(Mn,g)
Theorem 1.30 Assume (E, h, \7,.) (Boo(Mn,g)) and (Boo(E, \7)). Then
n 2,r(E, \7)
= n 2,r(E, D),
r
with
= 0,1,2, . ..
as equivalence of Sobolev spaces. We refer to [66] for the proof. 0 The (graded) Laplace operator ~ (~o""'~n) of (Mn,g) (with Weitzenboeck terms) is a special case of D2 a generalized Dirac operator D. This yields Theorem 1. 31 Let (Mn, g) be an open Riemannian manifold satisfying (Boo (Mn , g)). Then
n Q ,2,2S(M, \7)
= n Q,2,2S(M, ~),
0::; q ::; n,
s
= 0,1,2, . ... o
as equivalence of Sobolev spaces. Here the n's are Sobolev spaces of forms.
Theorem 1.32 Let (E, h, \7, .) --> (Mn, g) be a Clifford bundle satisfying (Boo(Mn,g)) and (Boo(E, \7)). If (Mn,g) is complete then
n2,r(E, \7) = n2,r(E, D),
r
= 0,1,2, . ..
as equivalent Sobolev spaces. Proof. We know from 1.29 that n2,r(E, \7) ~ n2,r(E, D), n 2,r(E, \7) ~ n 2,r(E,D). For complete (Mn,g) we have n 2,r(E, D) = n2,r(E, D), hence n2,r(E, \7) = n 2,r(E, D) = 2,r(E, D) 2 2,r(E, \7) 2 n 2,r(E, \7), so all these spaces must
n
coincide.
n
0
Non-linear Sobolev Structures
Corollary 1.33 Let (Boo(Mn, g)). Then
(Mn, g)
nq ,2,2S(M, \7) = nq,2,2s(M, 6.),
be
complete
q = 0, ... , n,
89
and
satisfy
s = 0, 1, ...
as equivalent Sobolev spaces.
D
1.33 has still a slight generalization. If (E, h, \7, .) - t (Mn, g) is a Clifford bundle and (F, hF, \7 F) - t (Mn, g) is another (Riemannian or Hermitean) bundle, then E @ F has a canonical Clifford bundle structure. Applying 1.33 to this bundle with E = Clifford bundle of graded forms, we obtain Corollary 1.34 Let (Mn,g) be complete, (F,hF' \7 F) - t (Mn, g) a Riemannian or Hermitean bundle, both satisfying (Boo). Then
nq ,2,2S(F, \7) = nq ,2,2S(F, 6.),
q = 0, ...
,n,
s = 0, 1, ...
as equivalent Sobolev spaces.
D
We finish at this point our short review of Sobolev spaces since we established what is needed in the sequel.
2 Uniform structures of metric spaces As we already indicated in the preface to this chapter, the key of our whole approach to define relative number valued invariants are Sobolev uniform structures. They allow to introduce natural intrinsic topologies in the considered set of geometric/analytic objects and to define admitted perturbations which are the elements of a generalized component. We start with a brief outline of the fundamental notions connected with uniform structures. Thereafter we present some
Relative Index Theory, Determinants and Torsion
90
basic examples as uniform structures of metric spaces, of sections of vector bundles, of conformal factors and of Riemannian metrics. Let X be a set. A filter F on X is a system of subsets which satisfies
(FI)
ME F, MI 2 M implies MI E F.
(F2) (F3)
MI'.··' Mn E F implies MI n··· n Mn E F. 0 ~ F.
A system it of subsets of X x X is called a uniform structure on X if it satisfies (FI)' (F2 ) and ~ C
(U2 )
Every U E it contains the diagonal V E it implies V-I E it.
(U3 )
If V E it then there exists W E it s.t. WoW C V.
(UI )
X xX.
The sets of it are called neighbourhoods of the uniform structure and (X, it) is called the uniform space. ~ C ~(X x X) (= sets of all subsets of X x X) is a basis for a uniquely determined uniform structure if and only if it satisfies the following conditions:
n V2 contains an element of~.
(B I )
If Vi, 112 E ~ then Vi
(Un (U~)
Each V E ~ contains the diagonal ~ C X x X. For each V E ~ there exists V' E ~ s.t. V' ~ V-I.
(U~)
For each V E ~ there exists W E ~ s.t. WoW C V.
Every uniform structure it induces a topology on X. Let (X, it) be a uniform space. Then for every x E X, it(x) = {V(X)}VEU is the neighbourhood filter for a uniquely determined topology on X. This topology is called the uniform topology generated by the uniform structure it. We refer to [68] for the proofs and further informations on uniform structures. We ask under which conditions it is metrizable. A uniform space (X, it) is called Hausdorff if it satisfies the condition The intersection of all sets E it is the diagonal ~ C X x X.
Non-linear Sobolev Structures
91
Then the uniform space (X, U) is Hausdorff if and only if the corresponding topology on X is Hausdorff. The following criterion answers the question above. Proposition 2.1 A uniform space (X, U) is metrizable if and only if (X,U) is Hausdorff and U has a countable basis ~. 0 Next we have to consider completions. Let (X, U) be a uniform space, V a neighbourhood. A subset A c V is called small of order V if Ax A c V. A system (5 c S:P(X) has arbitrary small sets if for every V E U there exists M E (5 such that M is small of order V, i.e. M x MeV. A filter on X is called a Cauchy filter if it has arbitrary small sets. A sequence (xv)v is called a Cauchy sequence if the associated elementary filter (= {xvlv 2:: vo}vo) is a Cauchy filter. Every convergent filter on X is a Cauchy filter. A uniform space is called complete if every Cauchy filter converges, i.e. is finer than the neighbourhood filter of a point. Proposition 2.2 Let (X, U) be a uniform space. Then there exists a complete uniform space (Xu, II) such that X is isomorphic to a dense subset of X. If (X, U) is also Hausdorff then there exists a complete Hausdorff uniform space (Xu, II), uniquely determined up to an isomorphism, such that X is isomorphic to a -udense subset of x. (X ,U) is called the completion of (X,U). For the proof we refer to [68], p. 126/127.
o
We define an co-locally metrized set. Let X be a set, co > o. X is called co-locally metrized if for x E X there exists d x (·, .), X x X ~ Ddx ---t [0, co[ such that (Ue:(x) = {yldAx, y) < c}, dx ) is a metric space, 0 < c < co and For given 0 < c < Cl there exists c' = c'(c) > 0 s.t. dy(y, x) < c' implies y E Ue:(x), for given 0 < c < Cl there exists 8 = 8(c) > 0 s.t. Y E U8(x) , z E U8(y) implies z E Ue:(X) , i. e. dx(x, y) < 8, dy(y, z) < 8 implies dx(x, z) < c.
(2.1)
(2.2)
92
Relative Index Theory, Determinants and Torsion
We admit the case Eo
= 00.
Proposition 2.3 Let X be Eo-locally metrized, 0 < 0 < Eo and set v" = {(x, y) E X2Id x (x, y) < o}.
Then ~ = {V8}o<8
da(x, y)
= inf
L dZ;_l
(Zi-l,
Zi)
(2.3)
i=l
where the infimum is taken over all finite sequences Zo, Zl, ... , zp s.t. Zo = x, zp = y and p = 1,2,3, .... Proposition 2.4 If in (2.3) dx(x, y) is defined then
~ dx(x, y)
::; da(x, y) ::; d x (x, y) .
(2.4)
o
Proof. This is just (2.6) in [68], p. 117.
l!:a.
da can be transformed into a metric d, e.g. by d = Let (Y, U y ) be a Hausdorff uniform space, X c Y a dense subspace. If X is metrizable by a metric e then e may be extended to a metric e on Y which metrizes the uniform space (Y, Uy ). In conclusion, if (X,U) is a metrizable uniform space and (Xe,II e ) or (Xli, II) are uniform or metric completions, respectively, then -li
-e
X =X
as metrizable topological spaces.
(2.5)
93
Non-linear Sobolev Structures
We present now some important examples. Let (E, h, \7) ----t (Mn, g) be a Riemannian vector bundle, 1 ~ p < 00, r > 0, J> O. Set
v"
{(
Ut,
E
C OO (E)211
1'\7'('1" -
'I')I~ dVO\'(9)) , d}.
Then it is evident that ~ = {V,,},,>o is a basis for a metrizable uniform structure IIp,T (Coo (E)), (COO (E), IIp,T (Coo (E))) is -=----:-c::::-:-up,r _
a uniform space. Let (COO (E) ,llP,r) be the completion. We started with the local metric d
= l(t2 - tl)<po - (t2 -
td
= l(t2 - tl)(<po -
We would be done if 1(t2 - t l ) (
o
-=---;""-=:::7".up ,r
Corollary 2.6 a) In Coo(E) components. -=---;""-=:::7"up ,r
b) Coo (E)
coincide components and arc
has a representation as a topological sum
iEI
94
Relative Index Theory, Determinants and Torsion
c) ---=:------;-::=:-iV' ,r
{'P' E Coo(E) II'P - 'P'I;~; < oo} 'P+??,r(E,\7), (2.6) z. e. each component is an affine Sobolev space.
Proof. a) and b) immediately follow from lemma 2.5. Consider compp,r('Po) , 'PI E compP,r('Po) and let {'Pt}o~t9 be an arc between 'Po and 'Pl. {'Pt }o~t9 is compact and can be covered by a finite number of E-balls Uc('Pta) = Uc('Po), Uc('Ptl)' ... , Uc('Pt m ) = Uc('PI), where E is chosen sufficiently small that (2.4) is valid. Let 'Pti,i+l E Uc('PtJ n Uc('Pti+J. We obtain l'Pti - 'Pti+llp,r = l'Pti - 'Pti,i+l + 'Pti,i+l - 'Pti+llp,r :::; l'Pti - 'Pti,i+11 + l'Pti'i+l - 'Pti+ll < E + E = 2E, altogether we obtain that l'Po - 'Pllp,r is defined and < mE < 00. The last equation (2.6) follows from 'PI 'Po + ('PI - 'Po)' 0
Corollary 2.7 On a compact manifold there is only one (arc) component) namely
compP,r(O)
= ??,r (E, \7).
Proof. On a compact manifold always holds l'Po - 'Pllp,r < 00 for 'Po, 'PI E Coo (E). This inequality extends to the difference of Cauchy sequences (since they are bounded) and to the limit by continuity. In particular this holds for 'Po = O. 0
Remarks 2.8 a) We see from (2.6) for'P = 0 that the zero component compP,r(O) coincides with the Sobolev space ??,r (E, \7). Insofar our approach yields a generalization of Sobolev spaces. b) It is very easy to see that the index set I is uncountable if (Mn, g) is open. "Each growth generalies its component." On
Non-linear Sobolev Structures
95
compact manifolds, there is only one growth, namely no growth, hence there is only one component. c) Let {XihEI be a family of disjoint metric spaces, di the metric on Xi. Then there exists a metric d on the topological sum X = LXi s.t. d induces the uniform structure on Xi which iEI belongs to di (cf. [68], p. 120). This is the situation in corollary 2.6 and c). 0 Let us consider other choices of V8. Set v" = {(O is a basis for a metrizable uniform structure llP,T(L1,loc(E)) which is already complete, (L1,loc(E), If,T (L1,loc)) = (L1,loc, llP,r(L1,loc(E)). The background for completeness is the fact that the Sobolev spaces of distributions are already complete. We write np,T(L1,loc(E)) for (L1,loc(E),llP,T(L1,loc)) which is a complete uniform space.
Proposition 2.9 a) np,r(L1,loc(E)) is locally arcwise connected. b) In np,T(L1,loc(E)) coincide components and arc components. c) np,T (L1,loc (E)) has a representation as a topological sum
np,T(L1,loc(E)) =
L compp,T(
d) {
o
96
We
Relative Index Theory, Determinants and Torsion
obtain
up,r (Cgo (E)) ,
(Cgo(EtP,r, Il,r(cgo(E)))
op,r (cgo (E)), locally arcwise connectedness, a topological sum representation and (2.8) But op,r (Cgo (E)) consists of one component as the following remar k shows.
Remark 2.10 If r.p E Cgo(E) then in op,r(cgo(E)) comp(r.p)
= compP,r(O) = op,r(E, '\7).
o Corollary 2.11 If (E, h, '\7) (Bk(Mn,g)), (Bk(E, '\7)) then
---t
(Mn , g)
op,r(L1,loc(E)) = op,r(CXJ(E)) for 0 :::; r :::; k
+ 2.
satisfies (1),
(2.9)
o
Remark 2.12 A long further series of equivalences for uniform spaces can be stated if we apply all of our invariance properties ~~~~~.
0
The advantage of this approach is that we can develop e.g. a Sobolev theory of PDEs without decay conditions for the sections. The classical theory is a theory between the zero components comp(O). Our framework allows a quite parallel theory as maps between other components. Clearly '\7 maps compp,r (r.p) into compp,r-l (r.p). If A is a differential operator which maps op,r(E) into op,r-m(F) then A maps compP,r(r.p) C op,r(L1,loc(E)) into compP,r-m(Ar.p) C op,r-m(L1,loc(F)). A necessary condition for the solvability of Ar.p = 'ljJ, r.p E compP,r(r.po) to find, 'ljJ E op,r-m(L1,loc(F)) or E op,r-m(coo(F)) given, is that Ar.po E compP,r-m( 'IjJ) etc. We will here not establish the complete PDE-theory for this setting. It should appear elsewhere.
Non-linear Sobolev Structures
97
A similar setting can be established for the Banach-Holder theory. Set m
Va = {(
E
c oo (E)21 b,ml
sup l\7i(
i=O
and SB
x
= {Va}8>O. SB is a basis for a metrizable uniform structure --=-----:-=::-:-,b, m
b,mu(coo(E)). Let (COO (E)
U
_
,b,mu) be the completion. Then
we get properties absolutely parallel to the assertions 2.5 - 2.9,
jEJ
b,m comp (
{
Similarly
Va = {(
E
COO(E)21 b,m,al
= {Va}8>O define b,m,aU(COO(E)) , the completion
b,m,an(COO(E)) =
L
b,m,acomp(
jEJ
b,m,acomp(
+ b,m,an(E, \7).
(2.10)
Here b,m,al
b,ml
+
sup
sup
X,yEM cEG(x,y)
m m IT(C)\7
where G(x, y) = { length minimizing geodesics joining x and y}, T(C) is parallel translation along c from 7f-l(X) to 7f-l(y) and d(x, y) is the distance from x to y.
Remark 2.13 Our Sobolev embedding theorems from section 1 induce embedding theorems for components of corresponding uniform spaces, e.g. if we have (1), (Bk)' r > ~ + m then
D
Relative Index Theory, Determinants and Torsion
98
We discuss another example, which is important in Teichmliller theory for open surfaces. That is the space of bounded conformal factors, adapted to a Riemannian metric g. Let
{
Pm (g)
E
GOO(M)
I
inf 0, sup
00,
xEM
IV'i
00,
r E Z+, c5 > 0,
"'8 = {(
E
Pm(g)211
(J tv 1
r 1
(V'); ('I' - '1") I:,x dvolx(g)
d},
Then ~ = {"'8}8>O is a basis for a metrizable uniform structure. Let JY:n ,l' (g) the completion,
Glp = {
I
inf 0, sup
xEM
xEM
and set P~( (g)
is locally contractible, hence locally arcwise connected and hence components coincide with arc components. Let
(2.11) and denote by comp(
P~((g).
Theorem 2.14
P~((g)
has a representation as topological sum
p~r(g) = Lcomp(
and
o
99
Non-linear Sobolev Structures
Remark 2.15 On a compact manifold there is only one component, the component comp(1). D
We introduce now uniform structures of Riemannian metrics. Let Mn be an open smooth manifold, M = M(M) be the space of all Riemannian metrics. We want to endow M with a canonical intrinsic topology either in the C m - or Sobolev setting, depending on the subsequent investigation. Let gEM. We define
bU(g)
=
{g' E M Iblg_g'l
:= sup
Ig-g'lg,x <
00,
blg_g'lg, < oo}.
xEM
Then, it is easy to see that bU(g) coincides with the quasi isometry class of g, i.e., g' E bU(g) if and only if there exist C, C' > 0 such that C . g' ::; 9 ::; C' . g' (2.12) holds in the sense of positive definite forms. In particular g' E E bU(g'). We introduced bU(g) in chapter I after remark 1.17 as US(g). For a tensor t we define bltlg := sup Itlg,x. 9 E bU(g') implies the existence of bounds
bU(g) if and only if 9 xEM
Ak(g, g'), Bk(g, g') > 0 such that, for every (r, s)-tensor field t with r
+s =
k,
Ak ·Itlg,x ::; Itlgl,x ::; B k · Itlg,x,
A k · bltl g ::; bltl g, ::; Bk . bltl g·
(2.13) To endow M with canonical topologies we use the language of uniform structures. Set for m 2: 1, 5 > 0, C(n, 5) = 1 + 5 + 5J2n(n - 1)
V8 = {(g,g') EM I g' E bU(g), C(n, 5)-1. 9 ::; g' ::; C(n, 5) . 9 m-1
and b,mlg - g'lg := big - g'lg
+ L bl(\7g)j(\7g - \79')lg < 5}. j=O
Proposition 2.16 The set ~ = {V,,},,>o is a basis for a metrizable uniform structure on M.
100
Relative Index Theory, Determinants and Torsion
Proof. The proof is already modelled by the proof of chapter I, lemma 1.1. and chapter II, theorem 1.19. But we present for completeness a complete proof. (B1) and (Un are trivial. We start to prove (U~). This would be proved if we could show b,ml g _ 9'l gl :s: P(bl g - g'lg, bl(\7 g)i(\7 g - \7gllg) = Pm,g, (2.14) where Pm,g is a polynomial in big - g'lg, bl(\7 g)i(\7 g - \7g/lg),
°:s:
:s: m - 1, without constant term. In fact, (2.14) implies as follows. Given 5 > 0, there exists 5' = 5'(5) such that for b,ml g - g'lg < 5', Pm,g < 5. According to (2.14), exchanging the role of 9 and g', b,ml g - g'lgl < 5' implies b,ml g - g'lg < 5, (g,g') E "\18, (g',g) E "V6- 1. Therefore we have to prove (2.14). g1 We set \7g = \7, \7 = \7'. From (2.2) follows i
(U~)
big - 9'l gl
:s: Co· big - 9'lg and bl\7 -
\7'l gl
:s: C 1 . bl\7 - \7'lg,
which gives the assertion for m = 0, 1. Set m = 2. Then, using (2.14),
1\7'(\7 - \7')l gl,x < 1(\7 - \7')(\7 - \7')l gl,x + 1\7(\7 - \7')lgl,x, bl\7'(\7 - \7')l gl < C2(bl\7 - \7'1; + 1\7(\7 - \7')lg), b,2lg _ g'lgl < Co· big - g'lg + C1 . bl\7 - \7'lg + C2 · (bl\7 - \7'1; + bl\7(\7 - \7')lg) = P2,g. Assume now for f1
:s: m -
1
b, J1 lg - 9'l gl :s: PJ1 ,g, set \7' - \7 = ry and consider \7'Ary, A :s: m - 1. Observe that \7,J1-1ry enters as highest derivative into b, J1 lg - g'lgl. Starting with \7,Ary = (\7'A - \7\7,A-1)rJ+ \7\7,A-1ry = (\7'- \7)\7,A-1ry+ \7\7,A-1ry, (2.15) one obtains by an easy induction A
L \7i-1(\7' -
\7)\7,A-iry + \7 Ary,
(2.16)
i=l
A
1\7'Arylgl,X <
L l\7i-1(\7' - \7)\7,A-irylgl,x + I\7Al gl,X' i=l
(2.17)
101
Non-linear Sobolev Structures
For i = 1 we are done in (2.6) according to our induction assumption (2.18) Therefore, we have to consider the terms
\7 i - 1(\7' - \7)\7,),,-i TJ ,
i> 1.
(2.19)
Iterating the procedure (2.15), (2.16) to \7,),,-1 in \7 i - 1(\7' \7) \7,),,-1 TJ , we have finally to estimate the expressions (2.20) i1 + ... + i)" = A, il ~ 1, i2 ~ 1, 0 ::; i)" ::; A - 1. By the Leibniz rule \7((\7' - \7)TJ) = (\7(\7' - \7))TJ + (\7' - \7)\7TJ each term of (2.20) splits into a sum of products (= compositions), where each factor can be estimated by
Taking sup we are done. xEM
Next, we have to prove (U~). Given 5 > 0, we have to show that there exists 5' = 5'(5) > 0 such that V8' 0 V8' c V8, i.e., if (gl, g2) E V8' 0 V8' = {(g~, g~) E M x M I there exists 9 such that (g~, g) E V8' and (g, g~) E V8'}, then
b,ml g1 - g2191 < 5. (U~) would be proved if we could show for gl, g, g2 with b,ml g1 -
gl91 <
00,
b,ml g - g219 <
00
(2.22) where Pm is a polynomial in b,il g1 - gI9 b,jlg-g219' i,j = 0, ... ,m without constant term. We start with 1)
Ig1 - g2191 bl g1 - g2191
< Ig1 - gl91 + Ig - g21 9 < Co' (bl g1 - gl91 + big - g219) 1)
Poo (bl g1 - g191' big - g219)'
(2.23)
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Relative Index Theory, Determinants and Torsion
where we used 9 E bU(gl)' We set \7 91 = \71, \7 92 = \72, \79 = \7. Since, according to our assumptions, for the pointwise norms
1191 we simply write
II
rv
1192
rv
119'
and multiply by constants. This gives
bl\7 1 - \7 21 ~ C· (bl\7 1 - \71
+ bl\7 -
\721) = Pll,
together with (2.23),
b,llg1 - g21 ~ P1(b,il g1 - g191) b,jlg - g219) = Po
+ Pll,i,j =
0, l. (2.24) Consider next b,21g1 - g21 which amounts, using (2.23), (2.24), to the estimate of \7 1(\7 1 - \7 2), We have
1\7 1(\7 1 - \72)1
~
C· (1\71(\7 - \72)1
+ 1\71(\7 1 -
\7)1)· (2.25)
The critical point in (2.14) is the estimate of 1\71(\7 - \72)1. But
1\71(\7 - \72)1 ~ Co' (1(\71 - \7)(\7 - \72)1 + 1\7(\7 - \72)1), bl\7 1(\7 - \72)1 ~ C1 . (bl\7 1 - \71· bl\7 - \7 21 +bl\7(\7 - \72)1), bl\7 1(\7 1 - \72)1 S; C2 . (bl\7 1(\7 1 - \7)1 + bl\7 1 - \71 .bl\7 - \7 21+ bl\7(\7 - \72)1) = P22 (bl\71 - \71, bl\7 1(\7 1 - \7)1, bl\7 - \721, bl\7(\7 - \72)1), together with (2.24)
b,21g1 -g2191 ~ P2(b,ilgl-gI91) b,jlg-g219) = H +P22 , i,j = 0,1,2. (2.26) In the general case
holds and we have to estimate \71(\7 - \72) above
== \7I1]. We start as
T
2: \7i- 1(\7 1 i=1
\7)\7~-i1]
+ \7T1].
(2.28)
103
Non-linear Sobolev Structures
Interating the procedure, we have to estimate expressions of the kind (2.29)
with i1 + ... + ir = r, ir = r. (2.18) splits into a sum of terms each of them can be estimated by (2.30)
where n1 +1+·· ·+n s +1 = r+1, ns < r, n1 + .. ·+n s-1 ~ r-1, s 2: 2. As an example, the term i = r coming from (2.17) leads to
We have two terms, each with s = 2. l\7 r - 1 (\7 1 - \7)11771 corresponds to n1 = r - 1, n2 = 0, 1\7 1 - \711\7r-l 77 1 to n1 = 0, n2 = r - 1. According to the proof of (U~)
where Qni is a polynomial in the indicated variables without constant terms. (2.27) - (2.31) yield b
i
Pr+1,r+1 ( 1\7 1 (\7 1 bl\7 j (\7 - \7 2 )lg),
Co· (bl g1 - gig!
-
\7) Ig1>
+ big -
g2lg)
m-1
+
L Pr+1,r+1 r=O
=
Pm (b,il g1 - glg1> b,jlg - g2Ig)·
Therefore, we established (U~). Denote by b,mU(M) the corresponding uniform structure. It is metrizable since it is 0 trivially Hausdorff and {V1/ n }n::::no is a countable basis. We see, the proof of 2.16 is parallel to that of 1.19, replacing b,mll by IIp,r.
104
Relative Index Theory, Determinants and Torsion
Denote ~M = (M, b,mU(M)) and by b,m M the completion. It has been proven in [202] that b,m M still consists of positive definite elements, which are of class
em.
Remark 2.17 We endowed by our procedure M with a canonical intrinsic em-topology without choice of a cover or a special 9 to define the em-distance. According to the definition of the uniform topology, for 9 E b,m M
o
is a neighbourhood basis in this topology. Proposition 2.18 The space b,m M is locally contractible.
For a proof we refer to [27], [35] and to proposition 1.9 of chapter VII. 0 Corollary 2.19 In b,m M components and arc components coincide. 0
Set
b,mU(g) = {g'
E
b,m M I big - g'lgl <
00,
b,ml g - g'lg <
oo}.
Lemma 2.20 g' E b,mU(g) if and only if 9 E b,mU(g'). Proof.
g'lg <
00.
Assume g' E b,mU(g). Then big - g'lgl < 00, b,ml g We have to show b,ml g - g'lgl < 00, i. e. to estimate \7,T (\7' - \7),
0::; r ::; m -
This has been done in the proof of b,m- 1 1\7'
_ \7l g < l
(U~).
oo,b,m
1.
We conclude
Ig' - glgl <
00.
All the arguments are symmetric in 9 and g' which proves the assertion. 0
105
Non-linear Sobolev Structures
Lemma 2.21 If g' E b,mU(g) then b,mU(g') = b,mU(g).
Proof. Assume h E b,mU(g'). Then big' - hlh < 00, b,ml g' hlgl < 00. By assumption big - g'lgl < 00, b,ml g - g'lg < 00. This implies big - hlh < 00 and, according to the proof of (U~), b,ml g _ hl g < 00, h E b,mU(g), b,mU(g') ~ b,mU(g). The other inclusion follows in the same manner. D
Corollary 2.22 If b,mU(g') b,mU(g).
n b,mU(g)
=1=
0 then b,mU(g') D
Proposition 2.23 Denote by comp(g) the component of 9 E b,mM. Then
comp(g) == b,mcomp(g) = b,mU(g). D
Proof. We start with comp(g) = b,mU(g). Let g' E comp(g) and {gt}o~t9 be an arc between 9 and g'. For any E > 0 this can be covered by a finite number of E-neighbourhoods with respect to the local metric b,ml . - . Ig, b,mUe(gtJ, cr = 0, ... , s, gto = g, gt = g'. We set gt" = gao Then b,mUe(go) n b,mUe(gl) =1= 0. According to corollary 2.22, gl E b,mU(go). Performing a simple induction, g8 = g' E b,mU(go) == b,mU(go), comp(g) ~ b,mU(g). Let g' E b,mU(g). Then {tg' + (1- t)g}O~t9 is an arc in b,mU(g) connecting 9 and g'. This follows from 3.2,3.10 in [35]. See also chapter VII section 1. D 8
By construction, b,mU(g) is open in b,m M and, according to proposition 2.23, closed. Therefore we have Theorem 2.24 The space b,m M has a representation as a topo-
logical sum b,m M =
L b,mU(gi).
(2.32)
iEI
D
106
Relative Index Theory, Determinants and Torsion
Remark 2.25 On a compact manifold M, the index set I consists of one element. One has only one component. On compact manifolds the notion of growth at infinity does not exist. 0
For later use we restrict ourselves additionally to metrics with bounded geometry. Let (Mn, g) be open. Consider the conditions (1) and (B k ) and
M(1) M(B k ) M(I,Bk)
{g E Mig satisfies (I)} , {g E Mig satisfies(Bk )}, M(1) n M(Bk)'
Lemma 2.26 If gEM (1), then (Mn, g) is complete. Proof. We have to show that every Cauchy sequence converges. Let r = rinj(M) > 0 and (Xi)i be a Cauchy sequence. There exists io such that e(Xi, Xj) < J for all i,j 2: io. Let U~(Xio) = exp(B~(OXio)) be the geodesic ball of radius ~ cen-
tered at Xio' Then, Xi E Ui(Xio) for all i 2: io· But Ui(Xio) C U~(Xio) is complete and (xik::io ----; X E Ui(Xio)' 0
Remark 2.27 There are several other proofs of this simple fact.
o It is trivially clear that (B k ) does not imply completeness. Nevertheless we have Corollary 2.28 If 9 E M(I, B k), then 9 is complete.
0
Proposition 2.29 Let 9 E M(B k), g' E M and b,k+ 2 Ig _ g'I9 < 00.
Then g'
E
M(Bk)'
We refer to [32].
o
Non-linear Sobolev Structures
107
Corollary 2.30 The space M(Bk) is open in ~M, m ~ k + 2. D
Now we can restrict our uniform structure b,mU(M) to M(Bk) x M(Bk) and obtain a completed metrizable space. But for m < k + 2 there doesn't exist a good description of components. Therefore it would be better to consider the case m ~ k + 2. Lemma 2.31 Let 9 E b,mM, comp(g) the component of 9 in b,m M and m ~ k + 2. If comp(g) contains a em-metric satisfying (B k ), then all metrics of comp(g) satisfy (Bk)'
This follows from propositions 2.34. Corollary 2.32 Let m as topological sum
~
k
b,m M(Bk)
+ 2. =
D
There exists a representation
L b,mU(gj), jEJ
c I an index set. Each component of b,m M(Bk) is a Banach manifold.
J
We refer to [32] for the proof that each component is a Banach manifold. Proposition 2.33 Assume 9 satisfies (1) and (B k ), k ~ O. Then there exists E > 0 such that g' satifies (1) for all g' E b,kUe(g) .
We refer to [32].
D
Proposition 2.34 Assume 9 satisfies (1) and (B k ), g' E comp(g) C b,m M, m ~ k + 2. Then g' satifies (1) and (Bk)'
108
Relative Index Theory, Determinants and Torsion
We refer to [32] for the proof. D Now we want to introduce Sobolev uniform structures into the space of metrics. Let now k 2: r > ~ + 1, 6 > 0, C(n,6) =
1 + 6 + 6J2n(n - 1),
V8 = {(g, g')
E
M(I, B k) x M(I, B k) I
C(n, 6)-1g :S g' :S C(n, 6)g and Ig - g'lg,p,r = 1'-1
+L
(J
(Ig -
g'I~,x 1
g1 g I(\7 )i(\7 - \7g)I~,x) dvolx(g)) P <
6}.
i=O
Proposition 2.35 The set {V8}.5>o is a basis for a metrizable uniform structure on M(I, Bk)'
The proof is quite analogous to that of 2.16. We replace bll by IIp,r and apply in (2.17) - (2.31) the module structure theorem. D
Denote M~(I, B k ) as (M(J, Bk),up,r(M(I, B k))) and by MP,r(I, B k ) the completion. It was proven [67] that the completion yields only positive definite elements, i.e. we still remain in the space of C 1 Riemannian metrics. For 9 E MP,r(I, B k)
{u:,r(g)}c:>o
=
{{g'
E
MP,r(I, B k) I big - g'lg <
big - g'lgl <
00,
00,
Ig - g'lg,p,r < E:} }c:>O
is a neighbourhood basis in the uniform topology. There arises a small difficulty. 9 E MP,r(I, B k) must not be smooth and hence Ig - g'lg,p,r must not be defined immediately. But in this case we use the density of M(J, B k) c MP,r(I, B k) and refer to [27] for further details. Proposition 2.36 The space MP,r(I, B k) is locally contractible.
109
Non-linear Sobolev Structures
For the proof we refer to [35], lemma 3.8 or the chapter VII. 0
Proposition 2.37 In MP,r(I, B k ) components and arc compo0 nents coincide. Set for 9 E MP,r(I, B k )
u:,r(g) = {g' E MP,r(I, B k ) I big - g'lg <
00,
big - g'lgl <
00,
Ig - g'lg,p,r < oo}. Lemma 2.38 We have: g' E up,r (g) if and only if 9 E up,r (g'). Proof. Assume g' E up,r(g). We have to show Ig - g'lg',p,r < Since Ig - g'lg,p < 00 and 9 and g' are quasi isometric, we conclude Ig - g'lg',p < 00. There remains to show 1\71t (\7' \7)l g"p < 00, 0 :S i :S r - 1. But this can literally be done as in the proof of (U~). All arguments are symmetric in 9 and g' which proves the assertion. 0 00.
Lemma 2.39 If g' E up,r(g) then up,r(g')
= up,r(g).
Proof. Assume h E up,r(g'). Then big' - hlgl < 00, Ig'hlg',p,r < 00. By assumption big - g'lgl < 00, Ig - g'lg,p,r < 00. This implies big - hlh < 00 and, according to the proof of (Un, Ig - hlg,p,r < 00, h E up,r(g), up,r(g') ~ up,r(g). The other inclusion follows quite similar.
o Corollary 2.40 If up,r(g) n up,r(g') =Iup,r(g).
0, then up,r(g')
Proposition 2.41 Denote by comp(g) the component of 9 E MP,r(I, Bk)' Then,
comp(g)
== compp,r (g) = up,r (g).
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Relative Index Theory, Determinants and Torsion
Proof. We start with comp(g) ~ up,r(g). Let g' E comp(g) and {gt}o::;t9 be an arc between 9 and g'. For any E > 0 this arc can be covered by a finite number of E-balls Uf,r (gt u ) ' a = 0, ... ,s, gto = g, gt s = g'. We set gt u = ga and assume without loss of generality Uf,r(ga) n Uf,r(ga+1) =1= 0. If this would not be the case, then we would choose a corresponding ordering. Then Uf,r(go) n Ur(gl) =1= 0. According to corollary 2.39, gl E Uf,r(go). An easy induction gives gs = g' E up,r(g), comp(g) ~ up,r(g). Then {tg' + (l-t)g }O::;t9 is an arc in up,r(g) connecting 9 and g'. This follows from lemma 3.8 of [35J or chapter VII. 0 By construction, up,r (g) is open and, according to proposition 2.36, closed. Therefore, we have
Theorem 2.42 Let Mn be open, k 2: r > ~ + 1. MP,r(I, B k ) has a representation as a topological sum
MP,r(I, B k ) =
Then
L up,r(gi). iEI
o We finish at this point the example of uniform structures of Riemannian metrics and turn to our last class of examples, uniform structures of metric spaces. We start with the GromovHausdorff uniform structure. Let Z = (Z, d z ) be a metric space, X, Y c Z subsets, E > 0, define Ue(X) := {z E Z I dist( z, X) < E}, analogously Ue (Y). Then the Hausdorff distance dH(X, Y) = dfI(X, Y) is defined as
drr(X, Y)
:= inf{E
I X c Ue(Y), Y c Ue(X)}.
If there is no such E then we set dfI(X, Y) := 00. Then dfI factorized by d(·,·) = 0 is an almost metric on all closed subsets, i.e. it has values in [O,ooJ but satisfies all other conditions of a metric. If Z is compact then dfI is a metric on the the set of all closed subsets. A metric space (X, d) is called proper if the closed balls Be(x) are compact for all x E X, E > O. This
111
Non-linear Sobolev Structures
implies that X is separable, complete and locally compact. In the sequel we restrict our attention to proper metric spaces. Let X = (X,d x ), Y = (Y,d y ) be metric spaces, Xu Y their disjoint union. A metric d on X U Y is called admissible if d restricts to dx and dy , respectively. The Gromov-Hausdorff distance dGH(X, Y) is defined as
dGH(X, Y)
:=
inf{dH(X, Y) I d is admissible on XU Y}.
Note that the Gromov-Hausdorff distance can be infinity. Originally Gromov defined dGH as
dGH(X, Y) := inf{d~(i(X),j(Y))i: X
-t
Z,j : Y
-t
Z
isometric embeddings into a metric space Z}. It is a well known fact and can simply be proved that both definitions coincide. Lemma 2.43 If X and Yare compact metric spaces and dGH(X, Y) = 0 then X and Yare isometric.
This follows from the definition and an Arzela-Ascoli argument.
o Remark 2.44 The class of all metric spaces, even only up to isometry, is not a set. But the class of isometry classes of proper metric spaces is, so we only consider proper metric spaces in this chapter. A proper metric space X can be covered by a countable number of compact metric balls of fixed radius. Each such ball is isometric to a subset of Loo([O, 1]) and we obtain X after identification of the overlappings, i.e. we can understand X as a subset of (IT:l Loo([O, 1]))2.
o Denote by 9J1 the set of all isometry classes [X] of proper metric spaces X and 9J1GH = 9J1/ rv where [X] rv [Y] if dGH([X], [Y]) =
O.
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Relative Index Theory, Determinants and Torsion
Proposition 2.45 dCH defines an almost metric on 9J1, i. e. it is a metric with values in [0,00]. D
We write in the sequel X = [X]CH if there cannot arise any confusion. Now we define the uniform structure. Let <5 > 0 and set v" := {(X, Y) E 9J1~H I dcH(X, Y) < <5}.
Lemma 2.46 Q3 = {V.s}.s>o is a basis for a metrizable uniform structure UCH (9J1). Proof. Q3 is defined by a local metric. Hence it satisfies all D desired conditions. Let 9J1CH be the completion of 9J1CH with respect to UCH and denote the metric in 9J1 by dCH .
Lemma 2.47 9J1CH = 9J1CH as sets and dCH and d CH are locally equivalent, i. e. for any X E 9J1CH = 9J1CH there exist equivalent neighbourhood bases by metric balls. Proof. Cauchy sequences with respect to dCH and UcH(9J1 CH ) coincide. But according to [56], proposition 10.1.7, p. 277, any Cauchy sequence in 9J1 with respect to dCH converges in 9J1CH . This proposition is formulated there for compact metric spaces, but the proof does not use compactness at any stage. The assertion concerning the local equivalence of dCH and dCH follows immediately from the formulae (2.4)-(2.6) in [68], 11.2.7, p. 117. We refer also to 2.3, 2.4. D Let X E 9J1CH = 9J1CH and denote by comp(X) and arccomp(X) the component and arc component of X in 9J1CH , respecti vely. A key role for all what follows is played by
Proposition 2.48 9J1CH = 9J1CH is locally arcwise connected.
Non-linear Sobolev Structures
113
We refer to [33] for the proof.
D
Corollary 2.49 In m GH components and arc components coincide. Moreover, each component is open and m GH = m GH is the topological sum of its components,
m= Lcomp(X
i ).
iEI
D
Proposition 2.50 Let X E m GH = m GH . Then comp(X) zs given by comp(X) = {Y E mGH I dGH(X, Y) < oo}.
Proof. Let Y E comp(X) = arccomp(X). Then there exists an arc between X and Y. For given c > 0 this can be covered by a finite number, say r, of c-balls. Hence dGH(X, Y) ~ 2rc < 00. If dGH(X, Y) = c < 00 then we can construct an arc from X to Y as given in the proof of proposition 2.48. D Hence we have a quite natural splitting of mGH = mGH into its components = arc components and a canonical topology and convergence inside each component. This is not - as usual until now - the pointed convergence of all metric balls but uniform convergence. We discuss this later. Any complete Riemannian manifold determines a unique component. First we give another characterization of components. We call a map : X --+ Y metrically semilinear if it satisfies the following two conditions.
1. It is uniformly metrically proper, i.e. for each R > 0 there is an S > 0 such that the inverse image under of a set of diameter R is a set of diameter at most S. 2. There exists a constant Gil> 2': 0 such that for all
Xl, X2
E X
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Relative Index Theory, Determinants and Torsion
Two metric spaces X and Yare called metrically semilinearly equivalent if there exist metrically semilinear maps : X - t Y, W : Y - t X and constants Dx and D y , such that for all x E X and y E Y d(x, wx) :S D x ,
d(wy, y) :S D y
.
Proposition 2.51 Y E comp(X), i.e. dCH(X, Y) < 00 if and only if X and Yare metrically semilinearly equivalent.
o We refer to [33] for the proof. The other class of uniform structures of particular meaning are Lipschitz uniform structures. A map : X ----t X is called Lipschitz if there is a constant C> 0 such that
for all Xl, X2 EX. Restricting now to Lipschitz maps, we create a local metric which takes into account the measure of expansivity. Define for a Lipschitz map : X - t Y dil :=
Xl~~~X
d( x I , X2) d(XI' X2) .
xli' x 2
Set dL(X, Y)
.- inf{max{O,logdil } +max{O,logdil w} + sup d(Wx, x) + sup d(wy, y) xEX
I : X
yEY -t
Y, W : Y
-t
X Lipschitz maps},
if { ... } =1= 0 and inf{ ... } is < 00 and set ddX, Y) = 00 in the other case. Then d L ~ 0, symmetric and dL(X, Y) = 0 if X and Yare isometric. Set ML = Mj "', where X '" Y if ddX, Y) = O. That", is in fact an equivalence relation follows from 2.52 below. Let is > 0 and define V" = {(X, Y) E MildL(X, Y) < oo}.
Non-linear Sobolev Structures
115
Proposition 2.52 ~ = {V8}o>o is a basis for a metrizable uniform structure tiL(M L). We refer to [33] for the proof.
0
Denote by 9J1~L the completion of 9J1L with respect to tiL, We come back to the topological properties of 9J1L below. Before doing this we introduce still another important Lipschitz uniform structure. Define for X, Y E 9J1
dL,top(X, Y)
:=
inf{max{O, log dil }
I <1>: X if { ... }
i- 0 and = 00
--t
+ max{O, log dil
-l} Y is a hi-Lipschitz homeomorphism}
in the other case.
Then dL,top(X, Y) ~ 0, symmetric and dL,top(X, Y) = 0 if X and Yare isometric. Set 9J1L ,top := 9J1/ ~ where X "" Y if dL,top(X, Y) = O. Then dL,top is an almost metric on 9J1L ,topo
Remark 2.53 Gromov defined in [39] dL,top(X, Y)
:=
inf{llogdil <1>1 + Ilogdil <1>-11 I <1>: X ---t Y is a bi-Lipschitz homeomorphism. }
But this does not work since this dL,top does not satisfy the triangle inequality. log(x· y) = logx + logy, log is monotone increasing, but Zl ~ Z2 + Z3 does not imply IZ11 ~ IZ21 + IZ31· There are explicit counterexamples to the triangle inequality of the local I log dil (.) I-metric. 0 Set for 8 > 0
V8 = {(X, Y) Proposition 2.54 ~ form structure tiL, top'
E
9J11,topldL,top(X, Y) < 8}.
= {Vo}o>o is a basis for a metrizable uni0
Denote by 9J1~~;;P the completion of 9J1 L,top with respect to tiL,top.
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Relative Index Theory, Determinants and Torsion
Proposition 2.55 9Jh,top is complete with respect to llL,top, i. e. 9JtL,top = 9Jt L,top.
o We refer to [33] for the proof. Next we return to 9Jt L. Above we discussed 9Jt L,top since the pro?fs for 9JtL are modelled by the proofs for 9Jt L,top- The next result is valid only for noncom pact proper metric spaces (= classes of spaces, exactly spoken). Such spaces have infinite diameter. The restriction to noncom pact proper spaces is no restriction for us since we have in mind them only. Denote by 9Jt L(nc) the (classes of) noncompact proper metric spaces. We proved in [33] Proposition 2.56 9Jt L(nc) is complete with respect to llL(9Jt L) restricted to 9Jtdnc). 0 Remark 2.57 If we restrict to compact metric spaces then by Arzela-Ascoli arguments dL,top(X, Y) = 0 or ddX, Y) = 0 al-
ways imply X isometric to Y. Hence the factorization by or
rv
L
rv L,top would not be necessary. For noncom pact proper metric
spaces the corresponding question is open (at least for us). If a sequence (
o The next step is to prove that 9JtL(nc) and 9Jt L,top are locally arcwise connected. We start with 9Jt L,top since the proof is easier and the proof for 9JtL(nc) is in certain sense modelled by that for 9JtL,top. Proposition 2.58 9Jt L,top is locally arcwise connected.
Non-linear Sobolev Structures
117
o
The proof is performed in [33].
Corollary 2.59 In 9J1 L,top components and arc components coincide. 0 Theorem 2.60 a) 9J1 L,top has a representation as a topological sum
b) compX = {Y
E 9J1 L,topldL,top(X, Y)
< oo}.
Proof. a) follows immediately from 2.59. b) Assume dL,top(X, Y) < E. The proof of 2.58 provides an arc {Xth between X and Y, i. e. Y E compX. Conversely, suppose X and Y can be connected by an arc {Xth in 9J1 L,top' For any <5 > 0 this can be covered by r <5-balls (w. r. t. dL,top). Then dL,top(X, Y) < 00. 0 A crucial step are the same assertions for 9J1 L which we prove now. We started with 9J1 L,top since the proofs for 9J1L are (slightly) modelled by them for 9J1 L,top. Proposition 2.61 9J1 L(nc) = 9J1 L(nc) is locally arcwise connected. We refer to [33] for the rather long proof. We immediately conclude from 2.61
o
Theorem 2.62 In 9J1 L and 9J1 L(nc) components coincide with arc components. 9J1L and 9J1L(nc) have topological sum representations
iEI
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Relative Index Theory, Determinants and Torsion
and compL(X) = {Y E 9JhlddX, Y) < oo}. In particular all compact spaces lie in the component of the 1point-space. 0 Finally we define still three further uniform structures which measure or express the homotopy neighbourhoods and, secondly, admit only compact deviations of the spaces inside one component. Define
dL,h(X, Y) .- inf { max{O, log dil }
+ max{O, log dil Ill}
+ sup d(llIx, x) + sup d(llIy, y) x y I : X -------t Y, III : Y -------t X are (uniformly proper) Lipschitz homotopy equivalences, inverse to each other} if there exist such homotopy equivalences and set dL,h(X, Y) = in the other case. Here and in the sequel we require from the homotopies to id x or id y , respectively, that they are uniformly proper and Lipschitz. dL,h ~ 0, dL,h is symmetric and dL,h(X, Y) = 0 if X and Yare isometric. Define 9J1 L,h = 9J1/~, X rv Y if dL,h(X, Y) = 0 and set v., = {(X, Y) E 9J1i,hldL,h(X, Y) < 5}. 00
That this rv is in fact an equivalence relation follows from 2.63 below which we will only state here. The proof consists only of a generalized triangle inequality and is completely parallel to that of 2.52. We only restrict the class of admitted maps to uniformly proper Lipschitz homotopy equivalences. Proposition 2.63 ~ = {V,s}8>O is a basis for a metrizable uni0 form structure llL,h(9J1 L,h). Proposition 2.64 9J1~~hh(nc) = 9J1 L,h(nc).
Non-linear Sobolev Structures
119
We omit the proof which is again parallel to that of 2.56.
0
Denote by arccompL,h(X) the arc component of X in 9Jt L,h.
Proposition 2.65 If Y E arccomPL h(X) then X and Yare (uniformly proper) Lipschitz homotopy equivalent. In particular dL,h(X, Y) < 00. Proof. Cover the arc between X and Y by a finite number of dL,h-balls and use transitivity of homotopy equivalence. 0 Let : X ~ Y, W : Y ~ X be Lipschitz maps. We say that and Ware stable Lipschitz homotopy equivalences at 00 inverse to each other if there exists a compact set K~- c X s.t. for any Ki c Kx there exists K y C Y s.t. lx\Kx : X\K x ~ Y \ K y is a Lipschitz h.e. with homotopy inverse WIY\Ky and W has the analogous property. Set
dL,h,rel(X, Y)
.- inf { max{O, logdil }
+ max{O, logdil w}
+ sup d(wx, x) + sup d( Wy, y) x
I : X
y
~
Y, W : Y ~ X are stable
Lipschitz homotopy equivalences at
00
inverse to each other} if there exist such maps , wand set dL,h,rel(X, Y) 00 in the other case. Then dL,h,rel ~ 0, symmetric and dL,h,rel = if X and Yare isometric. Set 9JtL,h,rel = 9Jt/~, X rv Y if dL,h,rel(X, Y) = and set
°
°
V" = {(X, Y) E 9Jti,h,relldL,h,rel(X, Y) < 8}. Proposition 2.66 ~ = {V,,},,>o is a basis for a metrizable uniform structure tiL,h,rel. 0 Proposition 2.67 If Y E arccompL,h,rel(X) then X and Y are stably Lipschitz homotopy equivalent at 00. In particular dL,h,rel(X, Y) < 00. 0
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Relative Index Theory, Determinants and Torsion
The last uniform structure tiL,top,rel is defined by dL,top,rel(X, Y), where we require that : X ----7 Y, W : Y ----7 X are outside compact sets bi-Lipschitz homeomorphisms, inverse to each other. We obtain 9J1 L ,top,rel. There holds an analogue of 2.66 and 2.67. We finish this considerations with a scheme which makes clear our achievements, where we refer to section 5 for the definition of coarse structures. One coarse equivalence class ./ splits into \.. many GH-components
one L-component
many L-components,
=
arc component
./ splits into \.. L,top,rel-arc components
L,h,rel-arc components
./ L,top-components = L,top-arc components components.
L,h-arc
It is now a natural observation that the classification of noncompact proper metric spaces splits into two main tasks 1. "counting" the components at any horizontal level, 2. "counting" the elements inside each component. A really complete solution to these two problems, i.e. a complete characterization by computable and handy invariants, is nowa-day hopeless. It is a similar utopic goal as the "classification of all topological spaces". Nevertheless stands the task to define series of invariants which at least permit to decide (in good
Non-linear Sobolev Structures
121
cases) nonequivalence. This will be the topic of the second part of this section, of sections 2 and 3. Finally we remark that GH-components (dCH(X, Y) < (0) and L-components (ddx, Y)) are very different. Roughly speaking, dCH is in the small unsharp and in the large relatively sharp, dL quite inverse. In section 5 we define invariants for the components of our uniform structures. It is clear that an invariant of comPti is also an invariant of compti' C comPti' U' finer than U.
3
Completed manifolds of maps
Our next class of examples for non-linear Sobolev structures are manifolds of maps and diffeomorphism groups. Let (Mn, g), (Nn', h) be open, complete, satisfying (1) and (Bk) and let f E COO(M, N). Then the differential f* = df is a section of T* M ® j*T N. j*T M is endowed with the induced connection j*\1 h which is locally given by
\1g and j*\1 h induce metric connections \1 in all tensor bundles TJ(M) ® j*T::(N). Therefore \1mdf is well defined. Since (1) and (Bo) imply the boundedness of the gij, lm, h/-lv in normal coordinates, the conditions df to be bounded and od to be bounded are equivalent. In local coordinates sup Idflx = suptrg(f*h) = supgijh/-lvojjlloir· xEM For (Mn, g), (Nn', h) of bounded geometry up to order k and m ::; k we denote by coo,m(M, N) the set of all f E COO(M, N) satisfying m-l
b,mldfl :=
L
sup 1\1/-ldflx < /-l=0 xEM
00.
Relative Index Theory, Determinants and Torsion
122
Assume (Mn, g), (Nn', h) are open, complete, and of bounded geometry up to order k, r :::; m :::; k, 1 :::; p < 00, r > ~ + 1. Consider f E C'Xi,m(M, N). According to theorem 1.16 b) for r>~+s
op,r (J*T N) <-+ b,sO(J*T N), b,slYI < _ D . IYI p,Tl
(3.1)
(3.2)
1
where IYlp,r = (ftol\7iY1PdVOl)P. Set for 8> 0, 8·D:::;
8N < rinj(N)/2, 1 :::; p < 00, 118 = {(J, g) E coo,m(M, N) x coo,m(M, N) I:3Y E O~(J*TN) such that 9 = gy = exp Y and IYlp,r < 8}. Proposition 3.1 ~
= {V,,}o<
o We refer to [31], [27] for the very long proof. up,r(coo,m(M, N)) is metrizable. Let mop,r(M, N) be the completion of coo,m(M, N). From now on we assume r = m and denote rop,r(M, N) = D,p,r(M, N). Theorem 3.2 Let (Mn, g), (Nn, h) be open, complete, of bounded geometry up to order k, 1 :::; p < 00, r :::; k, r > ~ + 1.
Then each component of D,p,r(M, N) is a Cl+ k - r -Banach manifold, and for p = 2 it is a Hilbert manifold.
o We refer to [31], [27] for the proof. The elements of op,r (M, N) are characterized by the following property. f E D,p,r (M, N) if and only if for every E > 0 there exist j E coo,r(M, N) and a Sobolev vector field X along j, X E D,p,r(j*T N), IXlp,r < E, such that f(x) = (exp X)(x) = exp!(x) X/(x) = (exp! X
0
j)(x).
(3.3)
In particular coo,r(M, N) is dense in op,r(M, N). A special case is given if we restrict to diffeomorphisms. Let (Mn, g), (Nn, h)
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Non-linear Sobolev Structures
as in the hypothesis of theorem 3.2. Define
1)p,r(M, N) = {f E np,r(M, N) I f is a diffeomorphism and there exists constants c, C > 0 such that c:::; inf Idflx :::; sup Idflx :::; C}. (3.4) xEM
x
(3.4) automatically implies the existence of constants such that C1 :::; inf Idf-1lx :::; sup Idf-1lx :::; C 1. x
C1,
C1 > 0
x
(3.5)
In fact, for diffeomorphisms (3.4) and (3.5) are equivalent. Moreover, (3.4) is an open condition in np,r(M, N). Hence we have Theorem 3.3 Suppose the hypothesis of 3.2. Then each component of1)p,r(M, N) is a cl+k-r -Banach manifold and for p = 2 it is a Hilbert manifold. 0
Corollary 3.4 Suppose for M = N the hypothesis of 3.2. Then each component of 1)p,r(M) is a cl+k-r -Banach manifold and for p = 2 it is a Hilbert manifold. 0
In the sequel, we need still a relative version of these manifolds of maps. Suppose (Mn,g), (Nn,h), k, r, p as in 3.2 and that there exist compact submanifolds KM C M, KN C N such that there exists f E np,r(M, N) with the following properties. a) fIM\KM maps M \ KM diffeomorphic onto N \ KN and b) there exist constants c, C > 0 such that
Then, automatically,
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Relative Index Theory, Determinants and Torsion
We denote for fixed K M, KN the subset c D,p,r(M, N) of these I by V~~~(M, N). Clearly, V~~~(M, N) depends on the choice of KM,KN . Finally, we need this construction still for Riemannian vector bundles. Let (Ei' hi, \7 hi ) ----t (Mr,9i), i = 1,2 be Riemannian vector bundles satisfying (1), (Bk(Mi , 9i)), (Bk(Ei , hi, \7i)), k 2: r > ~ + 1. If we endow the total spaces Ei with the Kaluza-Klein metric 9E(X, Y) = h(XV, yV) + 9M(-rr*X, 71"*Y) , Xv, yv vertical components, then E 1, E2 are again manifolds with bounded geometry (cf. [30]) and D,p,r(E1, E 2) is well defined. If we restrict to bundle maps (fE, 1M = 71" 0 IE 071"-1 then we obtain a subset D,~b(E1' E 2 ) c D,p,r(E1, E2)' Quite analogously to above we define V~{ (E1' E 2 ) c D,~: (E1' E 2 ) if the bundles are isomorphic and V~brrel(E1' E 2 ) if they are isomorphic over M \ KM and N \ K N . Here we require (3.3) and (3.5) both for IE and 1M' We apply this notations in the next section.
Uniform structures of manifolds and Clifford bundles 4
We introduce in chapters IV - VI relative index theory, relative eta and zeta functions, relative determinants and relative analytic torsion. The whole approach relies on the following construction. We endow the set of isometry classes of Clifford bundles (of bounded geometry) with a metrizable uniform structure, define generalized components gen comp( E) (= set of Clifford bundles E' with finite Sobolev distance from a given E), associate the corresponding generalized Dirac operators D, D' and make all constructions for the pair D, D', where D' is running through gen comp( E). The first step for doing this is the introduction of the corresponding uniform structure(s). This is the content of this section. The applications will be performed in chapters IV - VI. Denote by fJItn(ml, I, B k ) the set of isometry classes of n-dimensional Riemannian manifolds (Mn, 9) satisfying the con-
Non-linear Sobolev Structures
125
ditions (1) and (Bk)' We defined for (Mf,gl), (M:;,g2) E mn(mf, I, B k ) and k > ~ + 1 the diffeomorphisms DP,r(M1, M 2) and for appropriate compact submanifolds Ki C M i , M1 \ K1 9:: M2 \ K2 the maps V:~~(M1' M2) == D~~~(M1' M2; Kl, K 2) c np,r(M1, M 2) as at the end of section 3. Recall bldfl = sup Idflx. x The elements f of D~:Z, np,r(M1, M 2) are not smooth. For k ~ r > ~ + 2 they are C 2 . Hence f* 9 is a C 1 metric. This would cause some troubles if we would consider in the sequel only classical derivatives which would disappear if we work with distributional derivatives. Another way to work with the nonsmoothness of f*g is to work with smooth approximations of f. We decide to go this way and define cr,p,r(M1, M 2) = U E np,r(M1, M 2)lf E cr+1(M1' M 2) and bl\7idfl < 00, i = 1, ... , r}. Completing the uniform structure below, we end up with fs E D~~~, np,r, i.e. the restriction at the beginning to Ck fs implies at the end no further restriction. We restrict in the sequel to k -> r > !!:p + 2. Further we remark that the conditions c :::; blf*1 :::; C and C1 :::; blf;11 :::; C1 are equivalent: blf;11 ~ = C1 follows from blf*f;11 = 1 and blf*1 :::; C and blf;11 :::; C 1(c, C) follows from elementary matrix calculus. If f* is the induced map between twofold covariant tensors then 1 2 2 C2 = c :::; blf*1 :::; C = C2, similarly C3 :::; blf*- 1 :::; C 3 . Under these conditions, I\7 i f* I :::; d, 1 :::; i :::; v implies I\7 i f* I :::; d1 , 1 :::; i :::; v, and l\7if*- 1 1 :::; d2, 1 :::; i :::; v, where d1, d2 are continuous functions in c, C, d. All this follows from f;1 f* = id*, f*-1 f* = id* and 0 = \7id* = \7id*. Consider now pairs (Mf,g1),(M:;,g2) E mn(mf,I,Bk ) with this property: There exist compact submanifolds Kf C Mf, K'2 C M:; and an f E D~~~(M1' M 2, Kl, K2)' For such pairs
b
Relative Index Theory, Determinants and Torsion
126
define
d1j{diff,rel(M1 , 91)' (M2,92)) := inf { max{O,logbldfl} + max{O, log bldhl} + sup dist(x, hfx) XEMl + sup dist(y, hfy) + sup IV'idfl + sup IV'idhl yEM2 "'EMl "' EM2 l~i~r l$i~r +IUIMl\Kl)*92 - 91IMl\KlI9l,p,r If E r,p,r(M1 , M 2), hE r,p,r(M1 , M 2)
c
c
and for some Kl C M 1 , K2 C M2 holds flMl\Kl E ,])p,r(M1 \ KI, M2 \ K 2) and hM2\K2 =
UI Ml\Kl)-I},
if { ... } f= 0 and inf{ ... } < 00. d~4iff,rel((Ml' 91)' (M2' 92)) = 00. Set
v"
(4.1) In the other case set
= {((MI, 91), (M2' 92)) E (mn(mf,I, Bk))21 ~:diff,rel(Ml' 91), (M2' 92)) < 5}.
Proposition 4.1 Suppose r > ~ + 2. Then Q3 = {V,,},,>o is a basis for a metrizable uniform structure on mn(mf, I, B k )/ rv, where (Ml' 91) rv (M2,92) if d~:diff,rel(Ml' 91), (M2' 92)) = 0. Proof. We have to verify (U~) and (U~), i.e. the symmetry and transitivity of the basis (not of dt~ifJ,rel) and start with (U~). For this it is sufficient that
(4.2) implies such that
(4.4)
127
Non-linear Sobolev Structures
We consider the single numbers in the set (4.1). The first number, sum of 4 terms is symmetric in f and h. The second number sup l\7 i dfl + sup l\7 i dhl is symmetric in f and h too. Suppose yE M 2 l$i$r
yEMl l$i$r
1
(fIMl\Kl)*g2 - glIMl\K1191,P,T
== {
J
1(fIMl\Kl)*g2 -
Ml\Kl T-l
+
L
(\7 91 )i(\7 91
11
-
gll~l'x
\7r92)1~1,xdvolx(gl)
i=O
r; 1
< 61 . (4.5)
Now we have to estimate
We omit in the notation 1Mi\Ki since in the remaining part of the proof we restrict to this. Then
Now
Hence, in the case r = 0,
Consider now the case r
= 1. Then
1\7(f*-l(gl - j*g2))ly =
1\7(j*-l)(gl - j*g2) +j*-l\7(gl - j*g2)ly ::; bl\7(f*-l)l· Igl - j*g21 +blj*- l l . 1\7(gl - j*g2)1· (4.8)
128
Relative Index Theory, Determinants and Torsion
We briefly discuss the case r = 2, to indicate the general rule.
1\7[\7(j*-I(gl - j*g2))lI92,Y = 1\7[\7(j*-I)(gl - j*g2) + j*-I\7(gl - j*g2)lI92,Y = 1\7 2 (j*-I)(gl - j*g2) + \7(j*-I)\7(gl - j*g2) +\7 j*-I\7(gl - j*g2) + j*-1\72(gl - j*g2)192,Y :S bl\7 2j*- 11. Igl - j* g21 + bl\7 j*- I II\7(gl - j* g2) 1 +bl\7 j*- I II\7(gl - j*g2) 1 + blj*- 111\72(gl - j*g2)1. (4.9) Continuing in this manner, we obtain on the right hand side linear polynomials in l\7 i (gl - j*g2) 1 without constant term and where the coefficients can be estimated by is. Summing up (4.7), (4.8), (4.9) and integrating over Ml \ K 1 , we obtain
In particular, (4.11) implies (4.12) This proves (U~). Completely similar is the proof of the transitivity of the basis.
= = h
We assume (M1, gl)
h
(M2, g2)
hI
(U~),
(M3, g3), fi : Mi \Ki ~
i.e.
MHI \
h2
K H1 , i = 1,2, with the desired properties. The triangle inequality for the sum of the first 4 terms in the set (4.1) is just proposition 2.52. Consider the next to numbers in the set (4.1). Applying the Leibniz rule, immediately yields sup l\7i(!2*fh) 1+ sup l\7i(hhh 2*)1 "'EMI
zEM3
l~i~r
l~i:::;r
:S c[ sup l\7 i fhl . sup l\7i!2*1
+
",EM
yEM2
l:::;i:::;r
l:::;i~r
i
sup l\7ih2 *1· sup l\7 hh IJ, zEM3
yE M 2
l~i:::;r
l~i:::;r
(4.13)
129
Non-linear Sobolev Structures
where C essentially is an expression in binomial coefficients. (4.13) expresses the desired transitivity of the basis. The desired transitivity of the last number in the set (4.1) would be established if (4.14) would imply (4.15) We estimate by the triangle inequality for Sobolev norms
l(hh)*93 - 91IMj\Kj,9j,p,r
== If{U;93 - f{-1 91 19j,p,r
+ If{(92 - f{-1 91 )19l,p,r = If{U;93 - 92)19j,p,r + 1f{92 - 91)1 9l,p,r i :::; sup IV fhl·lf;93 - 92192,p,r + If{92 - 9119l,p,r :::; If{U;93 - 92)19j,p,r
xEMl
O~i~r
(4.16) D
Denote the corresponding uniform structure with ll~~iff,rel and 9Jt2',~iff,rel for the completion of 9Jtn(mf, I, B k ) with respect to this uniform structure. It follows again from the definition that d~~dif f,rel (( M 1, 91)' (M2,92)) < 00 implies dL((M1, 91)' (M2, 92)) < 00, where dL is the Lipschitz distance of section 2. Hence (M2,92) E comPL (MI, 91), where comPL denotes the corresponding Lipschitz component, i.e.
{(M2,92) E 9Jt2',~;ff,rel ~ compL(M1, 91)'
I ~~diff,rel(MI,M2) < oo}
For this reason we denote the left hand side { ... } by gen compt~iff,rel(M1' 91) = {... } = {... } n comPL(M1,91)
130
Relative Index Theory, Determinants and Torsion
keeping in mind that this is not an arc component but a subset (of manifolds) of a Lipschitz arc component, endowed with the induced topology. We extend all this to Riemannian vector bundles (E, h, V'h) ------t (Mn, g) of bounded geometry. First we have to define vp,r (E --> M). For this we consider as at the end of section 3 the total space E as open Riemannian manifold of bounded geometry with respect to the Kaluza-Klein metric and restrict the uniform structure to bundle maps f = (fE, fM)' Quite similar we define for Ei = ((Ei' hi, V'hi) ------t (Mr,gi)), i = 1,2, v p,r(E1 ,E2 ) by corresponding bundle isomorphisms and v r,p,r(E1 , E 2 ) = v p,r(E1 , E 2 ) n c r,p,r(E1 , E 2) c v p,r(E1 , E 2) as the C r+1 elements with r-bounded differential, i.e. for f E (fE, fM) E v r,p,r(E1 , E 2) there holds sup lV'idfMI :S d, sup lV'idfEI :S d. xEMl
eEEl
l~i~r
l~i~r
Quite analogously to fJJrI(mf, I, B k ) we denote the bundle isometry classes of Riemannian vector bundles (E, h, V') ------t (Mn,g) with (I), (Bk(g)), (Bk(V')) of rkN over n-manifolds by BN,n(I, Bk)' Set for k ~ r > ~ + 2, Ei = ((Ei' hi, V'hi) ------t (Mr, gi)) E BN,n(I, B k), i = 1,2 d~:diff(El' E 2 )
inf { max{O, log bldfEI}
+ +
+ max{O, log bldfE/I}
max{O, log bldfMI} + max{O, log bldfi/I} sup lV'idfMI + sup IV'idfE I xEM l
l~i$r
eEEl l~i$r
if { ... } i- 0 and inf{ ... } < 00. In the other case set 1 d~~iff(El' E 2) = 00. Here we remark that bldfEI, bldfE 1, bldfMI, bldfiil < 00 automatically imply the quasi isometry of hI, fih2 or gl, fl.Jg2, respectively. A simple consideration
Non-linear Sobolev Structures shows that d(E 1 , E 2 ) = 0 is an equivalence relation BZ'~ff(I, B k ) := BN,n(I, B k )/ rv and for 5 > 0 Vj
= {(Eb E2) E (BZ~ff(I, Bk))2
1
131
rv.
Set
d!i,~iff(El' E 2) < 5}.
Proposition 4.2 ~ = {Vj}8>0 is a basis for a metrizable uniform structure ll~,~if f . The proof is quite analogous to that of proposition 4.1 with
Kl = K2 = 0.
0
The set (4.17) contains some more terms as the set (4.1). The new terms are Ihl - fE h 21 9I,hI '\1 hI ,p,r and l\7 hI - fE \7 h2 19I ,hI '\1 hI ,p,r' For the symmetry we consider
Ih2 - f;;-lh I192,h2'\1h2,p,r
= If;;-IU;;h 2 - h 1)1 92,h2,'\1h2,p,r ~ b,rlf;;-11 ·If;;h2 - h 119I ,hI'\1hI,p,r ~ k1 (5) . 5 - - - t 0 8-.0
and
f;; \7 h2 ) 192,h2,'\1h2 ,p,r < b,rlf;;-11·I\7h l - f;;\7 h2 19j ,hI'\1hI,p,r < k2 (8)· 5 - - - t 0 If;;-1 (\7 hI
-
8-.0
The proof of the transitivity is completely parallel to (4.14) (4.16). Denote BZ~;hr(I, B k) for the pair (BN,n(I, Bk),ll~~diff) and BZ~rf(I, B k) for the completion. The next task would be to prove the locally arcwise connectedness of BZ~rr If we restrict to (E, h) - - - t (Mn, g), i.e. we forget the metric connection \7 k , then the corresponding space is locally arcwise connected according to 5.19 of [33]. Taking into account the metric connection \7\ the situation becomes much worse. Given (g, h, \7 h), (g', h', \7h') sufficiently neighboured, we have to prove that they could be connected by a (sufficiently short) arc {(gt, ht, \7 ht )}. Here \7ht must be metric w. r. t. ht .
132
Relative Index Theory, Determinants and Torsion
We were not able to construct the arc {V'ht h for given {hth. One could also try to set V't = (1 - t) V' + tV' and to construct h t from V't s. t. V't is metric w. r. t. ht. In local bases el, ... , en, 1 , ... , N this would lead to the system
= ri,iaht,"!(3 + ri,i(3ht,a"!, i = 1, ... ,n, a, (3 = 1, ... ,N, where h t,a(3 = ht(a, (3), V'~a = ri,ia"!. This is a system V'~iht,a(3
of n N(~+I) equations for the N(~H) components h a (3, i.e. it is overdetermined. With other words, we don't see a comparetively simple and natural proof for locally arc wise connectedness. B~~JF(I, B k) is a complete metric space. Hence locally and locally arcwise connectedness coincide. But to prove locally connectedness amounts very soon to similar questions just discussed. Consider for E = ((E, h, V'h) ---7 (M, g)) E BN,n(I, Bk) (4.18) The set is open and contains the arc component of E. If B~~JF(I, B k) would be locally arcwise connected = locally connected then we would have arccomp(E) = comp(E).
(4.19)
If we endow the total spaces E with the Kaluza-Klein metric gE(X, Y) = h(XV, yV) + gM(7r*X, 7r*Y) , Xv, yV vertical components,
then (E, gE) becomes a Riemannian manifold of bounded geometry, hence a proper metric space. It follows from the definition that {E' E B~~JF(I,Bk)
I d~~iff(E,E') < oo} ~ comPL(E).
(4.20) (4.18), (4.19) and the foregoing considerations are for us motivation enough to define the generalized component gen comp( E) by
gencomp~~diff(E)
:= {E' E
B~'~JF(I,Bk) I d~~diff(E,E') < oo}. (4.21)
Non-linear Sobolev Structures
133
In particular gencomp(E) is a subset of a Lipschitz component and is endowed with a well defined topology coming from ll~~diff' The next step in this section consists of the additional admission of compact topological perturbations, quite similar to the case above of manifolds. We consider pairs Ei = ((Ei' hi, \7 hi ) ----+ (Mi,9i)) E BN,n(I, B k ), i = 1,2, with the following property. There exist compact submanifolds Ki C Mi and f = (fE, fM) E cr,p,r(EI , E 2), flMl\Kl E vr,p,r(EIIMI\KI' E2IM2\K2)' For such pairs define d~~iff,rel(EI, E 2) = inf{ max{O, log bldfel}
+ max{O, log bldhEI} + max{O, log bldfMI} + max{O, log bldhMI} + sup d(hEfEel' el) el
+ sup d(fMhMX2, X2) + + eEEl sup
sup l'VidfMI xEMl
X2
l-:;i::;r
l'VidfEI
+
1
(fMI Ml\Kl)*92 - gIIMl\Kllgl,p,r
1-:; i::; r
+1(fEIEIMI\KI)*h2 - hIIElIMl\Kllgl,hl,V'hl,p,r +1 (fEIEIMI \KJ*'Vh2 - 'VhllEllMl \Kllgl,hl,V'hl ,p,r r 1 f = (fE,fM) E c ,p,r(EI ,E2), h = (hE, hM) E c r,p,r(E2, E I )}
bundle maps and for some KI C MI holds flEllMl\Kl E V;:z,r(EIIMI\KJl E2IfM(Ml\K l )) and
hI E2 If(Ml\Kl) = (fIEIMl\Kl)-I}
if { ... }
0 and inf { ... } <
(4.22)
In the other case set dt~iff,rel(EI' E 2) = 00. This definition seems to be quite lengthy but it is quite natural. It measures outside a compact set the distinction of "shape" and the geometric objects in question. Set =1=
00.
134
Relative Index Theory, Determinants and Torsion
Proposition 4.3 ~ = {1I8}c5>O is a basis for a metrizable uniform structure U~~iJJ,rel on BN,n(I, B k )/ where EI E2 if dt~iJJ,rel(EI' E 2) = o. f'V
f'V
The proof is completely parallel to that of 4.1 combined with that of 4.2. 0 Denote B~~rfrel(I, B k ) for the completed BN,n(I, B k ) endowed We have again that with this uniform structure. ~,~ifJ,rel(EI' E 2) < 00 implies ddEI, E 2) < 00 (here we consider E I , E2 as proper metric spaces). Hence E2 E comp L (EI ) , i.e. {E2 E
B~~rfrel(I, B k ) I d~~diJJ,rel(E1' E 2) < oo} ~
compL(EI ).
(4.23) For this reason we denote again the left hand side { ... } of (4.23) by gen comp~~iff,rel(EI) keeping in mind that this is not an arc component but a subset of a Lipschitz component endowed with the induced topology from Ut~iJ J,rel· In our later applications we prove and thereafter use the trace /2 class property of e- tD2 - e-tD . Here essentially enter estimates for D - D', coming from the explicit expression for D - D'. But in this expression only g, \7, ., g', \7', .' enter. This is the reason why we consider in some of our applications smaller generalized components, which are in fact arc components. Exactly speaking, we restrict in some of our applications to those uniform structures and components where hI = fEh2' i.e. the fibre metric does not vary. Nevertheless, the generalized components play the more important role as appropriate equivalence classes in classification theory. We prove the trace class property of tD/2 tD2 e_ ealso for generalized components.
135
Non-linear Sobolev Structures
Set now d~:diff,F(EI, E 2) = inf{ max{O, log bldfMI}
+max{O, logbldfMII} + sup l~idfMI xEMI l~i~T
f~ V h2 19I ,hI ,V'hl ,p,T f = (JE,jM) E VT,P,T(E I , E2)'
+IVhl I !E
-
fibrewise an isometry}
(4.24)
if { ... } =f 0 and inf{ ... } < 00. In the other case set dt~iff,F(EI, E2) = 00. d~:diff,F(-") = 0 is an equivalence relation rv. Set BZ~ff,F(I, B k ) = BN,n(I, B k )/ rv and for 5 > 0
Va = {(EI, E2) E (BZ~ff,F(I, Bk))2 I d~:diff,F(EI' E 2) < 5}. Proposition 4.4 ~ = {Va}c5>O is a basis for a metrizable uniform structure i1t~if f,F' 0 Denote Bz~r;>(I, B k) for the corresponding completion.
Proposition 4.5 a) BZ':lSfF(I, B k) is locally arcwise connected. b) In BZ~rfF(I, B k ) components coincide with arc components.
c) BZ~rfF(I, B k) =
2: comp~:diff,F(Ei)
as topological sum.
iE!
d) For E
E
BN,n
gen comp~:dif f,F (E)
{ E' E BN,n,p,T L,diff,F (I , B) k ~:diff,F(E, E') < oo}.
I
Proof. a) gt = (1 - t)gl + tf'Mg2, V t = (1 - t)V hl + tfE V h 2 yield an arc between EI and 1* E 2. Here we use hI = fEh2 and that V hl , V h 2 are metric. 0
136
Relative Index Theory, Determinants and Torsion
Quite analogously we define - based on hlIEIM\K · form space BN,n,p,r e um L,diff,F,rel (1 , B k ) an d·t 1 s f E* (h 2 IE'IM'\K' ) - t h generalized components gen comp~:dif f,F,rel (E) = {E'
I d~:diff,F,rel (E, E') < oo}.
Here dt~iff,F,rel(E, E') is defined as d~~iJf,F,rel,·) with the additional condition hIEIM\K = fE(h'IE,IM'\K'. Bf,';;J}>,rel is not locally arcwise connected. Now the classification of BN,n(I, B k ) amounts to two tasks. 1) Classification (i. e. "counting") the (generalized) components gen comp( E) by invariants, 2) Classification of the elements inside a component by invariants, where number valued invariants should be relative invariants. Until now g, h, \lh could be fixed independently, keeping in mind that \lh should be a metric connection with respect to h. The situation rapidly changes if we restrict to Clifford bundles. The new ingredient is the Clifford multiplication . which relates g, h, \lh. As we know from the definition, a Clifford bundle (E, h, \lh, .) ----+ (Mn, g) has as additional ingredient the module structure of Em over CLm(g) = CL(TmM, gm). A change of g, 9 ----+ g', changes point by point the Clifford algebra, CLm (g) ----+ CLm (g'). Locally they are isomorphic since by radial parallel transport of orthonormal bases in a normal neighbourhood U(mo) always CL(M, g)IU(mo) ~ U(mo) x CL(JRn ) ~ U(mo) x CL(M, g')IU(mo). (4.25) The same holds for bundles of Clifford modules if we fix the typical fibre, i. e. (4.26) Elu ~ E'lu but globally (4.26) in general does not hold although as vector bundles E and E' can be isomorphic. The point is the module structure which includes 9 (in CLm(g)) as operating algebra and
Non-linear Sobolev Structures
. : TmM ® Em
137
Em. Therefore for a moment we consider the following admitted deformations. Let (E, h, \lh, .) ----t (Mn, g) ----t
be a Clifford bundle of rkN. The vector bundle structure E ----t M (of rkN) shall remain fixed. We admit variation of g, hence of CL(T M), variation of . E Hom (T M ® E, E), hence variation of the structure of E as bundle of Clifford modules, compatible variation of h, \lh. Hom (T M ® E, E) ~ T* M ® E* ® E is for given g, h a Riemannian vector bundle. Including \lh, the notion of Sobolev sections is well defined. For fixed g, h, \lh the space r(mult, g, h, \lh) of Clifford multiplications· is a well defined subspace of r(Hom (T M ® E, E)) described invariantly by the conditions
(X.
-(
(4.27) (4.28)
where here (.,.) = h(·, .). We describe this space locally as follows. Let U(ma) C M and el(m), ... , en(m) be a field of orthonormal bases obtained from el (ma), ... ,en(ma) by radial parallel transport, similarly
-(
= 1, ... , N,
(\leiej) .
(4.29) (4.30)
If we write in the linear space Hom (Tm ® Em, Em) ei .
L
afj
k=1
equations between the afj' a~j - -aik' Hence the fibre multm(g, h, \lh) is an n· N 2 - nN(~+l) = nf/ (N -I)-dimensional affine subspace at any point m. This establishes a locally trivial fibre bundle mult(g, h, \lh). The charts for local trivialization arise from radial parallel transport P of . from m to ma since \l JL (P(ei .
138
Relative Index Theory, Determinants and Torsion
r(mult(g, h, ',\h)) now are those sections ofmult(g, h, V'h) which additionally satisfy (4.28) or (4.30). Consider the Riemannian vector bundle (1i = Hom (T M ® E, E) = (T* M ® E* ® E, h1t ) ---t (Mn, g)). Assume as always (Mn, g) with (1), (B k ), (E, h, V'h) with (B k ), k ~ ~ + 2 and (7r : E ---t M) E coo,k+l(E, M). Then with respect to the Kaluza-Klein metric g1t(X, Y) = h1t (X V , y V )+gM(7r*X, 7r*Y) , Xv, yV vertical components, the total space of 1i becomes a Riemannian manifold satisfying (1), (Bk)' The fibres 1im are totally geodesic submanifolds, moreover they are flat. The latter also holds for the affine fibres multm(g, h, V'h) of mult(g, h, V'h). If· and.' are sections of mult(g, h, V'h) satisfying (4.28), i. e. ',.' E CLM(g, h, V'h) then (1 - t) . +t·' E CLM(g, h, V'h), (4.31) Let k ~ r > ~
+ 2, 8 > O.
Vs = {(" .')
E
Set
CLM(g, h, V'h)2
1 I· - .' Ig,h,y>h,p,r < 8}.
Lemma 4.6 ~ = {Vs}.,>o is a basis for a metrizable uniform structure up,r(CLM(g, h, V'h)).
o Denote by CLMP,r(g, h, V'h) the completion. Proposition 4.7 a) CLMP,r(g, h, V'h) is locally arcw~se connected. b) In CLMP,r(g, h, V'h) components and arc components coincide. c) CLMP,r(g, h, V'h) = L compP,rh). iEI
d) compp,rO = {.'
1
,. -
.'
Ig,h,Y'h,p,r < oo}.
Proof. a) follows from (4.31), b) and c) follow from a), d) is a 0 simple calculation.
Non-linear Sobolev Structures
139
Remark 4.8 In the language of the intrinsic Riemannian geometry ofmult(g, h, V'h) and of r(mult(g, h, V'h)) we can rewrite I . - .' Ig,h,'Vh,p,r < 8 as .' = exp X 0 " X E r(T(mult(g, h, V'h))), IXlg,h,V'h,p,r < 8. Here Xm lies in the affine subspace mult m . 0
Denote by CLBN,n(I, B k ) the set of (Clifford isometry classes) of all Clifford bundles (E, h, V'h,.) ----t (Mn, g) of (module) rank N over n-manifolds, all with (I) and (Bk)' Lemma 4.9 Let Ei = ((Ei' hi, V'hi, 'i) ----t (Mt,gi)) E CLBN,n(I, B k), i = 1,2 and f = (fE, fM) E v p,r+1(EI, E 2) n c oo ,k+1(EI , E 2) be a vector bundle isomorphism between bundles of Clifford modules satisfying fE(X' l . Then 1*E2 := ((EI,fEh2,fEV'h2,IE'2) ----t (M1,fMg2)) E CLBN,n (I, Bk)'
Proof. The definitions of fEh2' fE V'h2, 1M g2 are clear. IE'2 is defined by X(fE'2) = f E 1(f*X '2 JE ~
+ 2 and define for
E 1, E2
E
CLBN,n(I, B k )
d!i~iff(E1' E 2) = inf{max{O, logbldfEI}
+ max{O, logbldfEII} + max{O, log bldfMI} + max{O, log bldf.M11} + sup IV'idfMI mEMl l~i:5r
+lg1 - fMg2Ig1,p,r
+ Ihi -
I Eh2Ig1 ,hl,'Vhl,p,r
fEV'h2Ig1,hl,V'hl,p,r + 1'1 - fE '2I g1,hl,'Vhl,p,r f = (fE, fM) E v r,p,r(E1, E 2) is a (r + 1) - bounded isomorphism of Clifford bundles}
+IV'hl -
::I 0
and inf{ ... } < 00. In the other case set d!i~diff(EI' E 2) = 00. dj;~diff is numerically not symmetric but nevertheless it defines a uniform structure which is by definition symmetric. Set for 8 > if { ... }
°
v., = {(El' E 2) E CLBN,n(I, Bk))2} I
dj;~diff(E1' E 2) < 8}.
140
Relative Index Theory, Determinants and Torsion
Proposition 4.10 ~ = {Vc5 }c5>O is a basis for a metrizable uni0 form structure U~~dif f (CLBN,n (I, B k )) . Denote CLB~~ff,r(I, B k) for the pair (CLBN,n(I, Bk),up,r) and CLB~~fF (I, B k) for the completion. By the same motivation as above we introduce again the generalized component gen comp(E) = gencomp~~iJj((E, h, V'h) -----+ (M, g)) c
CLB~~fF(I, B k ) by gencomp~~iff(E) =
CLB~~fF(I, B k) dj;,~iff(E, E') < oo}. {E'
E
I (4.32)
gencomp(E) contains arccomp(E) and is endowed with a Sobolev topology induced from Uf:~iJj. The absolutely last step in our uniform structures approach is the additional admission of compact topological perturbations. We proceed assuming additionally Ei = ((Ei' hi, V'h;, ·i) -----+ (Mr, gi)) E CLBN,n(I, B k ), adding still I(fEIEIM \K)* ·2 - ·1 IEIMl\KlI91,hl,V'hl,p,r and assuming f = (fE,fM)IMl\Kll h = (hE, hM ) IM2\K2=fM (Ml\Kl) vector bundle isomorphisms (not necessary Clifford isometries). Then we get d~~diff,rel(EI, E 2), define v", ~ = {Vc5}c5>O, obtain the metrizable uniform structure U~~diJj,rel(CLBN,n(I, B k )) and finally the completion
N,n,p,r 1XT t . CLB L,diff,rel. vve se agam gen comp (E)
gen comp~~dif f,rel (E)
CLB~~fFrel(I, Bk)) dt~iJj,rel(E, E') < oo}
= {E'
E
which contains the arc component and inherits a Sobolev topol(IP,r ogy f rom J.AL,diJj,rel. As in the preceding considerations we obtain by requiring additionally hI = fEh2 or hllEllMl\Kl = fE(h2IE2IM2\K) local distances d~~iff,F(-'·) or d~~diff,F,rel,·) and corresponding uniform spaces CLBf'~fFF(I, B k ) or CLBf~f:/F,rel(I, B k ) respectively. We obtain generalized components gen comp~~if f,F (E)
(4.33)
141
Non-linear Sobolev Structures
and (4.34) gen COmp~:diJ J,F,rel (E) as before. One of our main technical results in chapter IV will be that E and E' being in the same generalized component tD,2 into the Hilbert space implies that after transforming etD,2 tD,2 tD2 tD2 L2((M, E), g, h), e- _eand e- D_eD' are of trace class and their trace norm is uniformly bounded on compact tIntervalls lao, ad, ao > O. For our later applications the components (4.33), (4.34) are most important, excluded one case, the case D2 = .6. (g) , D,2 = \1(g'). In this case variation of 9 automatically induces variation of the fibre metric and we have to consider (4.32) and gen comp~:diff,rel (E). Perhaps, for the reader the definitions for the gen comp(E) look very involved. We recall, roughly speaking, the main points are as follows. The distance which defines gen comp ... (E) measures step by step the distance between the main ingredients of a Clifford bundle: the smooth Lipschitz distance between the diffeomorphic parts of the manifolds and the bundles and the Sobolev distance between the manifold metrics, the fibre metrics, the fibre connections and the Clifford multiplications. Remark 4.11 The gen comPL',diJ J,rez(- )-definition extended canonically to structures with boundary.
can
be D
5 The classification problem, new (CO-) homologies and relative characteristic numbers As we already indicated, we understand this treatise as a contribution to the classification problem for open manifolds. We proved in chapter I that meaningful number valued invariants for all open manifold do not exist. The way out from this situation is to introduce relative number valued invariants or to give up the claim for number valued invariants and to admit group valued invariants as e.g. in classical algebraic topology. We go
142
Relative Index Theory, Determinants and Torsion
both ways. The heart of this treatise are new number valued relative invariants like relative determinants, relative analytic torsion, relative eta invariants, relative indices. This will be the content of chapters IV - VI. Our general approach consists in two steps, 1. to decompose the class of manifolds/bundles under consideration into generalized components and to try to "count", to "classify" them, 2. to "count", to "classify" the elements inside a generalized component. Chapters IV - VI are exclusively devoted to the second step. Concerning the first step, we developed in [34] some new (co-) homologies which are invariants of the corresponding generalized component and hence represent steps within the first task above. In this section, we give a brief review of these (co-) homologies. In the second part, we give an outline of bordism theory for open manifolds and corresponding relative characteristic numbers. Let X and Y be proper metric spaces. We call a map : X coarse if it is
-+
Y
1. metrically proper, i.e. for each bounded subset Bey the inverse image -l(B) is bounded in X, and 2. uniformly expansive, i.e. for R > 0 there is S > 0 such that d(Xl,X2) ~ R implies d(Xl,X2)::; S. A coarse map is called rough if it is additionally uniformly metrically proper. X and Yare called coarsely or roughly equivalent if there exist coarse or rough maps : X -+ Y, 'It : Y -+ X, respectively, such that there exist constants D x, Dy satisfying
d('ltx, x) ~ D x ,
d('lty, y) ~ D y
.
Proposition 5.1 X and Yare coarsely equivalent if and only if they are roughly equivalent. We refer to [33] for the proof.
o
Non-linear Sobolev Structures
143
The equivalence class of X under coarse equivalence is called the coarse type of X. Let X be a proper metric space. Then we have sequences of inclusions (5.1) coarse type (X) :J compCH(X), coarse type (X) :J comPCH(X) :J arccompL,h,rel(X) :J :J arccompL,h(X) :J compL,top(X),
(5.2)
coarse type (X) :J compL(X) :J arccompL,h,rel(X) :J :J arccompL,top,rel(X) :J compL,top(X),
(5.3)
The arising task is to define for any sequence of inclusions invariants depending only on the component and becoming sharper and sharper if we move from the left to the right. Start with the coarse type which has been extensively studied by J. Roe. Given X = (X, d), xq+1 becomes a proper metric space by d((xo,,,.,xq), (Yo,,,.,Yq)) = max{d(xo,Yo),,,.,d(xq,Yq)}' Let ~ = ~q C xq+1 be the multidiagonal and set Pen(~,R)
= {y
E xq+lld(~,y):s
R}.
Then J. Roe defines in [63] the coarse complex (CX*(X), 8) = (CXq(X),8)q by
cxq(X) := {J: xq+1
I
f is locally bounded Borel function and for each R > 0 is supp f n Pen(~, R) relatively compact in Xq+l}, ---t
IR
q+l
8f(xo, ... ,Xq+1) := 2)-1)if(xo, ... ,Xi,'" ,Xq+l)
(5.4)
i=O
The coarse cohomology HX*(X) of X is then defined as
HX*(X) := H*(CX*(X)). Theorem 5.2 H X*(X) is an invariant of the coarse type, i.e.
coarse equivalences phisms.
:
X
---t
Y, \II : Y
---t
X induce isomor-
144
Relative Index Theory, Determinants and Torsion
o
We refer to [63] for a proof.
Corollary 5.3 H X* (X) is an invariant for all components right from the coarse type. Remark 5.4 It is well known that without the support condition supp f n Pen(b., R) relatively compact (5.5) the complex GX*(X) would be contractible. After fixing a base point x E X the map D : Gq ~ Gq-1,
Df(xl, ... ,Xq):= f(X,X1, ... ,xq) would be a contracting homotopy.
(5.6)
o
It is now possible to define in a canonical way a cohomology theory which is an invariant of comp L (.). One only has to choose the" right category". Let
Gt(X) = {f : Xq+1 ~ IR I f is Lipschitz continuous and supp f n Pen(b., R) is relatively compact for all R}. (5.7) Then, with 0 from (3.4), GL(X) = (Gt(X),O)q is a complex and we define
H'L(X) := H*(G'L(X)). If : X ~ Y is (u.p.) Lipschitz then induces t : Gt(Y) ~ GUX) by (t(X)f)(xo, ... , Xq) := f(xo, ... , Xq), f E Gt(Y), and 'L : HL(Y) ~ H'L(X).
Using Roe's anti Cech systems and uniqueness of the cohomology of uniform resolutions by appropriate sheafs as in [63], one easily obtains
Theorem 5.5 If Y E comPL(X) then there exist : X ~ Y, W : Y ~ X wich induce inverse to each other isomorphisms
w* 'blH'L(X) ~ H'L(Y). *L
Non-linear Sobolev Structures
145
But this approach is very unsatisfactory since we did in fact not define a really new invariant but the categorial restriction of a coarse invariant. The situation rapidly changes if we factorize or impose decay conditions. Let C1(X) as above, bC1(X) the subspace of bounded functions in C1(X) and CL(X) = C1(X)j bC1(X). Then 8 maps bC1(X) into bC1+ 1 (X), i.~. bCL is a sub complex and we obtain a complex CL b(X) = (C1 b(X), 8)q. Define "
HL,b(X) := H*(CL,b(X)), Any (u.p.) Lipschitz map : X ---t Y induces # : bCL(Y) bCL(X), hence # : bCL,b(Y) ---t bCL,b(X) and * : HL,b(Y)
---t
---t
HL,b(X),
Theorem 5.6 HL,b(X) is an invariant of comPL(X), Let Y E comPL(X), ddX, Y) < E, : X ---t Y, W : X, d(wx,x) < E, d(wy,y) < E and let [J] E HL(X). Then (wx) f(x)1 ~ C· d(wx, x) < C . E, i.e. (ww)*[g] = [g] for [g] E Hl,b(Y)' * and w* are inverse to each other isomorphisms.
Proof.
Y
---t
D
Remark 5.7 For compact X, HLb(X) = 0 as it should be. D , Let Z be a proper metric space, f : Z ---t IR Borel and locally bounded. We say f is uniformly locally bounded if for every D > 0 their exists 8 = 8f (D) > 0 S.t. If(z) - f(z')1 < 8 for all z' E BD(z), 8 independent of z. Let C~lb(X) C cxq(X) the subspace of all u.l.b. functions f E cxq(X). 8 maps C~lb(X) into C~~l(X). We obtain hence a complex CeJH(X) = (C6H(X) = cxq(X)jC~lb(X), 8)q and define
HCJH(X)
:=
H*(CcJH(X)),
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Relative Index Theory, Determinants and Torsion
If : X ~ Y is Borel and metrically semilinear then induces # : cxq(y) ~ cxq(X), # : C~lb(Y) ~ C~lb(X) (since I(# J)(x) - (# J)(x')1 = If(x) - f(x ' ) 1< 6j(D+Ccp), d(x, x') ~ d(x, x') + Ccp, x = (xo, ... , Xq)), # : CCJH(Y) ~ CCJH(X) and
Theorem 5.B HCJH(X) is an invariant of comPCH(X), Proof.
The proof is completely parallel to that of 5.6, Y E ~ X Borel
compCH(X), dCH(X, Y) < c, : X ~ Y, 1J! : Y and metrically semilinear, for [f] E HbH(X) (1J!
since If(Wx) - f(x)1 ~ 6j(d cH (X, Y)).
o
Remark 5.9 For X compact, HCH(X) = O.
o
Next we consider compL,h,rel(X), Denote by C*(X) = C*(S(X)) the finite singular chains and by C*,b,ulj(X) the complex of all uniformly locally finite bounded singular chains. Then C*(X) c C*,b,ulj is a sub complex and we define
and
H*,b,ulj,oo(X) = H*(C*,b,ulj,oo(X)), Similarly let C*,b,ulj (X) be the complex of uniformly locally finite bounded singular co chains , C~(X) the sub complex of co chains with compact support,
C*,b,ulj,oo (X) = C*,b,ulj (X) / C~ (X) and
H*,b,ulj,oo(X) = H*(C*,b,ulj,oo(X)).
147
Non-linear Sobolev Structures
Theorem 5.10 If Y E arccompL,h,rel(X) then there exist uniformly proper Lipschitz maps <1> : X - - t Y, W : Y - - t X which induce inverse to each other isomorphisms
~ H*,b,ulf,oo(X) ~ H*,b,ulf,oo(Y)
and H*,b,ulf,oo(y)
~
H*,b,ulf,oo(X).
w*
Proof. By assumption there exists u.p. Lipschitz maps <1> : X - - t Y, W : Y - - t X and compact subsets Kx eX, K y c Y S.t. <1>IX\Kx : X\K x - - t Y\Ky , WIY\Ky : Y\Ky - - t X\K x are inverse to each other Lipschitz homotopy equivalences. It follows immediately from the definition that <1>, W induce chain maps <1># : C*,b,ulf(X)
--t
C*,b,ulf(Y),
<1># : C*(X)
--t
C*(Y),
w# : C*,b,ulf(Y)
--t
C*,b,ulf(X),
w# : C*(Y)
--t
C*(X),
<1># : C*,b,ulf,oo(X)
--t
C*,b,ulf,oo(Y),
w# : C*,b,ulf,oo(Y)
--t
C*,b,ulf,oo(X),
and hence morphisms <1>* : H*,b,ulf,oo(X)
--t
H*,b,ulf,oo(Y),
w* : H*,b,ulf,oo(Y)
--t
H*,b,ulf,oo(X),
Any singular chain c E Cq,b,ulf(X) can be decomposed as C = Coo + CK x ' where supp Coo C X \ Pen(Kx , R), R = R(c). In particular every homology class [z] E Hq,b,ulf,oo(X) has a representation Zoo + Bq,b,ulf,oo(X), 8zoo E Cq_1(X). The refined prism construction in [6], p. 512 for uniformly proper Lipschitz homotopy equivalences now shows that W#<1># + Bq,b,ulf,oo = Zoo + Bq,b,ulf,oo,
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Relative Index Theory, Determinants and Torsion
similarly for #w#. Hence w** = id, *w* = id. Similarly for cohomology.
o Remark 5.11 It is also clear that , W induce isomorphisms of the usual end (co-) homology.
o Omitting the factorization by C*(X) or C;(X), respectively, we obtain the uniformly locally finite bounded homology H*,b,ulf(X) or cohomology H*,b,ulf (X), respectively. Theorem 5.12 If Y E arccomPL,h(X) then there exist u.p. Lipschitz maps : X - - - t Y, W : Y - - - t X which induce inverse to each other isomorphisms
~ H*,b,ulf(X) ~ H*,b,ulf(Y)
and H*,b,ulf(y)
~
H*,b,ulf(X).
w*
o Remark 5.13 It is completely clear that the classical homotopy invariants like singular (co-) homology etc. are invariants of arccomp L,h (.) too. 0
We can generalize 5.10 to 1 :::; p < Denote
00.
Cq,b,ulf(N) := {c E Cq,b,ulf I For all x E X and R > 0 is (#{ singular simplexes (Jq of c I supp (Jq c BR(x)})/R:::; N}.
149
Non-linear Sobolev Structures
Roughly speaking for all singular chains E Cq,b,ulf(N) simultanously holds that every metric ball of radius R contains at most R . N singular simplexes. From the definition Cq,b,ulf;;2 lim Cq,b,ulf(N) ----t
=
UCq,b,ulf(N). N
N
Set Cq,p,ulf(N)
{c = L C~(J E Cq,b,ulf(N)
I
~q
(Cq,p,ulf(N) , lip) is a normed space (nonseparable) and we have ... ~ Cq,p,ulf(N) ~ Cq,p,ulf(N + 1) ~ .... Denote Cq,p,ulf( (0) = lim Cq,p,ulf (N) with the inductive limit topol---->
ogy. Then 8 : Cq,p,ulf( (0) ----t Cq-1,p,ulf( (0) is continuous since 8 : Cq,p,ulf(N) ----t Cq-1,p,ulf(N) is norm-continuous. We obtain Hq,p,ulf( 00 )(X),
H q,p,ulf( 00 )(X),
Hq,p,ulf(oo)(X),
Hq,p,ulf(oo)(X),
(5.8)
where H denotes the reduced (co ) homology.
Theorem 5.14 The (co-)homologies of (5.8) are invariants of arccompL,h(')'
D
Corollary 5.15 (a) H*,b,ulf,oo and H*,b,ulf,oo are invariants of arccomp L,top,rel (.). (b) H*,b,ulf, H*,b,u1f and the (co-)homologies of (5.8) are invariants of compL,top(-)' D
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Relative Index Theory, Determinants and Torsion
The proof of 5.14, 5.15 follows from the fact that the admitted maps induce chain maps and chain homotopy equivalences between the corresponding complexes. There are many other classes of invariants which we did not consider explicit ely until now. These include the K -theory of C*-algebras, K*( C* X), and Kasparovs K-homology for locally compact spaces, K*X. We conclude this section with a brief review of bordism for open manifolds and relative characteristic numbers. We consider as before oriented open manifolds (Mn , g) satisfying
and
(1) (Bn+! , 9B) is a bordism between (Mr, gl) and (M2', g2) if it satisfies the following conditions.
1) (8B, gBI8B) ~ (Ml' gl) U (-M2' g2), 2) there exists <5 > 0 such that gBl u6(8B) ~ g8B + dt 2, 3) (B, gB) satisfies (B k ) and inf rinj(gB, x) > 0, XEB\U6(8B) 4) there exists R> 0 such that B c UR (M1 ), Be UR (M2 ). We denote (Ml' gl) (M2' g2). (Bn+!, gB) is called a bordism. f"V
b
Sometimes we denote additionally
f"V,
b,b g
bg stands for bounded
geometry, i.e. (1) and (Bk).
Lemma 5.16 a)
f"V
b
is an equivalence relation. Denote by [Mn, g]
the bordism class. b) [MUM',gUg'] = [M#M',g#g']. c) Set [M,g] + [M',g']:= [MUM',gUg'j = [M#M',g#g'j. Then + is well defined and the set of all [Mn, 9 j becomes an abelian semigroup. D Denote by n~c = n~c(1, B k ) the corresponding Grothendieck group. Similarly one defines n~C(X) generated by pairs ( (Mn, g), f : Mn ~ X), f bounded and uniformly proper.
151
Non-linear Sobolev Structures
Remarks 5.17 1) Condition 4) above looks like dCH(M, M') ::; R, where dCH is the Gromov-Hausdorff distance. But this is wrong. 2) There is no chance to calculate n~c. 3) One would like to have a geometric representative for and for -[M, g]. 0
°
The way out from this is to establish bordism theory for special classes of open manifolds or/and further restrictions to bordism. Our first example is bordism with compact support. Here condition 1) above remains but one replaces 2) - 4) by the condition There exists a compact submanifold C n +! C B n +1 such that (B \
c, gB\d
(B \
c, gBIB\C)
is a product bordism, i.e.
~ (M \ C x [0,1]' gM\C
+ dt 2 ).
(cs)
Then one gets a bordism group n~c (cs) (= b,cs Grothendieck group). At the first glance, the calculation of n~C( cs) or at least the characterization of the bordism classes seems to be very difficult. But we will see, that this is not the case. For this, we introduce still some uniform, structures. Denote by mn(mJ) := mn(mJ, nc) C mL the set of isometry classes of complete, open, oriented Riemannian manifolds. Consider pairs (Mf, gl), (M2,g2) E mn(mf) with the following property: We write
f'V.
There exist compact submanifolds Kr
c
Mr and K~
and an isometry Ml \ Kl ~ M2 \ K 2 .
c
M~
(5.9)
For such pairs, we define in analogy to sections 2 and 4
bdL,iso,rez((M1,gl), (M2,g2» := inf{max{O, logbldJI} + max{O, logbldhl} + sup dist(x, hJx) + sup dist(y, Jhy) I XEMI
yE M 2
J E COO(Ml' M 2), 9 E COO (M2' M1), and for some Kl C K,JIMl\Kl is an isometry and 9If(Ml\K = J-l}.
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Relative Index Theory, Determinants and Torsion
If (MI, 91) and (M2,92) do not satisfy (5.9), then we define bd L,iso,rel((M1,91), (M2,92)) = 00. We have bdL,iso,rel((M1,91), (M2,92)) = 0 if (M1, 91) and (M2,92) are isometric. Remarks 5.18 1) The notions Riemannian isometry and distance isometry coincide for Riemannian manifolds. FUrthermore, if ! is an isometry!, then we have bld!1 = l. 2) Any! that occurs in the definition of dL,iso,rel is automatically an element of C'X),m(M1, M 2) for all m. The same holds true for 9.
0
We write 9J1Lisorel(m!) = mn(mJ)j ",where (M1,91) '" (M2,92) if bd L,iso,rel((M1: 91), (M2, 92)) = O. Set
Va = {((M1,91), (M2,92)) E (9J1)'i,iso,rel(m!))2 bd L,iso,rel((M1, 9d, (M2, 92)) < 6}.
I
Proposition 5.19 .c = {Va}o>o is a basis for a metrizable uni0 form structure bUL,iso,rel. Denote by b9J1L ,iso ,rei (m J) the corresponding uniform space.
Proposition 5.20 If rinj(Mi , 9i) = ri > 0, r = min{rI, r2} and bd L,iso,rel((M1,91), (M2,92)) < r then Ml and M2 are (uniformly 0 proper) bi-Lipschitz homotopy equivalent. Corollary 5.21 If we restrict ourselves to open manifolds with injectivity radius 2': r, then manifolds (M1,91) and (M2,92) with bdL,iso,rel-distance less than r are automatically (uniformly proper) bi-Lipschitz homotopy equivalent. 0 Remark 5.22 If (MI, 91) satisfies (1) or (1) and (B k ) and bdL,iso,rel(M1, 91)' (M2, 92)) < 00 then (M2,92) also satifies (1) or (I) and (Bk). 0
Non-linear Sobolev Structures
153
We cannot show that b9J1L,iso,rel is locally arcwise connected, that components are arc components and bcompLisorel(M,g) = {(M',g')lbdL,iso,rel((M,g), (M',g')) < oo} is wrong. 'The reason is that we cannot connect non-homotopy-equivalent manifolds by a continuous family of manifolds. A parametrization of nontrivial surgery always contains bifurcation levels where we leave the category of manifolds. A very simple case comes from corollary 5.21.
Corollary 5.23 If we restrict bUL,iso,rel to open manifolds with injectivity radius ~ r > 0, then the manifolds in each arc component of this subspace are bi-Lipschitz homotopy equivalent. Proof. This subspace is locally arcwise connected and components are arc components. Consider an (arc) component and two elements (M1,gl) and (M2,g2) of it, connect them by an arc, cover this arc by sufficiently small balls, and apply 5.21. 0 By definition, we have
bd L,iso,rel((M1,91), (M2,92)) < 00 => dL((M1,9d, (M2,92)) < 00, where dL is the Lipschitz distance of section 2. Hence, (M2, g2) E comPL(M1,gl), i. e. {(M2,g2) E mn (mfWd L,iso,rel((M1,gl), (M2,g2)) < oo} <;;;; compL(M1, gl). (5.10) For this reason, we denote the left hand side { ... } of (5.10) by gen bcompL,iso,rel(M1 ,gl) = {... } = {... } n compL(M1 ,gl), keeping in mind that this is not an arc component, but a subset of (manifolds in) a Lipschitz arc component. If we fix (M1 , gl), then in a special case, we have a good overview of the elements in gen bcompL,iso,rel(M1 , gl).
Example 5.24 Let (M1 , gd (IRn, gstandard). Then gen bcompLiSorel(M1 ,gl) is in a i-i-correspondence with {(Mn,g)IMn'is a closed manifold and 9 is fiat in an annulus contained in a disk neighbourhood of a point }. 0
Relative Index Theory, Determinants and Torsion
154
This can be generalized as follows.
Theorem 5.25 Any componentgenbcompL,iso,rel(M, g) contains at most countably many diffeomorphism types. Proof. Fix (M, g) E genbcompL,iSo,rel(M, g) and an exhaustion K1 C K2 C ... , UK1 = M, of M by compact submanifolds, and let (M', g') E bcompL,iSo,rel(M, g). Then there are K' c M' and Ki C M such that M \ Ki and M' \ K' are isometric. The diffeomorphism type of M' is completely determined by that of the pair (K1 U K',K1 ), but the set of types of such pairs aKl~aKI
(after fixing M and Kl C K2 C ... ) is at most countable.
0
Thus, after fixing (M, g), the diffeomorphism classification of the elements in bcomp L,iso ,rei (M, g) seems to be reduced to a "handy" countable discrete problem. This is in fact the case in a sense which is parallel to the classification of compact manifolds. Now we connect the calculation of O~C(cs) with the generalized components genbcompL,iso,rel) C b9Jrl,iso,rel(mJ).
Remark 5.26 If (M1,gl), (M2,g2) E genbcompL,iso,rel(M,g), then, in general, (M1,gl)#(M2,g2) ~ genbcompL ,isorel(M, g). ,
o
Let O~c(cs, genbcompL,iso,rel(M, g)) C O~c(cs) be the subgroup generated by
{[M', g']csl(M', g') E genbcompL,iSO,rel(M, g)}. We
know
O~c(cs)
completely
if
we
know
all
O~c (cs, genbcomp L,iso ,rel (M, g)), and we know
O~c(Cs,b comPL ,iso,rel(M, g))
completely if we know a corresponding generating set. However, the elements of such a set are completely determined by their (relative) characteristic numbers.
Non-linear Sobolev Structures
155
Fix (Mn,g) E genbcompL isorel(Mn, g), where M is oriented. Assume that (M1 ,gl) E genbcompL,iSO,rel(Mn, g), and let : M\K - - - t Ml \K1 be an orientation preserving isometry. Define the (relative) Stiefel-Whitney numbers of the pair (Ml, M) by
wr
1
...
W~n(Ml'
M) := (w? ... w~n, [KID
+ (w? ... w~n, [KJ).
(5.11) Similarly, for (Ml' M) and n = 4k, we define the (relative) Pontrjagin numbers PIrl .. ,Pkrk(M1,
M)'.-
J
rl PI" ,Pkrk(M) 1
Kl
-
J
TI PI" ,Pkrk(M)
K
(5.12) and the (relative) signature by (5.13) Lemma 5.27 The numbers w? .. W~n(Ml' M), p~l ... p~k (Ml' M), and cr(Ml' M) are well defined, and we have (W~l
... W~n(Kl UK), [Kl
U
KJ), (5.14)
(p? .. . p~k(Kl U -K), [Kl U -KJ), (5.15) and
(5.16) Proof. The equations (5.11), (5.12) are clear. (5.13) comes from Novikov additivity of cr. Hence we have only to show the well definedness, i.e. the independence of the choice of K eM, Kl c MI' Start with (5.12). If K~ ::) K 1 , K' ::) K, IM\K' : M \ K' ~ Ml \ K~ orientation preserving isometric, then
r·-r~ J+J-j~'\KJ. r). +
Ki
K'
Ki\Kl
Kl
K
(5.17)
Relative Index Theory, Determinants and Torsion
156
But
J ... - J ... = °since K~ \ X
1
K~\KI
and K' \
X are iso-
K'\K
metric under by assumption. The analogous conclusion can o /I
a 1/
be done for Kr c K 1 , K" c K, : M\K - - t Ml \Kl already an isometry. In the general case K 1, K~, K, K' one considers K~ nKl , KnK' and reduces to the first two considerations after smoothing out K~ n K l , K' n K by arbitrarily small perturbations. The prooffor (5.11) is quite similar replacing integrations in (5.17) by application of co cycles to cycles. The independence of (5.13) comes again from Novikov additivity, applying it several times. 0
Theorem 5.28 Fix (Ml' gl), (M2' g2) E genbcompL,iso,rel(M, g). Then (Ml' gl) (M2' g2) if and only if all characteristic numf"V
b,cs
bers of (Ml' M) coincide with the corresponding characteristic numbers of (M2' M). (M2' g2). Choose C n+l C Bn+l large enough such that with oC n+l = OlC U 02C U 03C, OlC = Kl C M l , 02C = K2 C M 2, 03C ~ oKl X [0,1] there holds Ml \X l ~ M\X, M2 \X 2 ~ M\X. Then after smoothing out
Proof.
Assume (Ml' gl)
PI
f"V
b,cs
P2
by arbitrary small perturbations, oC is diffeomorphic ot Kl U P
= 0(2l1). Hence char.n.(Kl U -K2) = char.n.(Kl U P PI -K)+char.n.(K U -K2) = char.n.(Ml , M)+char.n.(M, M 2) =
-K2'
p;-I
char.n.(Ml , M) - char.n.(M2, M), i.e. char.n.(Ml , M) = char.n. (M2' M). Conversely, if char.n.(Ml , M) = char.n.(M2, M) then char.n.(Kl U -K2) = 0, Kl U -K2 is O-bordant, Kl U -K2 = P P P ocn+l. Form Kl U (OKI X [0,1]) U - K 2) which equals to oCn+l , P glue (Ml \ Xl) x [0, 1] ~ (M \ X) x [0, 1] ~ (M2 \ X2) x [0,1] and smooth out (the topology and the metrics). The result is a
Non-linear Sobolev Structures
157
bordism (Bn+l, gB) with compact support between (M1 , g) and
(M2 ,g).
D
Corollary 5.29 Description of all elements of n~e(cs) reduces
to "counting" the generalized components of bfJJrL ,iso ,rei (mJ).
D
= (Mlu8MI x [0,00[, gi), i = 1,2, MI compact, 8M~ = 8M~, (8M~ x [0,00[, gl,a,oo) isometric to (8M~ x [0,00[,g2,a,oo), gi,a,oo = giI8MIx[a,oo[' Let M = (Dn U sn x [0,00[, gstandard). If o-(MD i= O"(M~) then (M1 , g), (M2, g) E genbcomPL,iSO,rel(M, g) are not cs-bordant. D Example 5.30 Consider Mi
There is a simple approach to calculate the local algebraic structure of n~e(cs). Consider as above n~e(cs, M) .n~e(cs, genbcompL,iSO,rel(M)) and let nn be the usual bordism group of closed oriented n-manifolds. Then there exists a map
-Kd = char.n.(K' U -K), [K~' U -Kdnn = [K' U
-Klnn' There holds WM
Relative Index Theory, Determinants and Torsion
158
([NI], [N2])
1 [NI#N21 and
nnc(cs M) x nnc(cs M') n' n"
([MI' gI], [M{, g~])
1 [MI#M{,gI#g~l
Proposition 5.31 The diagrams n~C(cs,M)
x
n~C(cs,M')
1
(5.18)
and n~C(cs,
M) x
n~C(cs,
M')
1
commute.
o
Remark 5.32 It is important that we consider (5.18), (5.19) at bordism class level. In (5.19) e.g. (KI U -K)#(K~ U -K') =IKI #Kf U - (K #K'), but their bordism classes coincide. 0 The 1-1 property of M, WM moreover implies: Proposition 5.33 Modulo torsion {which is well defined at the nn-level} is n~C(cs, Mn) = 0 for n =I- 4k and for n = 4k are
[M#P2iI (C) x ... x P2i k (C), gM#p2il(C)x ... xP2ik(C)1 independent generators for n~C(cs, Mn) over Q, iI + ... + ik = k. 0
Non-linear Sobolev Structures
159
Remarks 5.34 1) 5.28, 5.29, 5.33 provide sufficient means to characterize cs-bordism classes and to calculate n~C( cs). 2) An analogous procedure can be applied to calculate e.g. n~c,SPin(cs, M), where bcomPL,iso,rez(M) is now a component consisting of Spin-manifolds. D As we pointed out, a contentful theory should be developed under three aspects. 1) A convenient characterization of bordism classes should be desirable. 2) It should be possible to exhibit sets of independent generators, at least for the intersections with gen-components. 3) A geometric realization of zero and the inverse should be desirable. The general bordism group n~c did not satisfy any of these three wishes. n~C(cs) satisfies the first two wishes. We develop below a bordism theory which satisfies the second and the third wish. This will be the bordism theory for manifolds with a finite number of ends, each of them nonexpanding. Let f be an isolated end of (M n , g). A ray in f is a geodesic "f defined on [0, oo[ which is a shortest geodesic between any two of its points and such that some neighbourhood of f contains up to a finite segment the whole of l"fl. Then the latter holds for any neighbourhood of f. Lemma 5.35 Let f be an isolated end of (Mn,g). a) Then there exists a ray in f. b) If (Mn , g) additionally satisfies (1) then there exists a ray in f with a uniformly thick neighbourhood. Proof. A proof of a) is e.g. contained in [37], p. 43. b) follows immediately from a) and (1). D
160
Relative Index Theory, Determinants and Torsion
We call an end E of (Mn, 9) nonexpanding if there exist a ray "I in E and an R = RM > and an element GEE such that G ~ UR(I"II), roughly written E ~ UR(bl). In the sequel we restrict to open manifolds satisfying (1), B(oo), with finitely many ends, each of them nonexpanding.
°
Examples 5.36 1) Consider the sphere radius rand
S~-l C IRn
c
IRn+l of
i.e. the closed half cylinder of radius r with a standard metric which should be the product metric of S~-l x [0, oo[ smoothly extended to the glued bottom D~ and with standard orientation. Then chcn(r) is an open manifold with one nonexpanding end, satisfying (1), (Boo). 9st
k
2)
# chc(ri) has finitely many nonexpanding ends. i=l
3) Any manifold (Mn = M' U 8M' x [0,00[, 9M) where M' is compact and 9Ml8M'x[a,oo[ = dt 2 + 98M' satisfying (1), (Boo) and has finitely many nonexpanding ends. 4) The same is true if we allow 9M of 3) to vary in comp P,T(9M) n
Coo. 5) If we consider M of smooth type of 3) and 9Ml8M'x[a,oo[ = dt 2 + f(t)298M' with C2 ~ f(t) ~ Cl > 0, f~) bounded for all v, t ~ a then M has finitely many nonexpanding ends. 0 We define now a slightly sharpened bordism relation. Let (Mn, 9), (Mtn, 9') be as above, each with finitely many nonexpanding ends El,'" ,Es or E~, ... ,E~" respectively. Let "1M, 1 , ... , "1M,s or "1M',!,· .. ,"IM',s' corresponding rays as above. From (M, 9) rv (M', 9') and all ends nonexpanding follows in bg
particular that for all sufficiently large compact C n+l C Bn+l
Non-linear Sobolev Structures
161
there exists R = RB > 0 S.t. s
B n+ 1
\
e
n+ 1
c
U UR(II'M,u I) , 1 s'
B n+ l
\
en+!
c
U UR(hM',u, l). I
We require additionally to this condition the additive compability of the inner I'-distance and the (B \ C)-distance for points x,)" Y')' on the I"s. There exist cn+1 C B n+! and c' > 0 s.t. for x,)" y')' E 11'1 \ holds
e
Here I' stands for I'M,I, . .. ,I'M,s, I'M',l,"" I'M',s' , respectively and d(., .) == dist(-, .). We denote (M, g) rv (M', g') if they are bg-bordant by means ne (B,gB) satisfying (GH). Remarks 5.37 1) The right hand inequality of (GH) trivially holds. We added it only for symmetry reasons. 2) It was essentially Thomas Schick who pointed out to the author the meaning of the condition (GH) or (GH I ) and who 0 proposed to include them into the definition of bordism. We consider instead of (G H) the condition There exist C n+ l C Bn+! and c' > 0 s.t. for all x, y E U(c:) holds
Here c: stands for C:I, ... ,c: s , c:~, ... hood of c:, U(c:) n C = 0.
,c:~,
and U(c:) for a neighbour-
Lemma 5.38 (GH) and (GH I ) are equivalent.
162
Relative Index Theory, Determinants and Torsion
Proof. Assume (CHI)' Then (CH) holds since for x-Y' Y-y E 11'1 c U(E), U(E) n C = 0, du(c) (x-y, Y-y) = d-y (x-y , y-y). If conversely x, Y E U(E) then there exists x-Y' Y-y E 11'1 C U(E) s.t. du(c) (x, x-y) ::; R M , du(c)(Y, Y-y) ::; R M . Then the assertion follows from
du(c) (x, y) - du(c) (x-y, Y-y) ::; du(c) (x, x-y) + du(c)(Y, Y-y), du(c)(x, y) - du(c)(x, x-y) - du(c)(Y' Y-y) ::; d-y(x-y, Y-y) = d-y (x-y , Y-y) - c' + c' ::; dB\C(x-y, Y-y) + c', du(c) - 2RM - c' ::; dB\c(x-y, Y-y), du(c) - 4RM - c' ::; dB\C(x, y).
o Remark 5.39 (CHI) immediately implies that dCH(B \ C, U(UEa)) < 00, where dC,He·) is the Gromov-Hausdorff disa
tance between proper metric spaces. This follows from the following facts. dCH(B \ C,U(UE a)) < 00 if we endow U(UEa) a
a
with the induced lengths metric and use (B \ C C UR(U(UEa)). a
Then we use dCH(U(E)), its own lengths metric, U(E), induced lengths metric < 00, which follows from (CHI)' As a matter of fact, we introduced (CH) to assure dCH(B \ C, U(E)) < 00. 0 Proposition 5.40
rv
ne
is an equivalence relation.
o We refer to [26] for the proof. O~C(ne) == onc(je, ne, bg ) is again defined as Grothendieck group. Next we develop geometric realizations for 0 and -[M, g]ne in O~C(ne).
Let (Mn, g) be as above, i.e. oriented, with (I), (Boo), finitely many ends Eb' .. , Es , each of them nonexpanding. Let E be one of them, C C M compact and so large that E is defined by one
163
Non-linear Sobolev Structures
of the components of M \ C, Ue C M \ C a neighbourhood, "I a ray in U(c). "I admits a tubular neighbourhood of radius 53 > O. Consider (B, gB) = (M x I, gM + dr 2). Then c x I =
{Uj(c) X I}jEJ is an end of M x I, U(c x I) = U(c) x I a neighbourhood disjoint to C MxI = C x I, and for 0 < 51 < 1, the curve "101 = "I x {5tl = b,(1 ) is a ray in U(c x I). c x I is nonexpanding. "101 admits a tubular neighbourhood with a radius 52 > 0, To 2 bo1)' Theorem 5.41 8T02 ("101) has bounded geometry, one nonexpanding end and there holds
We refer to [26] for the proof. 0 n Next we shall see, (chc (5),gst) will play the role of our zero in n~C(ne).
Lemma 5.42
chc n (r2)'
a)
For r1 < r2 is chcn(rl)
b)
[i~lchCn(ri)Le = [chcn(r)lne for r > rl + ... + rk·
rv
ne
(5.20)
(5.21)
Proof. a) is immediately clear (or follows from b)). Set for b) r = r1 + ... + rk + 5, place chcn (r1) u· .. U chcn(rk) all with parallel [0, oo[ direction into int(chcn(r)), where int(chcn(r)) coresponds to b~ x ]0,00[. Then CL(int( chcn(r)) \int( chc n(r1) u· .. U
chcn(rk))) defines the desired ne-bordism.
0
Theorem 5.43 For any oriented manifold (Mn, g) of bounded geometry and a finite number of ends, each of them nonexpanding, there holds (5.22)
164
Relative Index Theory, Determinants and Torsion
Proof. We must construct a ne-bordism between (Mn, g) and -((Mn,g) U (chcn(r),gst)). Let (Bn+l,gB) = (M x [0, l],gM + dt 2 ), E be an end of M, 'Y a ray in E, form 'Y,h = b, (h) c M x [0,1], T"2bch), 62 < inf{~,rinj(M)/2} and set B-y = Bn+1 \ intT"2 ('Y"1) with the induced metric. From our assumption rinj > 0 follows easily that aT"2 b"J has a smooth collar U" (aT). Endow U§. with the product metric g§. and form on U" - U§. the 2 2 2 smooth bg-convex combination of g§. and gB getting gB-y' En2 dow aTy2 b"1) with the induced orientation. Then (R(lgB-y) is a bg, ne-bordism between (Mn,g) and (Mn,g) U (aT"2b"1),gaT)' Theorem 5.41 yields
o
Theorem 5.44 n~C(ne) == n~C(bg, ne) is an abelian group with -[Mn,g] = [(-Mn,g)] and 0 = [chcn(r),gst]. 0
Our next goal is to produce independent generators of n~C(ne). As we shall see in the sequel, infinite connected sums of complex projective spaces (or their cartesian products) supply such elements. We prepare this by several assertions Lemma 5.45 Let (M[t, gi), i = 1,2, be open, oriented of bounded geometry and with a finite number of ends, each of them non expanding. Let further (Bn+l, gB) be a ne-bordism between them and K c B compact such that the ends of B coincide with the components of B \ K. Let Ce C B \ K a component of B \ K and Xo E Ce. Then there exists a constant C 1 > 0 such that the diameter of any metric sphere
is ~ C1 • Here we understand the diameter with respect to the induced length metric dB of B.
Non-linear Sobolev Structures
We refer to [26] for the proof.
165
o
Now we recall once again the chopping theorem of Cheeger / Gromov (cf. [17]) which is a consequence of Abresch's habilitation (cf. [1]) and was our I 1.33.
Theorem 5.46 Suppose (Mn, g) open, complete with bounded sectional curvature IKI :::; C. Given a closed set X c Mn and o < r :::; 1, there is a submanifold, un, with smooth boundary,
8U n, such that for some constant c( n, C)
Xc U c Tr(X), vol(8U) :::; c(n, C)vol(Tr(X) \ X)r- 1 , III(8U)1 :::; c(n, C)r-l. Moreover, U can be chosen to be invariant under I(r, X) group of isometries of Tr(X) which fix x.
0
In our case, X = X(2 = B(2(xo) c Bn+l. To apply 5.46, we form (vn+l, gv) = (Bn+l U Bn+l, gB U gB) which is well defined and smooth since we assumed the Riemannian collar gBlcoll ar = gaB + dt 2 . Now we set Xv = X U X and apply 5.46. Fix 0< r :::; 1. Then we get Uv , H(2,v,r = 8Uv .
Xv c Uv C Tr(Xv)(= {x E Vldv(x, Xv) :::; r}),(5.23) vol(H(2,v,r) = vol(8Uv ) (5.24) :::; c(n + 1, C)vol(Tr(Xv) \ Xv)r- 1 1 (5.25) III(8Uv ) 1 :::; c(n + 1, C)rand Uv is invariant under I(r,Xv). The main idea of the proof consists in considering the distance function F = d(·,X v ) where for points E V \ Xv, d(·,X v ) = d(·, X(2) = d(·,8(2). Then one applies Yomdin's theorem to F in Abresch's smoothed out metric. All constructions are invariant under the metric involution and this involution remains an isometry also with respect to Abresch's smoothed out metric.
166
Relative Index Theory, Determinants and Torsion
Restricting the obtained Uv , 8Uv to B, we obtain the desired result for X = Bg(xo) c B. Restricting for (J large to Co and using the construction of U as pre image under the smoothed F, we obtain in Co a hypersurface H = Hg which decomposes Co into a compact and noncompact part Co,e and Co,ne, respectively. Under our assumptions (8 B is totally geodesic) it is possible to arrange that Hn intersects 8B transversally under an angle> 5 and that there exists a constant C 1 independent of (J such that (5.26)
We infer from (5.23), bounded curvature and lemma 5.45 that for fixed 0 < r :::; 1 there is a constant C 2 > 0 such that (5.27)
for all (J. Moreover, H; has bounded geometry (at least of order 0) according to (5.24) and to the bounded geometry of B. Now we are able to present independent generators of n4~(ne). Let p2k(C) be the complex projective space with its standard orientation and with its Fubini-Study metric, fix two points ZI, Z2 and form by means of fixed spheres about ZI, Z2 the infinite connected sum 00 M4k = (M4k, g) = #P2k(C), (5.28) 1
always with same glueing metric. Then (M4k, g) is oriented, has bounded geometry, one end which is nonexpanding. Theorem 5.47 M4k
00
#p 2k (C) defines a non zero bordism 1
class in n4~(ne). Proof. Suppose [M4k] = O. Then there exists a bordism (Bn+\gB), 8B = M4k U -chc4k (r), gBlu,,(8B) = g8B + dt 2, UR (M4k) "2 B, UR (chc 4k (r)) "2 B and dB ~ dM-c, dB ~ dchc-c. We choose Zo E Plk(C), K = 0 and obtain for any (J > 0 a compact hypersurface H: k c B = B \ 0 = Co which decomposes B into a compact and noncompact part Be and B ne , respectively, and which satisfies (5.26), (5.27) and has bounded
167
Non-linear Sobolev Structures
geometry at least of order 0 with constants independent of (2. Then 8B4k+1 = (8B4k+l n M4k) U He U (8B4k+l n chc4k ) . Here c c c a(8Bdk+1 n chc 4k ) = O. a(8Bdk+1) must be zero since it is 0bordant (if one wants, after smoothing out). Hence (5.29)
But
a(H;k) =
J+ L
"7(8H;k)
+
H~k
J
expression(II(8H;k)). (5.30)
aH~k
The first expression on the r.h.s. of (5.30) is bounded by a bound independent of (2 according to (5.27) and (Bo) for H;k. The same holds for the second expression according to
1"7(8H;k) I ::; C3 vol(8H;k) and for the third expression according to (5.26), (5.27). On the other hand, choosing (2 sufficiently large, a(8B4k+l n M4k) can be made arbitrarily large. This contradicts (5.29). D Looking at the proof of theorem 5.47, we immediately infer Theorem 5.48 Let (M4k, g) be open, oriented, of bounded geometry and with a finite number of ends, each of them nonexpanding. If for any exhaustion Ml C M2 C ... by compact submanifolds, U Mi = M, there holds
lim a(Mr)
= 00
t---+OO
then [M4k, g]
i- 0 in n~Z(ne).
D
00
Corollary 5.49 #P2k(C) , or, more general, P 2h(C) x ... 1
p2i r1 #p 2jl (C) X ... X p 2j r2 # ... , i 1 + ... iT! k, . .. are not torsion elements in n~k( ne).
= k,
jl
+ ... + jr2
X
= D
168
Relative Index Theory, Determinants and Torsion
A special case for theorem 5.48 is given by manifolds M4k of the type
vol(Mi4k ) :::; G1 , !K(9i)! :::; G2 , rinj(9i) 2: G3 > 0, C7(Mfk) 2: o for i 2: io and > 0 for infinitely many i 2: io. Then, in particular, 7t 2k,2(M 4k ) is infinitedimensional and [M4k, 9] i=- 0 in n~k(ne), i.e. adding a finite number of closed manifolds with negative signature and an infinite number of closed manifolds with zero signature (such that the bg, ne-end struture remains preserved) does not transform a nonzero element into zero in n~k(ne). A finer characterization of nonzero elements in n~k(ne) will be presented at another place. Moreover there are very interesting specializations of the theory developed until now and generalizations, e.g. the restriction to manifolds with warped product structure at infinity or with prescribed volume growth of the ends etc .. This will be the topic of another investigation.
III The heat kernel of generalized Dirac operators Substantial estimates for the operator e- tD2 are more or less equivalent to estimates for the corresponding heat kernel. We present in the first section those estimates which are needed in the sequel and establish some invariance properties of the spectrum which we apply in chapters IV, V and VI.
1 Invariance properties of the spectrum and the heat kernel We start with an absolutely fundamental theorem.
Theorem 1.1 Let (E, h, \7 i) ----t (Mn, g) be a Clifford bundle, (Mn, g) complete and D the generalized Dirac operator. Then all powers Dn, n ~ 0, are essential self-adjoint.
o
We refer to [20] for the proof.
Corollary 1.2 Let (E, h, \7) ----t (Mn, g) be a Riemannian vector bundle, (Mn, g) complete and b. q the Laplace operator acting on q-forms with values in E. Then (b.q)n, n = 1,2, ... are essentially self-adjoint. In particular this holds for the Laplace operator acting on ordinary q-forms. Proof. b. q
= D2 for the Clifford bundle A*T* M
® E.
0
In what follows, we always consider the self-adjoint closure Dn and write Dn == Dn.
Corollary 1.3 There is a spectral decomposition
169
170
Relative Index Theory, Determinants and Torsion
where (Je denotes the essential and (Jpd the purely discrete point spectrum. In particular,
o A E (Je if and only if there exists a Weyl sequence for A. Properties of Weyl sequences imply very important invariance properties for the spectrum. Proposition 1.4 Let (E,h,'\lh,.) --> (Mn,g) be a Clifford bundle, Mn open and complete, K c M a compact subset, DF(EIM\K) Friedrichs' extension of Dlc,?O(EIM\K)' Then there hold (Je(D) = (Je(D F) = (Je(DF(EIM\K)) (1.3)
and
Proof. We start with (1.3) and (Je(D) ~ (Je(DF(EIM\K)). Let A E (Je(D), ('l/Jv)v be an orthonormal Weyl sequence for A, D'l/JvA'l/Jv --> O. Then (wv)v, Wv = 'l/J2v+l -'l/J2v is still a Weyl sequence for A. Let E CC: (M), 0 ::; ::; 1, = 1 on a neighbour hood U = U(K) of K. According to the Rellich chain property of Sobolev spaces (with real index) on compact manifolds, ('l/Jv) v contains an L 2 -convergent subsequence which we denote again by ('l/Jv) v' This yields w v --> 0 and grad . Wv --> 0 in L 2 . ((1 -
D F(EIM\K)(l - wv)) - Awv = Dw v - AWv - (Dwv - AWv ) - grad • Wv
--> v-->oo
0
since by assumption and construction DWv-AWv --> 0, grad . Wv --> O. Hence (Je(D) ~ (Je(DF(EIM\K)). VDp(EIM\K) ~ V Dp and every Weyl sequence for A E (Je(DF(EIM\K)) is also a Weyl sequence for A E (Je(D). This finishes the proof of (1.3). (1.4)
Heat Kernel of Generalized Dirac Operators
171
is an immediate consequence of (1.3) by means of the spectral 0 theorem but it can also similarly be proven.
Corollary 1.5 The essential spectrum of D and D2 remains invariant under compact perturbations of the topology and the metric. In particular this holds for the Laplace operators acting 0 on forms with values in a vector bundle. As for compact manifolds, we can define the Riemannian connected sum for open Riemannian manifolds, even for Riemannian vector bundles (Ei' hi, '\1 hi ) -+ (Mt, gi), where at the compact glueing domain the metric and connection are not uniquely determined. Another corollary is then given by Proposition 1.6 Suppose (Ei' hi, '\1 hi ) - + (Mt,gi), i 1, ... ,r Riemannian vector bundles of the same rank, (Mt, gi) complete, and let .6. = .6.q be the Laplace operator acting on q-forms with values in Ei (resp. E). Then
O"e.6. q (
i~l
r
(Ei -+ Mi )) =
UO"e(.6. (Ei -+ Mi)). q
(1.5)
i=l
o 1.4 can be reformulated as the statement that the essential spectrum for an isolated end E is well defined. We denote it by
O"e(DF(E)), O"e(D~(E)). Proposition 1. 7 If (Mn, g) is complete and has finitely many ends El,' .. , Er then r
r
i=l
i=l
172
Relative Index Theory, Determinants and Torsion
Proposition 1.8 Assume the hypothesis of 1.4. Suppose A E
CTe(D). Then there exists a Weyl sequence ('Pv)v for A such that for any compact subset K
c
M
(1.7) For every A E CTe(D2) there exists a Weyl sequence ('Pv)v satisfying (1. 7) and (1.8)
Proof. Start with (1.7). Let ('l/Jv)v be a Weyl sequence for A E CTe (D), Kl C K2 C ... C Ki C KH 1 C "', UKi = M, an exhaustion by compact submanifolds. By a Rellich compactness argument there exists a subsequence ('l/J~l»)v of ('l/Jv)v ('l/J~O»)v converging on K 1. Inductively, there exists a subsequence ('l/J~Hl»)v of ('l/J~i»)v converging on K H1 . Set ('Pv)v = (('l/J~~~~l) - 'l/J~~v»)/V2)v. Then ('Pv)v is a Weyl sequence for A E CTe(D) satisfying (1.7). For A E CT e(D2) with Weyl sequence ('l/Jv) v, we choose the subsequence ('l/J~Hl»)v of ('l/J~i»)v such that ('l/J~Hl»)v and (D'l/J~Hl»)v converge on KHI (in L 2, as always). 0 1.8 means that w.l.o.g. Weyl sequences should "leave" (in the sense of the L 2-norm) any compact subset, i.e. there must be "place enough at infinity" .
Proposition 1.9 Let (E, h, \7, .) - - t (Mn, g) be a Clifford bundle with (1), (B r- 3 (M,g)), (B r- 3 (E, \7)), r > ~ + 1 and \7' a second Clifford connection satisfying 1\7' - \71V',2,r-l < 00. Then for D = D(\7) and D' = D(\7') there holds (1.9) and
(1.10)
(Mn,g) is complete, D and D' are self-adjoint. 1JD = 2 n ,I(E, D) = n2,1(E, \7) = n2,1(E, \7') = n2,1(E, D') = 1J D , Proof.
173
Heat Kernel of Generalized Dirac Operators
according to II 1.25 and II 1.32. We write \1' = \1 + ry. Then D' = Lei . \1~i = Lei' (\1 ei + ryei (.)) = D + ryOP, where the i
i
operator ryOP acts as ryOP (
L
ei . ryei (
i
Sobolev embedding theorem since r - 1 > ~. Then, pointwise, IryOPlx C1 'Irylx, C 1 independent of x. Given E' > 0, there exists a compact set K = K(E') eM such that
:s:
sup XEM\K
E'
Irylx < -C ' 1
i. e.
sup IryOPlx < xEM\K
(1.11)
E'.
Assume now ,\ E (Je(D), (
:s:
(D' - '\)
+ (D -
'\)
By assumption, (D - '\) 0 arbitrarily be given, choose E' < !:2 and K = K(E') such that (1.11) is satisfied. Then
I(D' - D)
C1Iry
:s: C 1 xEM\K sup Irylx' 1
Hence (D' - '\)
Corollary 1.10 Assume the hypotheses of 1.9. i E IN
Then for all ( 1.12)
and (1.13)
174
Relative Index Theory, Determinants and Torsion
Proof. This follows from the self-adjointness of D on V D .
0
Remark 1.11 For n2 ,1(E, \7) = n2 ,1(E, D) (Bo) would be sufficient which easily follows from the Weitzenboeck formula, but we need moreover the Sobolev embedding theorem for n2,r-l, in particular 2 ,r-l = n2 ,r-l, hence (B r - 3 ). One could try to estimate 11'TJlx '1'PlxIL2 instead of sup 1'TJlx 'l'PIL2' i. e. one could try to
n
x
work without the Sobolev embedding theorem, but in this case we would need some module structure. Hence an assumption (B i ), i > 0, seems to be inavoidable in any case. 0 For the Laplace operator on forms, 1.9 is not immediately applicable since if we replace for (Mn, g) the Levi-Civita connection \7g by another metric connection then we lose many of the standard formulas, i. e. we should consider a change 9 ----t g'.
Proposition 1.12 Let (Mn, g) be open, complete, with (1) and (Br (Mn , g)), r > ~ + 1, g' another metric satisfying the same conditions and suppose g, g' quasi isometric and Ig' - glg,2,r = r-l (J(lg' - gl~,x + L 1(\7 g)i - \7gl~,x) dvolx(g))! < 00. Then i=O
v ~q(g) = V ~q(g')
as equivalent Hilbert spaces
(1.14)
and (1.15)
Proof. We write ~ = ~q(g), ~' = ~q(g'), \7 = \7g, \7' = \7g'. Then, according to II 1, V~ = nq,2,2(~, g) = nq,2,2(~Bochner' g) = n q,2,2(\7 , g) ~ n q,2,2(\7' , g') = nq,2,2(~'Bochner' g') = n q,2,2 (~', g') = V~,. Denote by R the Weitzenboeck endomorphism, ~
= \7*\7 + R.
(1.16)
175
Heat Kernel of Generalized Dirac Operators
We write t::..' - t::.. = \1'*'\1' - \1*\1 + R' - R. Let A E oAt::..) and (wv)v be a Weyl sequence for A as in proposition 1.8, for any K (1.17) and (1.18) We infer as above from (wv)v bounded, t::..wv - AWv (wv)v, (t::..wv)v are bounded, hence
(Wv)v is a bounded sequence in But under our assumption r >
~
----+
nq ,2,2(t::.., g).
+ 1, according to II
0 that
(1.19) 1.27,
n ,2,2(t::.., g) = n ,2,2(\I, g) ~ n ,2,2(\I', g') = n ,2,2(t::..', g') Q
Q
Q
Q
(1.20) as equivalent Sobolev spaces. Hence
(\lwv)v, (\l 2wv)v" (\I'Wv)v' , (\I,2wv )v' , (t::..'wv)v are bounded (1.21) and (1.22) To be very explicit, we choose a u. l. f. cover of (Mn,g) by normal charts with respect to g, U = {(Ua, ua)}a and an associated partition {'Pa}a of unity with bounded derivatives as in I 1. iq Then for a q-form wlu" = '" W·H ... 2q. dU i1a 1\ ... 1\ du a 6 h<···
-
ijn n )
9 v j vi Wil ... iq
= g'ij\lj(\I~ _ \li)Wh ... iq + ,ij i· , , +(g - 9 J)\l j \liWil ...iq + +gij(\lj - \lj)\liWh ...i q.
(1.23) (1.24) (1.25) (1.26)
176
Relative Index Theory, Determinants and Torsion
If we write \7iWiI ... iq (1.24) as
+ f~PWij ... iq
then we can rewrite
j /OP _ fOP)w· . 9 li \7'-(f J t t tj ... tq
(1.27)
=
8~iWij ... iq
L:
Insert now WI/ =
Then
a.
j _ fOP) w. . 9 li \7'j (flOP i i
(1.28)
) (f'OP -_ 9 lij (n v j<po. i
(1.29)
+
lij n l (f'OP Vj
i
-
fOP)
-
i
fOP) i
WI/,ij ... i q +
(1.30)
WI/,ij ... iq·
We obtain from \7j
L9
j li \7'j (\7 'i -
\7.){/") . t yo. W· I/,tj, ... ,tq
=0
(1.31)
K
where 11K = II L 2(MIK)' K c M compact. Here we used that in normal coordinates w. r. t. g, (gij) and (gij) are bounded. Moreover, for K = K (E) sufficiently large, 1\7 / (f /OP -
sup
fOP)I,
xEM\K
sup If lOP
-
fOPlx
(1.32)
XEM\K
become arbitrarily small. Together with (1.21) we infer from (1.32) and (1.31) lim
L
- fOP) W· . 9 lij \7j' (f'OP i i
1/-+00
a.
= O.
(1.33)
L2
Quite similar we infer from (1.22) lim 1/-+00
L(9 lij -
j W· . 9 i )\7'\7. j t
a.
=0
(1.34)
= O.
(1.35)
L2
and from (1.21) lim 1/-+00
L9 a.
Ij (f'OP j
fOP) \7 t
177
Heat Kernel of Generalized Dirac Operators
The essential point is that for K = K(E) sufficiently large Ig g'l g,x, Ig-1 - g,-ll g,x, Ir'oP - rap Ig,x, \7(r'op - rap) , 1\7'(r'°Prop) Ig,x become uniformly arbitrarily small outside K. But this follows from our assumption and the Sobolev embedding theorem. We conclude from (1.33) - (1.35) lim (Ll' - Ll)wv = 0,
>.
v-->oo
(Ll'). Exchanging the role of 9 and g' yields the other inclusion. 0 E
(J"e
We state without proof the generalization to forms with values in a vector bundle. Proposition 1.13 Let (E,h, \7 h) ----t (Mn,g) be a Riemannian vector bundle satisfying (1), (Bk(Mn,g)), (Bk(E, \7)), k ~ r > ~ + 1, and let g' be a second metric, h' a second fibre metric with metric connection \7h', g, g' and h, h' quasi isometric, respectively,
Ig - g'lg,2,r < 00, Ih - h'lh,y>h,g,2,r < h h' g2r-1<00. I\7 -\7 Ihy>h , " ,
00,
(E,h', \7h') ----t (Mn,g') also satisfying (1) and (Bk). Then there holds for the Laplace operators Ll = Llq(g, h, \7 h), Ll' = Llq(g', h', \7h') acting on forms with values in E Vt, = Vt" as equivalent Hilbert spaces
(1.36)
and (1.37) The proof is quite similar to that of 1.12, taking the difference of the Weitzenboeck formulas and proceed as in the proof of 0 proposition 1.12. Now we collect some standard facts concerning the heat kernel 2 of e-tD • The best references for this are [9], [27]. We consider the self-adjoint closure of D in L2(E) = HO(E), +00
D =
J >.E>..
-00
178
Relative Index Theory, Determinants and Torsion
Lemma 1.14 {eitDhER defines a unitary group on the spaces
HT(E), for 0 :S h:S r holds (1.38)
o We can extend this action to H-T(E) by means of duality. Lemma 1.15 e- tD2 maps L 2 (E) r > 0 and
== HO(E)
------+
HT(E) for any (1.39)
Insert into e- tD2
Proof.
= J e- t ).,2 dE>.. the equation
J +00
t
e- >..2 =
V~7rt
ei>"se-ft. ds
-00
and use
o Corollary 1.16 Let r, s E Z be arbitrary.
HT(E)
------+
Then e- tD2
HS(E) continuously.
Proof. This follows from 1.15, duality and the semi group property of {e- tD2 h;:::o. 0
e- tD2 has a Schwartz kernel W E f(R x M x M, E tD2 W(t, m,p) = (6(m), e- 6(p)) ,
[8]
E),
where 6(m) E H-T(E) 0 Em is the map W E HT(E) ------+ (6(m), w) = w(m), r > The main result of this section is the fact that for t > 0, W(t, m,p) is a smooth integral kernel in L2 with good decay properties if we assume bounded geometry. Denote by C(m) the best local Sobolev constant of the map W ------+ W(m), r > i, and by a-(D2) the spectrum.
i.
Heat Kernel of Generalized Dirac Operators
179
Lemma 1.17 a) W(t, m,p) is fort> 0 smooth in all Variables. b) For any T > 0 and sufficiently small E > 0 there exists C > 0
such that IW(t, m,p)1 :::; e-(t-e)inf(T(D2) . C· C(m) . C(p) for all t E]T, 00[. (1.40)
c) Similar estimates hold for
(D~D~W)(t, m,p).
Proof. a) First one shows W is continuous, which follows from (8(m), -) continuous in m and e- tD2 8(p) continuous in t and p. Then one applies elliptic regularity. b) Write 2 1(8(m), e- tD2 8(p)) 1 = 1((1 + D2)-~8(m), (1 + D2Ye-tD (1 + D2)~8(p))1
c) Follows similarly as b.
D
Lemma 1.18 For any E > 0, T > 0,8 > 0 there exists C > 0 such that for l' > 0, m E M, T > t > 0 holds
J
(r_e)2
IW(t, m,p)1 2 dp:::; C· C(m) . e- (4+O)t.
(1.41)
M\Br(m)
A similar estimate holds for D~D~W(t, m,p). D
We refer to [9] for the proof.
Lemma 1.19 For any E > 0, T > 0,8 > 0 there exists C > 0 such that for all m, p E M with dist( m, p) > 2E, T > t > 0 holds
IW(t,m,p)1 2
(diBt(m,p)-e)2
:::;
C· C(m)· C(p)· e-
(4+o)t
(1.42)
A similar estimate holds for D~D~W(t, m,p). We refer to [9] for the proof.
D
180
Relative Index Theory, Determinants and Torsion
Proposition 1.20 Assume (M n , g) with (/) and (B K ), (E, \7) with (B K ), k 2: r > ~ + 1. Then all estimates in (1·40) - (1.42) hold with uniform constants. Proof. From the assumptions Hr (E) ~ wr (E) and sUPm C(m) = C = global Sobolev constant for wr(E), according to II 1.4, II 1.6. D
Let U c M be precompact, open, (M+, g+) closed with U c M+ isometrically and E+ --t M+ a Clifford bundle with E+lu ~ Elu 2 isometrically. Denote by W+(t, m,p) the heat kernel of e-tD+ . Lemma 1.21 Assume E > 0, T > 0, J > O. Then there exists C > 0 such that for all T > t > O,m,p E U with B2c(m), B 2c (p) C U holds .2
IW(t,m,p) - W+(t,m,p)1 ::; C· e-(4+8)t
We refer to [8] for the simple proof. Corollary 1.22 trW(t, m, m) has for t totic expansion as fortrW+(t,m,m).
(1.43) D
--t
0+ the same asympD
2 Duhamel's principle, scattering theory and trace class conditions 2 -,2 We want to prove the trace class property of e- tD - e- tD , where fy is a perturbation of D. The key to get convenient - 2 expressions for e- tD2 - etD' is Duhamel's principle. For closed manifolds, this is a very well known fact. We establish it here for open complete manifolds. In principle, it follows from Stokes' theorem, or, what is the same, from partial integration. Having established Duhamel's principle, the proof of the trace class property amounts to the estimate of a certain number of operator valued integrals. Their estimate occupies the whole 30 pages of chapter IV.
Heat Kernel of Generalized Dirac Operators
181
The trace class property is the key for the application of scattering theory. We give an account on those facts of scattering theory which are of great importance in chapters V and VI. First we establish Duhamel's principle and make the following assumptions: D and D' are generalized Dirac operators acting in the same Hilbert space,
where
'fl
= 'flop
is an operator acting in the same Hilbert space.
Lemma 2.1 Assume t > O. Then
J t
e-tD2 - e _tD,2 --
2 e -SD (D,2 - D2) e -(t-s)D,2 ds.
(2.1)
o
Proof.
(2.1) means at heat kernel level
W(t,m,p) - W'(t,m,p)
JJ t
=-
(W(s, m, q), (D2 - D,2)W'(t - s, q,p))q dq ds,
o M (2.2) where (,)q means the fibrewise scalar product at q and dq = dvolq(g). Hence for (2.1) we have to prove (2.2). (2.2) is an immediate consequence of Duhamel's principle. Only for completeness, we present the proof of (2.1), which is the last of the following 7 facts and implications. 1. For
t > 0 is W(t, m,p)
2. If <1>, W E formula).
Vb
E L 2(M, E,
dp)
n Vb
then J(D 2<1>, w) - (<1>, D2W) dvol = 0 (Greens
3. ((D2+ %7)<1>(T, g)W(t-T, q))q-(<1>(T, g), (D2+~)W(t-T, q))q = = (D2( <1>( T, q), W(t-T, q) )q- (<1>( T, q), D 2w(t-T, q) )q+ (<1>( T, g),
tr
W(t-T,q))q.
Relative Index Theory, Determinants and Torsion
182
f3
4. J J((D 2 + tr)
+ ft)1J!(t-
oM
T, q))q dq dT = = J[(
5.
= W(t, m, q), 1J!(t, q) = W'(t, q, p) yields
f3
- J J(W(T,m,q), (D2
+ ft)W'(t - T,q,p) dq dT =
oM
= J[(W(,8, m, q), W'(t -,8, q,p))q M
-(W(a,m,q), W'(t - a,q,p))q] dq . 6. Performing a
---t
0+,,8
---t
r in 5. yields
t
- J J (W(s, m, q), (D 2+ft)W'(t-s, q,p))q dq ds = W(t, m,p)OM
W'(t,m,p). 7. Finally, using D2+ft = D2-D,2+D,2+ft and (D,2+ft)W' o we obtain
=
t
W(t, m,p) - W'(t, m,p) = - J J(W(s, m, q), (D2 - D,2)W'(tOM
s, q, p) )q dq ds which is (2.2).
183
Heat Kernel of Generalized Dirac Operators
If we write
D2 -
=
D,2
D(D -
D')
-J = - J -J J +J
+ (D -
D')D'
then
t
e- sD2 (D2 -
D,2)e-(t-S)D
I2
ds
o
t
e-
sD2
D(D -
D')e-(t-S)D
I2
ds
o
t
e-sD\D -
D')D'e-(t-S)D
I2
ds
o
t
e- sD2 DfJ e -(t-S)D
I2
ds
o
t
e-
fJ D ' e-(t-s)D
,2
ds,
o is an operator which will specified in chapter IV.
where fJ
= fJOP
We split
J = J + J, o
~
t
sD2
0
t
t
"2
t
J "2
e-
tD2
_ e- tD'2
=
e- sD2 DfJ e -(t-s)D
I2
ds
(II)
o t
J +J "2
+
e-
sD2
fJ D ' e-(t-s)D
I2
ds
(I2)
ds
(h)
ds.
(I4)
o
t
e-
sD2
DfJ e -(t-s)D
I2
t
"2
J t
+
e-
t
"2
sD2
fJ D ' e-(t-s)D
I2
184
Relative Index Theory, Determinants and Torsion
We will show in chapter IV that each integral (II) - (I4) is a product of Hilbert-Schmidt operators and estimate their Hilbert-Schmidt norm. 0 Now we introduce scattering theory and recall the standard decomposition of the spectrum O"(A) of a self-adjoint operator A: VA ----+ X. Every Borel measure m on IR admits a decomposition m = mpp+ me = mpp+mac+msc, where mpp is a pure point measure, me = m - mpp is characterized by mc( {p}) = 0 for all points p, mac is absolutely continuous with respect to the Lebesgue measure and msc is singular with respect to the Lebesgue measure, i.e. msc(5) = 0 for a certain set 5 such that IR \5 has zero Lebesgue measure. We consider for x E X the positive Borel measure [a, b] ~ (E[a,b]X, x) and set Xpp = {x E Xlmx = mx,pp}, Xac =
{x E Xlmx = mx,ac}, Xsc = {x E Xlmx = mx,sc}, Xc = {x E Xlmx = mx,c} and O"pp(A) := dAIXpp) , O"c(A) := O"(AIXc), O"ac(A) := O"(AIXac ), O"sc(A) := O"(AIXsc). Then O"c(A) , O"ac and 0" sc are called the continuous, the absolutely continuous and the singular continuous spectra. Finally we define O"pd(A) := {A E O"p(A)lmult(A) < 00 and A is an isolated point in O"(A)}, called the purely discrete spectrum, and O"c,R(A) = O"(A) \ O"p(A) , O"p(A) the point spectrum. Theorem 2.2 Let A : VA
----+
X be self-adjoint.
a) X = Xpp EEl Xac EEl Xsc. b) O"(A) = O"pd(A) U O"e(A), O"pd(A) n O"e(A) = 0. c) O"e(A) = O"c(A) U O"p(A)1 U {A E O"p(A) Imult(A) = oo}. d) O"c(A) = O"ac(A) U O"sc(A). e) dA) = O"pp(A) U O"c(A) = O"p(A) U O"c(A). Here Ml denotes the set of accumulation points of M.
Remark 2.3 In general, O"p(A) C O"pp(A) , hence O"p(A)UO"ac(A)U O"sc(A) C O"(A), but O"p(A) = O"pp(A). This shows also that in
Heat Kernel of Generalized Dirac Operators
185
One of the main topics are invariance questions concerning the components of the spectrum, in particular the invariance of the absolutely continuous spectrum under perturbations. These questions are essentially settled by the wave operators, their completeness and the scattering matrix. We present them now. Let A and B be self-adjoint operators in a Hilbert space X and let Pac(B) be the projection onto the absolutely continuous subspace of B. The wave operators W ± for the pair A, Bare defined as strong limits
W ± = W ±(A,B) =
S -
· eiAt e -iBt Tn ac (B) 11m
t--->±oo
if they exist. Denote by EA(I) or EB(I) , I an interval, the spectral projections of A or B, respectively. We collect some standard properties of
W±: Proposition 2.4 Suppose that W±(A, B) exist. Then a) W± are isometric on (Pac(B))(X), b) EA(I)W±(A, B) = W±(A, B)EB(I), c) e-iAsW±(A, B) = W±(A, B)e- iBs , d) if additionally W±(B, C) exist, then W±(A, C) exist and W±(A, C) = W±(A, B)W±(B, C). D
2.4 b) immediately implies RanW± = im W± ~ (Pac(A))(X). We say the W± are complete if RanW± = (Pac(A))(X).
Proposition 2.5 Suppose that W±(A, B) exist. Then they are complete if and only W±(B, A) exist. In this case W±(B, A) = W±(A,B)*. D
186
Relative Index Theory, Determinants and Torsion
The central question are considerations which assure the existence and completeness of the wave operators. There are several classes of theorems. We restrict here ourselves to the conditions of Birman-Kato-Rosenblum. Theorem 2.6 Let A and B self-adjoint operators in X. Then the wave operators W±(A, B) exist and are complete in the following cases. a) A and B are positive operators with A2 - B2 of trace class, b) A and B positive operators with (A2 + 1)-1 - (B2 + 1)-1 of trace class, c) A, B 2: -a + I and (A + a) - -k -(B + a)-k of trace class for some k, d) e- A - e- B of trace class, e) (A + i)-l - (B + i)-l of trace class, f) A and B positive and e- tA A - etB B of trace class. We refer to [58], [72], [73] for the proofs of 2.3 - 2.5. 0 All of our efforts in chapter IV will be concentrated to the task to satisfy on of the conditions above. Suppose W±(A, B) exist. Then we define the scattering operator S by
Remark 2.7 Completeness is not needed to define S. S commutes with B. IfRanW± = (Pac(A»(X), i.e. ifW± are complete then S is unitary on (Pac(B»(X). 0 Next we will introduce the spectral decomposition of the scattering operator S. For this we need the notion of a direct integral of Hilbert spaces. The direct integral
J +00
EBX(>')du(>')
-00
(2.3)
Heat Kernel oj Generalized Dirac Operators
187
is defined as the set of all vector-valued functions J, J(>..) E X(>..) which are measurable and square-integrable with respect to the measure m. The scalar product in (2.3) is defined as +00
(1, g):= j (J(>..), g(>..))x(>,)dm(>"), -00
where (J(>..),g(>..))x(>.) is the scalar product in the Hilbert space
X(>..). We say, a Hilbert space X has a decomposition as a direct integral if there is a unitary mapping +00
F: X
7
j EBX(>..)du(>..). -00
We denote this by +00
X
~
j EBX(>")du(>"). -00
A special case of such a representation is given by the spectral resolution for a self-adjoint operator A,
-00
which induces a decomposition of type (2.3) in which the operator F AF* acts as multiplication by>... Let 8 c IR be a Borel set. Then FEA(8)F* reduces to XB, and we obtain
(EA(B)J,g) = j((FJ) (>..), (Fg) (>..))dm(>..). B
If we apply these considerations to the self-adjoint operator B (of the pair A, B above) and to Xac = Xac(B) == (Pac(B))(X),
Relative Index Theory, Determinants and Torsion
188
then we get
Xac(B)
J
~
EBX>.(B)d)..:= X(B)(ac).
(2.4)
&(B)
Here o-(B) is a so-called core of (J"(B), i.a. a Borel set of minimal measure such that EB(lR \ o-(B)) = O. The right hand side of (2.4) diagonalizes the operator Bac = BlxaJB). The scattering operator S = W~ W - commutes with B ac , hence under the correspondence (2.4) its action goes over into multiplication by an operator valued function S()") : X>.(B) ----t X>.(B). S()") = S()..; A, B) is called the scattering matrix. If we consider the left hand side of (2.2) then we obtain the version S
=
J
(2.5)
S()..)dEB ()..)
&
instead of FSF* =
J
(2.6)
S()")d)"
Both representation are equivalent. In chapter V and VI, the spectral shift function (S S F)~ will playa central role. We introduce it now. The main goal is to introduce a function ~()..) such that
tr(
J
(2.7)
lR
where A, B are self-adjoint in X,
=
e-27ri~(>'), a.e. ).. E
(J"(B).
(2.8)
First we must show that the left hand side of (2.8) makes sense.
Heat Kernel of Generalized Dirac Operators
189
Proposition 2.8 Let T be a trace class operator in X and {Udi a complete orthonormal system (ONS) in X. a) Then
exists and does not depend on the choice of the ONS.
b) It is a continuous functional on the ideal of trace class operators.
c) There exists a representation as absolutely convergent infinite product
det(I
+ T) = II (1 + Ak(T». k
d) There holds
+ T*) det((I + T 1 )(I + T2 » = det(I + T I T 2 ) det(I
e) I
+T
det(I + T), det(I + T 1 ) . det(I + T2 ), det(I + T 2 T I ) ,
ist invertible if and only if det(I
+ T)
=1=
0, and then
holds
det(I
+ T)
. det(I + T)-I
= 1,
(2.9)
f) if T( z) is a CI-family of trace class operators then dd det(I Z
+ T(z» = det(I + T(z»tr
((I
+ T(Z»-I d~~Z»)
.
We refer to [72], [73] for details and proofs. 0 For A, B closed operators in X, VA = VB with (A - B)Rz(B) of trace class, we define the perturbation determinant 6(z)
== 6A/B(Z)
:= det(I
The resolvent equations
+ (A -
B)Rz(B),
z E g(B).
190
Relative Index Theory, Determinants and Torsion
imply ~A/B(Z) ~(z)
= det(A -
zI)(B - zI)-l),
z E e(B).
(2.10)
is holomorphic on e(B) and
~/(Z)/ ~(z)
z E e(B) n e(A).
= tr(Rz(B) - Rz(A)),
(2.11)
We infer from (2.9) and (2.10) ~B/A(Z)~A/B(Z) = 1,
z E e(A) 1\ e(B)
(2.12)
and from 2.8 d) ~B/A(Z)~B/C(Z)
=
~A/c(Z),
Z E e(A)
1\
dC),
(2.13)
if (A - B)Rz(B) and (B - C)Rz(C) are trace class. Now we are able to state the first theorem of Klein.
Theorem 2.9 Suppose A and B self-adjoint, A - B of trace class. Then log ~A/B(Z) =
J
~(t)
--dt, t -
Z
im Z
# 0,
(2.14)
IR
with ~
= ~(.; A, B) = ~ E ~(,\)
Ll (1R)
= 7[-1 lim arg ~(,\ + ic:), a. e. ,\ E lR,
J~(t;
e~O+
A, B)dt
= tr(A -
B)
IR
and
JI~(t;
A, B)ldt "5: IA - Bll'
IR
where
ITII = L('\k(T*T))~ k
is the trace norm.
(2.15)
Heat Kernel of Generalized Dirac Operators
191
o We refer to [72], [73] for the proof. To state the second theorem of Krein, we introduce the Wiener class WI (lR) WI(lR)
=
{cp E Cl~clcpl().) =
J
e-iAtdO"(t) ,
IR
d,) a complex-valued finite Borel measure on lR}. Theorem 2.10 Suppose the hypthesis of 2.9 and cp E WI (lR). Then cp(A) - cp(B) is of trace class and
Icp(A) - cp(B)h ::; 1001(lR)IB - All and tr(cp(A) - cp(B)) =
Jcp'()')~()')d)'.
(2.16) (2.17)
IR
o We will make extensive use of the spectral shift function in the forthcoming chapters.
IV
Trace class properties
In this chapter, we prove the trace class property for e- tD2 e- tfY2 , D a perturbation of D as defined in II 4, i.e. the defining data of Dare E gencomp (defining data of D). We introduced in II 4 a hierarchy of perturbations, admitting step by step larger perturbations, i.e. the goal of this chapter is to prove that
is for t > 0 of trace class where D' is an appropriate transform of D' into the Hilbert space of D. We decompose the perturbations into several steps, 1) '\7 ----t '\7', all other fixed, 2) h, '\7 ----t h', '\7', " 9 fixed, 3) h, '\7" ----t h', '\7', J, 9 fixed and finally 4) h, '\7, " 9 ----t h', '\7', .', g'. The last step consists in even admitting compact topological perturbations.
1
Variation of the Clifford connection
The first and simplest case is settled by Theorem 1.1 Assume (E, '\7) ----t (Mn,g) with (1), (B k ), (E, '\7) with (B k ), k ~ r > n + 2, n ~ 2, '\7' E comp('\7) n CE(Bk) c C~r(Bk)' D = D(g, '\7), D' = D(g, '\7') generalized Dirac operators. Then
are trace class operators for t > 0 and their trace norm is uni0 formly bounded on compact t-intervalls lao, ad, ao > O.
Here '\7' E compl,r('\7) means in particular 1'\7 - '\7'IV',l,r < and both connections satisfy (Bk(E)). Denote '\7 - '\7' = 'rJ.
192
00
Trace Class Properties
193
As we indicated in III 2, we have, writing D2 - D,2 D') + (D - D')D', to estimate
J = - J -J J +J
=
D(D -
t
e- tD2 _ e- tD'2
= _
e- sD2 (D2 - D,2)e-(t-S)D
I2
e- sD2 D(D - D')e-(t-S)D
I2
ds
o
t
ds
o
t
e- SD2 (D - D')D'e-(t-s)D
I2
ds
o
t
e- sD2 DT/e-(t-S)D
I2
ds
o
t
e- sD2 T/D' e-(t-s)D
,2
ds,
o n
where T/ = T/op is defined by T/oP(w)lx =
2: eiT/ei(w) and IT/OPlop,x
::;
i=l
C . IT/Ix, C independent of x. We split
~
t
t
J = J + J,
o
0
t
:I
t
J :I
e- tD2 _ e- tD'2
=
e- sD2 DT/e-(t-s)D
I2
ds
(h)
o t
J +J :I
+
e-sD2T/D'e-(t-s)DI2 ds
(h)
o
t
e- sD2 DT/e-(t-s)D
t
:I
I2
ds
(Is)
194
Relative Index Theory, Determinants and Torsion
J t
+
e- SD\,D'e-(t-S)D'2 ds.
t
'2
We want to show that each integral (II) - (14) is a product of Hilbert-Schmidt operators and to estimate their HilbertSchmidt norm. Consider the integrand of (1 4 ),
There holds
Write
Here f shall be a scalar function which acts by multiplikation. The main point is the right choice of f. e-~D2 f has the integral kernel s (1.1) W('2' m,p)f(p) and f-le-~D27] has the kernel (1.2) We have to make a choice such that (1.1), (1.2) are square integrable over M x M and that their L 2-norm is uniformly bounded on compact t-intervals.
Trace Class Properties
195
We decompose the L 2-norm of (1.1) as
JJIW(~,m,pWlf(mW = J J IW(~, + J J IW(~,m,p)12If(mW
(1.3)
dm dp
MM
m,pWlf(mW dp dm
(1.4)
dp dm
(1.5)
M dist(m,p)'2:c
M dist(m,p)
We use the fact that for any T > there exists C > such that
°
IW(t,m,p)1 ~
and sufficiently small
e-(t-c)infa(D
for all t E]T, oo[ and obtain for
(1.5)
°
S
).
C1If(mWvolBc(m) dm
M
>
°
C· C(m). C(p)
EH, t[
J
~
2
E
~ C2
° °
J
If(m)12 dm
M
Moreover, for any E > 0, T > 0, 0 > there exists C > Osuch that for r > 0, m E M, T> t > holds
J
M\Br(m)
which yields
J J ~J
IW(~,m,pWlf(mW dp dm
M dist(m,p)'2:c
C3 e- -~+8)2 ~ If(mW dm
M
~ C3 . e- _(~+;)2 ~
J
If(m)12 dm,
M
c>
E.
(1.6)
196
Relative Index Theory, Determinants and Torsion
Hence the estimate of
J J IW(~, m,p)1 2If(m)1 2dpdm
for s E
MM
[~, t] is done if
J
If(mW dm <
00
M
and then le-~D2 fl2 :S C 4 . IfIL2' where C 4 = C4 (t) contains a factor e- at , a > 0, if inf 0"( D2) > O. For (1.2) we have to estimate
JJIf(m)I-21(W(~,
m,p), 7]0P(p)')pI2 dp dm
(1.7)
MM
We recall a simple fact about Hilbert spaces. Let X be a Hilbert space, x E X,x 1= O. Then Ixl = sup l(x,y)l, IYI=l
Ixl 2 =
(sup
IYI=l
l(x,Y)lr.
(1.8)
:s
This follows from I(x, y) I Ixl . Iyl and equality for y = I~I' We apply this to E ---t M, X = L 2 (M, E, dp), x = x(m) = W(t, m,p), 7]0P(p).)p = W(t, m,p) o7]°P(p) and have to estimate
N(Cf» =
sup E Cgo(E) 1IL2 = 1
1(8(m), e- tD2 7]0PCf»IL2
sup
(1.9)
E Cgo(E) 1IL2 = 1
The heat kernel is of Sobolev class,
W(t, m,·)
IW(t, m, ')I H 2 :s C5 (t).
E H~(E),
(1.10)
Hence we have can restrict in (1.9) to sup
N(Cf»
(1.11)
E Cgo(E) 1IL2 = 1 II H ;, :::: C5
In the sequel we estimate (1.11). For doing this, we recall some simple facts concerning the wave equation aCf>s as
.
= 'l,DCf>s,
Cf>o
=
Cf>,
Cf> C
1
with compact support.
(1.12)
197
Trace Class Properties
It is well known that (1.12) has a unique solution CPs which is given by (1.13) and supp CPs C Uisl (supp cp)
Uisl =
lsi -
(1.14)
neighbourhood. Moreover,
We fix a uniformly locally finite cover U = {Uv},J = {Bd(Xv)}v by normal charts of radius d < rinj (M, g) and associated decomposition of unity {'Pv}v satisfying
l\7i 'Pvl
:S C for all v, O:Si:Sk+2
(1.16)
Write
N(CP)
I(8(m), e- tD2 ryOPcp) I
J -l +00
1 J41ft
1(8(m),
e
eiSD(ryOPIP)
ds)1
L2(dp)
-00
J -4~2 +00
1 J41ft
I
e
(eisDryoPIP)(m)
dsl
.
(1.17)
L2(dp)
-00
We decompose (1.18) v
(1.18) is a locally finite sum, (1.12) linear. Hence (1.19) v
Denote as above in particular
198
Relative Index Theory, Determinants and Torsion
Then we obtain from (1.15), (1.16) and an Sobolev embedding theorem
(1.21) !: - '!! r - 1 > !: 2 > i for r > n + 2 and since r - 1 - '!!t > 2 2' 2' IIH~ ::; 0 5 . This yields together with the Sobolev embedding the estimate
l/
mE U.(U.,)
< 0 11
, l/
mE U.(U.,)
0 12 , vol(B2d+lsl(m)) . (
vo
IB 1 ( ) ·17J11,r-1,B2d+ I•I(m») . 2d+lsl m (1.22)
There exist constants A and B, independent of m s. t.
vol(B2d+lsl(m)) ::; A. eBI • I . Write .2
e- 4t ·
9 .2 vol(B2d+lsl(m)) ::; 0 13 , e-W4t,
0 13 = A. elOB
2 t,
(1.23)
thus obtaining
0 14
=
0 12 .013
=
0 12 . A . e lOB2t .
Now we apply Buser/Hebey's inequality in chapter II, proposition 1.9,
Jlu - ucl M
dvolx(g) ::; O· c
Jl'\lul M
dvolx(g)
Trace Class Properties
199
for U E W1,1(M)nC1(M), c EjO, R[, Ric (g) ~ k, C and
uc(x)
:=
VOI~c(x)
J
= C(n, k, R)
u(y) dvoly
Bc(x)
with R
= 3d + s and infer
J
vo
M
IB 1
( ). 11J!t,r-l,B2d +181(m) dm
2d+lsl m
:::; 11J!t,r-l + C(3d + s) . (2d + s)I\71Jh,r-l :::; 11Jll,r-l + C(3d + s) . (2d + s)I1Jll,r-l.
(1.24)
C (3d + s) depends on 3d + s at most linearly exponentially, i. e.
C(3d+s)· (2d+s):::; A1e B1S • This implies
J J :::;= J : :; J 00
2
e- 10984t"
o
vo
M
IB 1 ( ) ·11Jll,r-l,B2d +181(m) dm ds (1.25) 2d+lsl m
00
e-!oft (11Jh,r-l + C(3d + s) . (2d + s)I1Jll,r-l) ds
o
00
e- ilift ds(I1Jll,r-l + AlelOBrtl1Jh,r
o
= Vi· ~J57i=(I1Jll,r-l + AlelOBrtl1Jll,r) < The function JR+
X
00.
M ---. JR, 2
(s, m) ---. e- 109 4t" 8
(
VO
IB 1 ( ) ·11Jll,r-l,B2d+181(m) ) 2d+lsl m
is measurable, nonnegative, the integrals (1.24), (1.25) exist, hence according to the principle of Tonelli, this function is 1summable, the Fubini theorem is applicable and
200
Relative Index Theory, Determinants and Torsion
1= 0)
is (for TJ
everywhere
=1=
0 and 1-summable. We proved
(1.26) Now we set
(1.27)
I(m) = (fj(m))! and infer I(m)
=1=
1/-1e-~D2
0 everywhere, 0
1 E L2
and
TJIL
JJl(m)-21((W(~, : ; Jfj(~) J
=
m,p), TJOP)pI2 dp dm
MM
fj(m)2 dm =
M
fj(m) dm
M
2 In 1 ~(I 1 ::; C12 · A· elOB sV8' "2v57f TJ 1,r-1
2 + Ale lOB s1 TJ 11,r) 1
(1.28)
::; C15Vse10B2sITJI1,r, i. e.
1/- 1e- 2D 8
!
2
0
!
TJI2 ::; C 125 '8 4 . e5B s . ITJlr,r' 1
2
(1.29)
Here according to the term A1e10Brs, C 15 still depends on We obtain
IIL2 . 1/-10 e-~D2 C4 1/1L2 . C1!25 • 8 41 . e5B2 s.
le-~D2
::;
0
TJI ! ITJIf,r 0
::; C4 . C15VselOB2SITJiI.r = C 16 . Vs' e This yields e- sD2 0 TJ is of trace class,
le- sD2 TJl1 ::; e- sD2 le- SD2
0
0
TJ TJ
0
0
le-~D2
0
8.
lOB2s
ITJiI,r' (1.30)
112 '1/-1e-~D2TJI2 ::; C16VselOB2SITJ11,r, (1.31)
,2
D' 0 e-(t-s)D is of trace class, D' 0 e-(t-S)D
1 \
< le- sD2 TJl1
. ID' e-(t-s) DI2 Iop
< C16VIn8e lOB2s1 TJ 1,r' C . 1
'
1
~'
t-8 (1.32)
Trace Class Properties
I
201
t
(e-
SD2
O'T) 0
D'
0
e-(t-s)D
I2
ds
t
2"
I t
:S
le-
sD2
'T} 0
D' e-(t-S) DI2 11 ds
t
2"
:S C 16 • C , . elOB
2
tI I It (t _ s )! S
'T} 1,r .
(1.33)
ds,
t
2"
[VS(t - s)
t
2s - t
t
t7r
tn
2
22
2 2
-- + -- = -( - -
I
t
+"2 arcsin -t-l~ 1)
'
t
(e-
SD2
o'T) 0
D'
0
e-(t-s)D
,2
ds
t
2"
1
:S C 16 . C , . e lOB t . ( 2"n - 1) '"2t I'T} Il,r 2
=
C17 e lOB2t . t·
I'T} Il,r·
(1.34)
Here C17 = C 17 (t) and C17(t) can grow exponentially in t if the volume grows exponentially. (1.34) expresses the fact that (14) is of trace class and its trace norm is uniformly bounded on any t-intervall lao, all, ao > O. The treatment of (II) - (13) is quite parallel to that of (14). Write the integrand of (1 3 ), (1 2 ) or (II) as (1.35) or ( 1.36)
Relative Index Theory, Determinants and Torsion
202
or (1.37) respectively. Then in the considered intervals the expressions [ ... J are of trace class which can literally be proved as for (14). The main point in (14) was the estimate of j- I e- TD2 'T/. In (1.36), TDI2 (1.37) we have to estimate expressions 'T/ej-l. Here we use the fact that 'T/ = 'T/OP is symmetric with respect to the fibre metric h: the endomorphism 'T/ei (.) is skew symmetric as the Clifford multiplication ei' which yields together that 'T/0P is symmetric. Then the Lrestimate of ('T/OP . W' ( T, m, p), .) is the same as that of W'(T,m,p),'T/°P(p)') and we can perform the same procedure as that starting with (1.6). The only distinction are other constants. Here essentially enters the equivalence of the D- and D'-Sobolev spaces i.e. the symmetry of our uniform structure. The factors outside [... J produce on [~, tl, and on [0, ~J (up to constants). Hence (Id - (h) are of trace class with uniformly bounded trace norm on any t-intervall [aD, all, aD > O. This finishes the proof of the first part of theorem 1.1. We must still prove the trace class property of
Js
e-tD2D -e -
vbs
tD,2 D'
.
Js
(1.38)
Consider the decomposition
e-tD2D -e -
.
tDI2 D'
(1.39)
tD2
tD,2 .
Accordmg to the first part, e-"2 - e-"2 IS for t > 0 of trace class. Moreover, e-~D2 D = De-~D2 is for t > 0 bounded, its operator norm is ::; ~. Hence their product is for t > 0 of trace class and has bounded trace norm for t E [aD, all, aD > O. (1.39)
Trace Class Properties
203
is done. We can write (1.40) as
t
J '2
+
e- sD\,D'e-(&-s)D
/2
/ ds](D'e-&D \
(1.41)
o
Now (1.42) (1.42) is of trace class and its trace norm is uniformly bounded on any lao, all. ao > 0, according the proof of the first part. If t
'2
we decompose
i
4
i
2
J = J + J then we obtain back from the integrals o
0
t
4
in (1.41) the integrals (II) - (I4), replacing t ---t ~. These are /2 done. D'e-&D generates C / yt in the estimate of the trace norm. Hence we are done. 0
2
Variation of the Clifford structure
Our procedure is to admit much more general perturbations than those of 'V = 'V h only. Nevertheless, the discussion of more general perturbations is modelled by the case of 'V -perturbation. In this next step, we admit perturbations of g, 'Vh,., fixing h, the topology and vector bundle structure of E ----+ M. The next main result shall be formulated as follows.
Theorem 2.1 Let E = (E, h, 'V = 'V h, .) ----+ (Mn, g) be a Clifford bundle with (1), (Bk(M,g)), (Bk(E, 'V)), k ~ r+1 > n+3, E' = (E, h, 'V' = 'V,h, .') ----+ (Mn, g') E gencomp~~d~Jf,F(E) n
204
Relative Index Theory, Determinants and Torsion
CLBN,n(I, B k ), D = D(g, h, \7 = \7 h, .), D' = D(g', h, \7' = \7,h, .') the associated generalized Dirac operators. Then for
t>O (2.1)
is of trace class and the trace norm is uniformly bounded on compact t-intervalls lao, al], ao > O. Here D'L is the unitary transformation of D,2 to L2 = L2 ((M, E), g, h). 2.1 needs some explanations. D acts in L2 = L 2((M, E), g, h), D' in L; = L 2((M, E), g', h). L2 and L; are quasi isometric Hilbert spaces. As vector spaces they coincide, their scalar products can be quite different but must be mutually bounded at the diagonal after multiplication by constants. D is self adjoint on V D in L 2 , D' is self adjoint on V D , in L; . L 2. H ence e- tD,2 an d e- tD2 - e- tD,2 but not necessan'1y m are not defined in L 2. One has to graft D2 or D,2. Write dvolq(g) == dq(g) = a(q) . dq(g') == dvolq(g'). Then
o < Cl
:S a(q) :S C2, a, a-I are (g, \79) and (g', \79') - boundedup to order 3,
la -
119,1,r+1,
la -
119',1,r+1
<
00,
(2.2)
since g' E compl,r+l(g). Define U : L2 ~ L;, U = a~. Then U is a unitary equivalence between L2 and L;, U* = U-l. D~2 := U*D'U acts in L 2, is self adjoint on U-l(V D,), since U is a unitary equivalence. The same holds for D'L = U* D,2U = (U* D'U)2. It follows from the definition of the spectral measure, the spectral integral and the spectral repreA2 dE~, e- tD ,2 = e- t )..2 dE~ that D'L = sentations D,2 = U* D,2U = U* A2 dE~U = A2 d(U* E~U) and
e- tD'I 2 =
!
J
J
J
e- t)..2 d(U* E~U)
J
= U*(! e- t)..2 dE~)U = U*e- tD'2u.
(2.3) In (2.1) e- tD'I 2 means e- tD'I 2 = e- t(U*D'U)2 = U*e- tD ,2 U . We obtain from g' E compl,r+1(g), \7,h E COmpl,r+l(\7hg), .' E l I d ' , compl,r+l(.), D - a-"2D'a"2 = D - D' - 9r~aa. and (2.2) the following lemma concerning the equivalence of Sobolev spaces.
205
Trace Class Properties
Lemma 2.2 W 1,i(E, g, h, 'Yh) = W 1,i(E, g', h, 'Y,h) as equivalent Banach spaces, 0 ::; i ::; r + 1.
o Corollary 2.3 W 2,i(E, g, h, 'Yh) alent Hilbert spaces, 0 ::; j ::;
r!l.
= W 2,i(E, g', h, 'Y,h) as equiv0
Corollary 2.4 Hj(E, D) ~ Hj K(E, D'), 0::; j ::;
r!l.
0
2.2 has a parallel version for the endomorphism bundle EndE. Lemma 2.5 n1,1,i(EndE, g, h, 'Yh) O::;i::;r+1. Lemma 2.6 0< J' < r+1. 2
e- tD ''i2
W'
L2
:
== WL"
----+
n1,1,i(EndE, g', h, 'Y,h) o
C::!
n1 ,2,j (EndE, g', h, 'Y,h) o
L2 has evidently the heat kernel
Our next task is to obtain an explicit expression for
2
tD2
n1,2,j(EndE, g, h, 'Yh)
C::!
tD
e- e- ''i 2 • For this we apply again Duhamel's principle. The steps 1) - 4) in the proof of lemma 2.1 in chapter III remain. Then we set 4>(t,q) = W(t,m,q), w(t,q) = WL(t,m,q) and obtain f3
- j j hq(W(T, m, q), (D2 Q
+ :t)WL(t -
M
= j[hq(W({3, m, q), W£2(t - (3, q,p) M
-hq(W(a, m, q), W£2 (t - a, q,p)] dq(g).
T, q,p)) dq(g) dT
206
Relative Index Theory, Determinants and Torsion
Performing a yields
----t
0+, (3
----t
t and using dq (g)
= a (q) dq (g')
t
- j j hq(W(s, m, q), (D2
+ !)W'(t -
s, q,p))dq(g)ds
°M
t
= - j j[hq(W(S, m, q), (D2 -
D'L)W~2(t -
°M
= W(t, m,p)a(p) - WL(t, m,p).
s, q,p)dq(g)ds (2.4)
(2.4) expresses the operator equation t
- j e- sD2 (D2 - D'L)e-(t-S)D'i2 ds. e- tD2 a _ e- tD'i 2 e- tD2 _ e- tD'i 2
°2 e-tD (a -1) + e- tD2 - e- tD'i2, hence _e-tD 2(a - 1) t
- j e- SD2 (D2 _ D'i 2)e-(t-S)D'i 2 ds.
°
(2.5)
Jt:tt;, -
As we mentioned in (2.2), (a - 1) = ::(~}) - 1 = 2 1 E 0°,1,1'+1 since 9 E COmp1,1'+1(g). We write e-tD (a - 1) = (e-~D2 f)(f-1e-~D2 (a -1)), determine f as in the proof of theorem 1.1 from T/c. = a-I and obtain e- tD2 (a - 1) is of trace class with trace norm uniformly bounded on any t-interval lao, a1], ao > O. Decompose D2 - D'L = D(D - D~J + (D - DL)DL. We need explicit analytic expressions for this. D(D - DL) = D(D - a-!D'a!) = D(D - D') - D gr~'aJ, (D - DL)D~2 = ((D - D') - gra;a'aJ)a-!D'a!. If we set again D - D' = -TJ J then we have to consider as before with gr;~ 'a = gra;a'a where
Trace Class Properties
grad'
==
grad
207
g'
grad 'a)D' e-(t-s)D'L 2 d s 2a L2 t
J +J
e -SD2D( 'TJ -
+
grad 'a) e -(t-s)D,2L2 ds 2a
t
"2
t
e- SD2 ('I1- grad'a)D' e-(t-s)D'L d '/ 2a L2 2 s,
t
"2
It follows immediately from g' E COmpl,r+l (g) that the vector field gra:' a E nO,l,r (T M). If we write 'TJr! = a.' then 'TJgP is a zero order operator, l'TJolr < 00 and we literally repeat the ad' , procedure for (11) - (14) as before, inserting 'TJo = - gr a a· for 'TJ there. Hence there remains to discuss the integrals
gr:'
J t
J t
e- sD2 D'TJ e -(t-S)D'L2 ds
°
+
e- sD2 'TJD~2e-(t-s)D'L2 ds.
(2.6)
°
The next main step is to insert explicit expressions for D - D'. Let mo E M, U = U(mo) a manifold and bundle coordinate neighbourhood with coordinates Xl, ... , xn and local bundle basis
ax:; \7<Pa = dx i ®rfa
-b .'
_ ,ik D 'm. 'i.' g
axa , n'm. k '
Y i'i.'.
(2.7)
Relative Index Theory, Determinants and Torsion
208
This yields
8
ik
-(D - D')Cf>
"("'7
9 -. [( 9
ik
- 9
+glik
if-.
V i'¥ -
8xk
lik)
9
8
lik
8
- k.
1 "("'71 if-.
V·'¥
8x
"("'7
8x k ' v i
t
+ 9 lik 8x8 k
'
("("'7
8~k (. - ./)\7~lCf>,
"("'7/)
v i-V i
(2.8)
i. e. we can write -(D - D')Cf> = ('r/fP
+ 'r/i + 'r/~P)Cf>,
(2.9)
where locally (9
9
ik
lik
glik
Here
(glik)
- 9
8
lik)
8x k '
8
("("'7
8~k (. -
"("'7
if-.
8x k ' v i '¥, "("'7/) if-.
(2.10)
v i - V i '¥,
(2.11)
\7~Cf>.
(2.12)
./)
= (g}Z)-l. We simply write 'r/v instead 'r/~, hence
J + J t
(2.6)
=
e-
sD2
D('r/1
+ 'r/2 + 'r/3)e-(t-S)D'i 2 ds + (2.13)
o
t
e-
sD2
('r/1
+ 'r/2 + 'r/3)D~2e-(t-S)D'i2
ds. (2.14)
o We have to estimate
J t
e- sD2 D'r/v e -(t-s)D''i 2 ds
(2.15)
D' e-(t-s)D''i 2 ds e -sD2'n 'IV L2 •
(2.16)
o and
t
J o
209
Trace Class Properties
t
Decompose
~
t
J = J + J which yields ~Ot
"2
t
J "2
e- sD2 D'fJv e -(t-S)D''i 2
ds,
o t
"2
J J
e- sD2 'Yl D' e-(t-s)D''i 2
·,v
L2
ds ,
o
t
e- sD2 D'fJv e -(t-S)D''i 2
ds,
t
"2
J t
e-
sD2
'fJvD~2e-(t-S)D''i2
ds.
t
"2
(Iv,l) - (Iv,4) look as (II) - (14) as before. But in distinction to that, not all 'fJv = 'fJ':! are operators of order zero. Only 'fJ2 is a zero order operator, generated by an EndE valued I-form 'fJ2. 'fJl and 'fJ3 are first order operators. We start with 1/ = 2, 'fJ2' /'fJ2/1,r < 00 is a consequence of E' E comp~~dtff(E) and we are from an analytical point of view exactly in the situation as before. (h,d-(I2,4) can be estimated quite parallel to (I1)-(I4) and we are done. There remains to estimate (Iv,j), 1/ i- 2, j = 1, ... ,4. We start with 1/ = 1, j = 3 and write
(2.17) De-~D2 and
are bounded in [~, tJ and we perform their estimate as in section 1. e-~D2 . f is Hilbert-Schmidt if f E L 2 . There remains to show that for appropriate f e-(t-s)D,2
f-le-~D2 'fJ1
210
Relative Index Theory, Determinants and Torsion
is Hilbert-Schmidt. Recall r + 1 > n + 3, n 2: 2, which implies r.2 > !!2 + 1' r - 1 - n -> r.2 - ! !2' r - 1 > r. 2 -> i. If we write in - 2' the sequel pointwise or Sobolev norms we should always write IWlgl,h,m l , IwIW(E,DI), Iw lgl,h,'V",2,¥, Ig - g'lgl,m, Ig - g'lgl,l,r etc. or the same with respect to g, h, V, D, depending on the situation. But we often omit the reference to g', h, V', D, m, g, h . .. in the notation. The justification for doing this in the Sobolev case is the symmetry of our uniform structure. Now
To estimate
n
2
L I~ I k=l x g,m
more concretely we assume that
xl, ... ,xn are normal coordinates with respect to g, i.e. we assume a (uniformly locally finite) cover of M by normal charts of fixed radius::; rinj(M, g). Then 1~1;,m = 9 (~,~) = gkk (m), and there is a constant C 2 = C2 (R, rinj (M, g)) s. t. n
(
2) ~ ::; C
~ IVilh,m
2.
Using finally
IVx I ::;
IXI .
IV I,
we
obtain (2.19) (2.19) extends by the Leibniz rule to higher derivatives IVk1h lm' where the polynomials on the right hand side are integrable by the module structure theorem (this is just the content of this theorem). (2.18), (2.19) also hold (with other constants) if we perform some of the replacements 9 ~ g', V ~ V': We remark that the expressions D(g, h, V h,., D(g', h, vh, .) are invariantly defined, hence
[D(g, h, vh,·)
- D(g, h, V h, ·)](lu) = ((gik - g'ik)8k)· Vi(lu). (2.20)
Trace Class Properties
211
We have to estimate the kernel of
hp(W(t, m, p), r/(·)
(2.21)
in L 2 ((M, E), g, h) and to show that this represents the product of two Hilbert-Schmidt operators in L2 = L2((M, E), g, h). We cannot immediately apply the procedure as before since r/fP is not of zero order but we would be done if we could write (2.21) as (2.22)
710J
71~~ of first order,
tV
of zeroth order. Then we would replace by 710 (p)W(t, m,'p), apply k ~ r + 1 > n + 3, and obtain
710 W(t, m,')
E H~ (E),
IW(t, m, ')I H 2 ::; C(t)
(2.23)
and would then literally proceed as before. Let E C':' (U). Then
j (W(t, m, p), 71((p) (p))p dvolp(g) = j(((gik _ glik)8k). V'iW,
j (W, 71((L
L j(V'et,i W , ((g~k - i:)8et ,k'
- L et
J
(W, (V' et,i((g: -
g'~)8k)) .
Relative Index Theory, Determinants and Torsion
212
J(L -J L =-
'\l Q,i W,
g'~)aQ,k) . if»
-
(2.24)
Q
(W,
g/~)ak)) . if»
(2.25)
Q
Using (2.24), (2.25), we write
I(5( m), etD2 1]?if» IL2(M,E,dp) I(W (t, m, p), 1]?if»p IL2(M,E,dp) I(1]0(p)W(t, m,p), 1]~:oif»p +
N( if»
(W(t, m,p, 1]~,Oif»)pIL2(M,E,dp).
+
(2.26)
Now we use l'\l xxi :s: IXI . l'\lxl, that the cover is u.l.f. and I'\lWI :s: C 1 . (lDWI + W) (since we have bounded geometry) and obtain N(if»
< C· (I(DW(t, m,p), 1]~:Oif>IL2(M,E,dp) +IW(t, m,p, 1]~,oif>IL2(dp) C· (N1(if» + N2(if»).
(2.27)
Hence we have to estimate sup
N1(if» =
E Cg"(E) 1IL2 = 1
sup E Cg"(E) 1IL2 = 1
(2.28) and sup
N 2(if» =
E Cg"(E) 1IL2 = 1
sup E Cg"(E) 1IL2 = 1
(2.29) According to k > r
+ 1 > n + 3,
D(W(t, m, .), W(t, m,·) E H~ (E), I(D(W(t,m, ')I H2 , IW(t,m, ')I H2 and we can restrict in (2.28), (2.29) to sup E Cg"(E) 1IL2 = 1 IIH~ :<::: Cl(t)
Ni(if».
:s: C1(t)
(2.30)
(2.31)
213
Trace Class Properties
7}~, , 7}~~ , ,0 are of order zero and we estimate them by
C· Ig - g'19,2,~ < C'lg - g'lg,1,r-1 (2.32) D '1\7(g - g')19,2,~ < D'I\7(g - g')lg,1,r-1 < D"I'I (233) 9 - 9 g,l,r .
and
respectively. As we have seen already, into the estimate (2.33) enters 1\77}h,r-b i. e. in our case 1\72(g-g')lr_1 Ig-g'lr+1. For this reason we assumed E' E compi,~tlf,F(E). In the expression for N1 (
+oo
=
N 1 (
1
1 r;c;y47ft 2t
J
l
s· e
_~ isD 4te
op
(2.34)
7}lo(m) ds . '
00
We estimate in (2.34) s· e-isft by a constant and write 17. 2
e- 184t . vol(B 2d+s (m))
~
9.2
C· e- W4t
and proceed now for N1 ( o. (I1,3) is done. (I1,4) is absolutely parallel to (h,3), even better, since the left hand factor D is
vb which
missing. ID~2e -(t-s) Dt2Iop now produces the factor is integrable over [~, t]. Write the integrand of (h,l) as
(2.35) We proceed with (2.35) as before. Here 7}1 already stands at the right place, we must not perform partial integration. Into the estimate enters again the first derivative of W'. De- sD2 generates the factor which is intgrable on [0, We write (I1,2) as
Js
H
t
"2
J
e- sD [(7}l e-
o
2
~D'2 4
L2f-1)(Je-
~D'2 4
L2)]e-
(t-.) 2
D'2 , L2D12 ds (2.36)
214
Relative Index Theory, Determinants and Torsion
and proceed as before. Consider finally the case v OPif..
'fl3 '±'
= 3, locally
= 9 lik axak (' - . ')'r'7' V i'±', if..
The first step in this procedure is quite similar as in the case v = 1 to shift the derivation to the left of Wand to shift all zero order terms to the right. Let X be a tangent vector field and <1> a section. Lemma 2.7 X(· - ./)'\7~<1>
=
'\7~(X(·
- .')<1»
+ zero order terms.
X(· - .')'\7~<1> = [X(· - ./)'\7~<1> - '\7~(X(- - .')<1»] + - .')<1». We are done if [... ] on the right hand side contains no derivatives of <1>. But an easy calculation yields
Proof. '\7~(X(·
[X(· - ./)'\7~<1> - '\7~(X(- - ./)<1»]
= X . ('\7~ - '\7i)<1> +('\7~
=
('\7~
- '\7i)(X . <1» - '\7i)X ., <1> + ('\7 i X) (.1 - .)<1>.
(2.37)
o Hence for <1>, W E Cr: (U)
J a: ./)'\7~<1»pdp(g) J '\7~(glik a: +J a: ('\7~ ('\7~ h(w, glik
=
k (. -
h(w,
h(w, glik
+('\7~ -
k (. - ./)<1»p dp(g)
k
- '\7i)<1> -
'\7i)X ., <1> + ( '\7ig
,ik
a:
+
- '\7 i )glik
(2.38)
a:
k . <1»p +
k ) (.1 - .)<1»p dp(g). (2.39)
(2.38) equals to
Jh('\7~*W, a: glik
k (. - ./)<1»p dp(g).
(2.40)
215
Trace Class Properties
If is Sobolev and W = W then we obtain again by a u.l.f. cover by normal charts {UoJa and an associated decomposition of unity {'Pa}a
J =J ~ g'~ a~k (. - ·')V'~,i('Pa
dp(g) =
a
.')
a
h(W,
+ (2.41)
a
dp(g),
(2.42)
where 7]~o is the right component in h(·,·) under the integral (2.41), multiplied with 'Pa and summed up over a. Now we proceed literally as before. Start with
J J(De-~D2) e-~D2 t
(13,3) =
e- SD2 D7]ie-(t-S)Dt2 ds
=
t
"2
t
[(
f) (f-le-~D2 7]i)]e -(t-S)Dt2 ds.
t
"2
(2.43) We want that for suitable f E L 2 , f-le~D27]~P is Hilbert-Schmidt. For this we have to estimate h(W(t,m,p),7]~P.)p and to show it defines an integral operator with finite L 2 ((M, E), dp)-norm. We estimate tD2 N( )pIL2((M,E),dp)' (2.45) Using (2.41) and (2.42), we write Ih(W(t,m,p), 7]i
N(
Ih(7]~l W(t, m,p), 7]~;o
+
h(W(t, m,p), 7]~;o,o
(2.46)
216
Relative Index Theory, Determinants and Torsion
Now we use IV'~ *xl :S C11V'~xl :S C2IXI·IV"xl :S C3 IXI(IV'xl + Ixl), that the cover is u.l.f. and IV'WI :S C4 (IDWI + IWI) and obtain
N(
< C(lhDW(t, m,p), 1]~;0
Here we again essentially use the bounded geometry. Hence we have to estimate sup
N 1 (
sup E Cgo(E) 1IL2 = 1
E Cgo(E) 1IL2 = 1
(2.47) and sup
N 2 (
E Cgo(E) 1IL2 = 1
sup
1(<5 (m), e-tD21]~;0,0
E Cgo(E) 1IL2 = 1
(2.48) According to k > r
+ 1 > n + 3,
DW(t, m, .), W(t, m, .)
E H~(E),
IDW(t, m, ·)I H 5, IW(t, m, ')I H 5 :S C1 (t)
(2.49)
and we can restrict on (2.48), (2.49) to sup
(2.50)
E Cgo(E) 1IL2 = 1 1 IH 5 ::; C1(t)
1]~0, 1]~;0,0 are of order zero and can be estimated by
Col' - ., 12,~ :S C11. - ., 11,r-1 and
respectively.
(2.51)
217
Trace Class Properties
Now we proceed literally as for (h,3), replacing (2.34) by
Nl(
1
1
;-c:;y4~t2t
J +oo
l
8
2
.
se-TtetSDr/fo(m) ds . '
(2.53)
00
(13,3) is done, (13,4), (13,1), (13,2) are absolutely parallel to the case /J = 1. o This finishes the proof of theorem 2.1. Theorem 2.8 Suppose the hypotheses of 2.1. Then
is of trace class and the trace norm is uniformly bounded on compact t-intervalls lao, ad, ao > O. Proof. The proof is a simple combination of the proofs of 1.1 and 2.1. 0 Now we additionally admit perturbation of the fibre metric h. Before the formulation of the theorem we must give some explanations. Consider the Hilbert spaces L 2 (g, h) = L 2 ((M, E), g, h), L 2 (g', h) = L 2 ((M, E), g', h), L 2 (g', h') = L 2 ((M, E), g', h') L~ and the maps
i(g',h},(g',h'} : L 2 (g', h) U(g,h},(gl,h} : L 2 (g, h) where dp(g)
D~2(9,h)
---t ---t
L 2 (g', h'), i(gl,h),(gl,hl} = L 2 (g', h), U(g,h},(g',h} = a~
= a(p)dp(g'). Then we set D~2 := U(g,h), (gl ,h) i(gl ,h}, (g',h') D' i(gl ,h),(g' ,hi} U(g,h},(gl ,h} == U*i* D'iU. (2.54)
i;
Here i* is even locally defined (since g' is fixed) and = dual"hlo i' odualh', where dualh((p)) = hp (" (p)). In a local basis field 1 , ... ,N, (p) = e(p)i(p), (2.55)
218
Relative Index Theory, Determinants and Torsion
It follows from (2.55) that for h' E COmp1,r+1 (h) i*, i*-l are bounded up to order k,
i* - 1, i*-l - 1 E n,0,I,r+I(Hom((E, h', 'Vh') ----+
----+
(M,g'), (E, h, 'V h) ----+ (M,g')))
(2.56)
and i* -1,i*-1 -1 E n,0,2,!:f-(Hom((E,h', 'Vh')----+ ----+
(M, g'), (E, h, 'V h )
----+
(M, g'))).
(2.57)
D' == D' is self adjoint on DD' = ego (E) I lv, , where JJ~, = JJi,2 + JD'Ji,·2 i: L 2 (g', h) ----+ L 2 (g', h') == L~ and i* : L 2 (g', h') ----+ L 2 (g', h) are for h' E compl,r+l(h) quasi isometries with bounded derivatives, they map er;:(E) 1-1 onto er;:(E) and i* D'i is self adjoint on ego(E)lli*D'i = Di*D'i C L 2((M, E), g', h) == L 2 (g', h). We obtain as a consequence that e- t(i*D'i)2 is defined and selfadjoint in L 2 ((M, E), g', h) = L 2 (g', h), maps for t> 0 and i,j E Z Hi(E,i*D'i) continuously into Hj(E,i*D'i) and has the heat kernel W~"h(t, m,p) = (o(m), e- t(i*D'i)2 o(p)), W'(t, m,p) satisfies the same general estimates as W(t, m,p). By exactly the same arguments we obtain that e-W*(i* D'i)2U = e-t(U*i* D'iU)2 = U*e-t(i* D'i)2U is defined in L2 = L 2((M, E), g, h), self adjoint and has the heat kernel W{ 2 (t, m, p) = W g' , h (t, m, p) = a-! (m)W~"h(t, m,p)a(p)!. Here we assume g' E compl,r+l(g). Now we are able to formulate our main theorem.
Theorem 2.9 Let E = ((E, h, 'V = 'Vh,.) ----+ (Mn, g)) be a Clifford bundle with (1), (Bk(M, g)), (Bk(E, 'V)), k ~ r + 1 > n+3, E' = ((E,h,'V' = 'V h',.') ----+ (Mn,g)) E gencompZ~d;;f (E)nCLB N ,n(1, B k), D = D(g, h, 'V = 'V h, .), D' = D(g', h, 'V' = 'V h' ,.') the associated generalized Dirac operators, dp(g) = a(p)dp(g'), U = a!. Then for t > 0 e -tD2 - U* e -t(i* D'i)2U
(2.58)
is of trace class and the trace norm is uniformly bounded on compact t-intervalls lao, all, ao > O.
219
Trace Class Properties
Proof. We are done if we could prove the assertions for e- t (UD'u*)2 _ e- t (i*D'i)2 = Ue- tD2 U* _ e- t(i*D'i)2 (2.59) since U*(2.59)U = (2.58). To get a better explicit expression for (2.59), we apply again Duhamel's principle. This holds since Greens formula for U D 2U* holds,
J
hq(UD 2U*if?, w) - h(if?, UD 2U*w) dq(g') = O.
We obtain t
-JJhq(a~(m)W(s,m,q)a-~(q), o M
(UD 2U*
+ %t) W~"h(t -
JJhq(a~
s,q,p)) dq(g') ds
t
=-
o
(m)W(s, m, q)a-~ (q),
M
(U D 2U* - (i* D'i)2)W~"h(t - s, q, p) dq(g')) ds
= a~ (m)W(t, m, q)a-~ (q) - W~"h,(t, m,p) = Wg',h(t, m,p) - W~"h(t, m,p).
(2.60)
(2.60) expresses the operator equation e-t(U DU*)2 -e -t(i* D'i)2
-J -J t
e- s(U*DU)2((UDU*)2 - (i*D'i)2)e-(t-S)(i*D'i)2 ds
o
t
e- s(UDu*)2 UDU *(UDU* - i* D'i)e-(t-S)(i*D'i)2 ds
o
(2.61)
J t
e-s(UDU*)\U DU* - i* D'i)(i* D'i)e-(t-s)(i* D'if ds.
o
(2.62)
220
Relative Index Theory, Determinants and Torsion
We write (2.62) as
-J t
a~e-sD2 Da-~(a~Da-~ - i* D'i)e-(t-S)(i*D'i)2 ds
o t
J = - Ja~e-SD2Da-~i*((i*-1 a 12 e- sD2 D a _12 (D
=-
-
'*D" ~
~
-
o
grad a.) e -(t-s)(i* D'i)2 ds 2a
t
- l)D
+ (D -
D')
o
_i*-l grad a· )e-(t-s)(i* D'i)2 ds 2a
Ja~e-sD2 t
=
D('r/o
+ 'r/l + 'r/2 + 'r/3 + 'r/4)e-(t-s)(i*D'i)2
ds,
o
'r/o
=
= -a-~i*'r/i(2), i = 1,2,3, 'r/l(2) = (2.10), = (2.11), 'r/3(2) = (2.12), 'r/4 = a-~i*-l(i* - l)D. Here grad3 a',
'r/i
2a2"
'r/2(2) 'r/o and 'r/2 are of zeroth order. 'r/l and 'r/3 can be discussed as in (2.18)-(2.53). 'r/4 can be discussed analogous to 'r/b 'r/3 as before, i.e. 'r/4 will be shifted via partial integration to the left (up to zero order terms) and a-~i*(i* -1) thereafter again to the right. In the estimates one has to replace W by DW and nothing essentially changes as we exhibited in (2.35). We perform in (2.62) the same decomposition and have to estimate 20 integrals, t
J 2
a~ e- sD2 D'r/ve-(t-S)(i* D'i)2 ds,
o t
2
J J
-(t-s)(i* D'i)2 ds, a 12 e -sD2 'r/v ('*D") ~ ~ e
o
t
a~e-sD2 D'r/v e -(t-s)(i*D'i)2 ds,
t
2
Trace Class Properties
J
221
t
a! e- sD2 'TJv(i*D'i)e-(t-s)(i*D'i)2 ds,
(Iv,4)
t
'2 1/ = 0, ... ,4 and to show that these are products of HilbertSchmidt operators and have uniformly bounded trace norm on compact t-intervals. This has been completely modelled in the proof of 2.1. 0
Finally we obtain Theorem 2.10 Assume the hypotheses of 2.9. Then for t > 0
is of trace class and its trace norm is uniformly bounded on compact t-intervalls lao, al]' ao > O.
o The operators i* D,2 i and (i* D'i)2 are different in general. We should still compare e- ti*D'2i and e-t(i* D'i)2 . Theorem 2.11 Assume the hypotheses of 2.9. Then for t > 0
is of trace class and the trace norm is uniformly bounded on compact t-intervalls lao, al]' ao > O.
Relative Index Theory, Determinants and Torsion
222
Proof. We obtain again immediately from Duhamel's principle ,2 e -ti* D i - e -t(i* D' i)2 =
J =- J J t
e- S(i*D '2 i)(i* D,2 i - (i* D'i)2)e-(t-s)(i*D i)2 ds
=-
1
=
o
t
e- S(i*D '2 i)i* D'(l- ii*)D'ie-(t-S)(i*D' i)2 ds
=
o
t
I2 e- S(i*D i)(i*D'i)C 1(1_ ii*)i*-1(i*D'i)e-(t-s)(i*D'i)2 ds.
=-
o
(2.63)
W:_
In [~, t] we shift i* D'i again to the left of the kernel
S
(i*D 2 i)
via partial integration and estimate I2 I2 (i* D' ie-~(i* D i») [( e-1W D i») f) U-1e-1(i*DI2i)i-1(1_ ii*)i*-l)] (( i* D' i)e-(t-s)(i* Dli)2)
as before. In [0, ~] we write the integrand of (2.63) as (e-S(i* D'2 i)i* D' i)[( (i*it 1e- t 4s (i* D'i)2 f-1) U e- t 4
8
(i* Dli)2) 1
(e- t;s (i* D ' i)2 (i* D'i))
and proceed as in the corresponding cases.
o
Theorem 2.12 Assume the hypotheses of 2.9. Then for t > 0
is of trace class and the trace norm is uniformly bounded on any t-intervall lao, a1], ao > o.
Proof. This immediately follows from 2.9 and 2.11.
0
223
Trace Class Properties
3 Additional topological perturbations Finally the last class of admitted perturbations are compact topological perturbations which will be studied now. Let E = ((E, h, \7 h) ~ (Mn, g)) E CLBN,n(I, B k) be a Clifford bundle, k 2:: r + 1 > n + 3, E' = ((E, h', \7h') ~ (Mm, g')) E compi:~dthrel(E) n CLBN,n(I, Bk)' Then there exist K c M, K' C M' and a vector bundle isomorphism (not necessarily an isometry) f = (JE,jM) E 15 1,r+2(EIM\K,E /IM'\K') s. t.
gIM\K and f'Mg'IM\K are quasi isometric,
(3.1)
hIEIM\K and f~h/IEIM\K are quasi isometric,
(3.2)
IgIM\K - f'Mg ' IM\Klg,l,r+1 <
(3.3)
00,
IhIEM\K - f~h/IEIM\Klg,h,Y'h,1,r+1 < 00, h l\7 IEIM\K - f~ \7h'IEIM\K Ig,h,Y'h,1,r+1 < I . IM\K -
f;'; ., IM\Kl g,h,Y'h,l,r+1 <
(3.4) (3.5)
00,
(3.6)
00.
(3.1) - (3.6) also hold if we replace f by f- 1 , M \ K by M' \ K' and g, h, \7h,. by g', h', \7h', .1. If we consider the complete pull back fE( E'l M'\K' ), i.e. the pull back together with all Clifford datas, then we have on M \ K two Clifford bundles, EIM\K, fE(E/IM'\K' which are as vector bundles isomorphic and we denote f;';(E/IM'\K' again by E' on M \ K, i.e. g~ew = (JMIM\K)*g~ld etc .. (3.1) - (3.6) and the symmetry of our uni1,r+1 . 1 form st ruc t ure UL,dij j,rel 1mp y
W1,i(EIM\K) ~ W1,i(E/IM\K), W2,j(EIM\K)
0:::; i :::; r
~ W2,j(E/IM\K), 0:::; j
Hj(EIM\K, D)
~ Hj(E/IM\K, D'),
:::;
r; r;
0:::; j :::;
n1,1,i(End(EIM\K)) ~ n1,1,i(End(E/IM\K)), n1,2,j(End(EIMW))
+ 1,
1,
(3.7)
1,
0:::; i :::; r
~ n1,2,j(End(E/IM\K)), 0:::; j
:::;
+ 1,
r;
1.
224
Relative Index Theory, Determinants and Torsion
Here the Sobolev spaces are defined by restriction of corresponding Sobolev sections. We now fix our set up for compact topological perturbations. Set'H = L 2((K,EIK),g,h) EB L2((K',E'IK,),g',h') EB L 2 ((M \ K, E), g, h) and consider the following maps iL2,K' : L 2 ((K', E'IK'), g', h') iL2,K,(, C
1
:
'H,
L 2((M' \ K', E'IM'\K,),g', h') ---+
i-1
---+
L 2((M' \ K', E'IM'\K'), g', h),
= ,
U* : L 2 ((M' \ K', E'IM'\K'), g', h) ---+
L 2 ((M' \ K', E'IM'\K'), g, h),
U* = a-! ,
where dq(g) = a(q)dq(g'). We identify M \ K and M' \ K' as manifolds and E'IM'\K' and EIM\K as vector bundles. Then we have natural embeddings L 2 ((M, E), g, h) ---+ 'H, i L2 ,K' EB U*i- 1 : L 2((M',E'),g',h') ---+ 'H, (iL 2,K' EB U*C1) = i L2 ,K,XK' + U*i-1XM'\K,. i L2 ,M
:
The images of these two embeddings are closed subspaces of 'H. Denote by P and P' the projection onto these closed subspaces. D is defined on V D C im P. We extend it onto (im P)l.. as zero operator. The definition of (the shifted) D' is a little more complicated. For the sake of simplicity of notation we write U*i- 1 == i L2 ,K' EB U*i- 1 = id EB U*i- 1, keeping in mind that i L2 ,K' fixes XK, and the scalar product. Moreover we set also iUXK, = U*i*XK' = XK,. Let E VD" XK, + U*i-1XM,\K, its image in 'H. Then (U*i* D'iU) (XK, + U*i-1XM'\KI, = XK'D' + U*i*XM'\K,D'. Now we set as VU*i* D'iU C H
Trace Class Properties
225
It follows very easy from the selfadjointness of D' on 'DD' and (3.7) that U*i* D'iU is self adjoint on 'DU*i*D'iU, if we additionally set U*i* D'iU = 0 on (im P').l. Remark 3.1 If 9 and h do not vary then we can spare the whole i - U-procedure, i = U = id. Nevertheless this case still includes interesting perturbations. Namely perturbations of \7,. and compact topological perturbations. 0 We set for the sake of simplicity fy = U*i* D'iU. The first main result of this section is the following Theorem 3.2 Let E = ((E, h, \7 h ) ---+ (Mn, g)) E CLBN,n (I, B k ), k 2: r + 1 > n + 3, E' E gencomptd~;f,rel(E) n CLBN,n (I, Bk). Then for t > 0
and
2 - 2 e-tD P -e -tD' p'
(3.9)
2 - 2 e-tD D -e -tD' D-'
(3.10)
are of trace class and their trace norms are uniformly bounded on any t-intervall [aD, al]' aD > O. For the proof we make the following construction. Let V c M\K be open, M \ K\ V compact, dist(V, M \ K\(M\K)) 2: 1 and denote by B E L('H) the multiplication operator B = Xv. The proof of 3.2 consists of two steps. First we prove 3.2 for the restriction of (3.9), (3.10) to V, i.e. for B(3.9)B, thereafter for (1 - B)(3.9)B, B(3.9)(1 - B) and the same for (3.10).
226
Relative Index Theory, Determinants and Torsion
Theorem 3.3 Assume the hypotheses of 3.2. Then
B(e- tD2 P - e- uY P')B, B(e- tD2 D - e- uY [Y)B, B(e- tD2 P - e- tlY P')(1 - B), (1 - B) (e- tD2 P - e- uY P')B, B(e-
tD2
D - e- tiY 15')(1 - B),
(1 - B) (e- tD2 D - e- uY [Y)B, 2 tfy2 (1 - B) (e-tD P - eP')(1 - B), 2 tlY (1 - B) (e-tD D - e15')(1 - B)
(3.11) (3.12) (3.13) (3.14) (3.15) (3.16) (3.17) (3.18)
are of trace class and their trace norms are uniformly bounded on any t-intervall lao, all, ao > o. 3.2 immediately follows from 3.3. We start with the assertion for (3.11). Introduce functions
E(t, m,p) := '"Y(m)W(t, m,p)(1-
+ 'ljJ(m)W'(t, m,p)
Applying Duhamel's principle yields f3
- j j hq(W(s,m,q), (%t
+D2) E(t-s,q,p))Xv(p) dq(g) ds
M
a
= j[hq(W((3, m, q), E(t -
(3, q,p)) - hq(W(a, m, q),
M
E(t - a, q,p))]Xv(p) dq(g).
(3.19)
Trace Class Properties
Performing a
-t
0+, (3
r
-t
f3
- j j hq(W(s,m,q), (D2+
227
in (3.19), we obtain
!) E(t-s,q,p))Xv(p) dq(g) ds
M
Q
= i~~- j[hq(W((3, m, q), E(t -
(3, q,p))Xv(p) dq(g)
M
(3.20)
-E(t, m,p)xv(p). Now we use
Xv(p)(1 -
(3.21)
and obtain
i~~- j
[h q(W ((3, m, q), E (t - (3, q, p) ) Xv (p) dq (g)
M
= i~~ j[hq(W((3,m,q),'lj;(q)W'(t - (3,q,p))
= W(t,m,p) _
- 2
since W'( T, q, p) is the heat kernel of e- rD'
.
This yields
t
- j j hq(W(s,m,q), (D2+ %t) E(t-s,q,p))Xv(p) dq(g) ds o M t
= - j j hq(W(s, m, q), (D2'lj;(q) _ 'lj;(q)fy 2) o M
W'(t - s, q,p))Xv(p) dq(g) ds = [W(t,m,p) - W'(t,m,p)]' Xv(p).
(3.22)
(3.22) expresses the operator equation t
(e- tD2 P _ e- tlY P')B = - j e- sD2 (D2'lj; - 'lj;1}2)e-(t-s)iY B ds o
(3.23)
Relative Index Theory, Determinants and Torsion
228
in '}-{ at kernel level. We rewrite (3.23) as in the foregoing cases. t
(3.23)
-J
e- SD2 (D(D - fy)'IjJ
+ (D -
fy)fy'IjJ
o +fy2'IjJ _ 'ljJfy 2)e-(t-S)IY ds
-[!
e-,v,(D(D - jj'),pe-I'-')D" ds
(3.24)
t
J J J J '2
+
e- sD2 (D - D')D''ljJe-(t-S)IY ds
(3.25)
e- sD\D,2'IjJ - 'ljJD,2) ds
(3.26)
e- sD2 D(D - D')'ljJe-(t-s)JY ds
(3.27)
e- sD2 (D - D')D''ljJe-(t-s)JY ds
(3.28)
o
t
'2
+
o
t
+
t
'2
t
+
t
'2
Write the integrand of (3.27) as (e-~D2 D) [(e-tD2 f)(f-le-tD2 (D _ D')'IjJ)]e-(t-s) D,2,
2 le-~D2 Dlop::; cs ' le-(t-s)D, Iop ::; c' and [... ] is the product of two Hilbert-Sc(~idt operators if f can be chosen E L2 and such that f- 1 e-t D2 (D-D')'IjJ is Hilbert-Schmidt. We know from the
229
Trace Class Properties
preceding considerations, sufficient for this is that (D - f)')~ has Sobolev coefficients of order r + 1 (and p = 1).
(D - f)')~
=
_ (D
grad'a,• -z'*D') 2a
-
= i*
-
Z.*
(D - a-!i* D'ia!)~ of, 'f/
((i*-l -1)D + (D - D') _
gr~~'a.') ~
~ i' [i'-1 (grad ,p . +,pD) + grad 'li(. +( grad
~-
grad '~) .' +~(D - D') -
!)
1
grad 'a.' ~ . 2a
i* is bounded up to order k, i*-l -1 is (r+ 1)-Sobolev, grad~, grad '~ have compact support, 0 ~ 1, gr~ 0;.' is (r + 1)-
:s: :s:
Sobolev and ~(D - D') is completely discussed in (2.9) - (2.53). Hence (3.27) is completely done. Write the integrand of (3.28) as
[(e~D2 J)(f-IdD\D - D'))] (D'e-(t-s)i5'\ [... ] is the product of two Hilbert-Schmidt operators with bounded trace norm on t-intervalls [aD, all, aD > O. An easy calculation yields
fY~ = a-!i* D'ia!~ = i* grad ~ .' +~f)', hence - 2
115'~e-(t-s)D'
lop
- 2
I(i* grad '~ .' +~f)')e-(t-s)D'
C' < C+ y't=S' t-s (3.28) is done.
lop
230
Relative Index Theory, Determinants and Torsion
Rewrite finally the integrands of (3.24), (3.25) as
(e- SD2 D) [( (D _ 15')'l/Je- t;8 IY f- 1)(Je- t;8 15,2)] = e- sD2 Di* [(((i*-1 - 1)( grad 'I/J . +'l/JD) + + grad 'I/J(. - .1) + ( grad 'I/J - grad ''I/J) ., +
+'I/J(D - D') _ grad a .' 'I/J)e- t;8 15,2 f- 1)(Je- t ;8 15 ,2)] 2a and
e-sD2 i*[((D _ 15')'l/Je-t~8 15,2 f-l)(Je- t~8 15,2)] (15'e-t;8 15,2) = e- sD\*[(((i*-1 - 1)( grad 'I/J . +'l/JD) + grad 'I/J(. - .1)
+
(
grad 'I/J - grad I 'I/J ) .I +'I/J ( D - D ') -
grad 2a a ., of,) 'f/
e- t~8 15,2 (J-l )(Je- t~8 15,2)] (15'e- t;8 15'\
respectively, and (3.24), (3.25) are done. The remaining integrals are (3.26) and (3.29). We have to find an appropriate - 2 - 2 expression for D' 'I/J - 'I/J D' .
1512
(a -! i* D'ia!)( a-! i* D'ia!)
(3.30)
2 a -l'*D 22 "2* ( gradl'a'I +a 1DI)
2.*
(DI a
2a2
_1 2
1DI)
'a'') 2.* ( gradl 'a' I +a 2 + grad3
2a2
2a2
'*D"*D '*D * grad 'a. , 2 2 ' + 2 "2 2a .* grad Ia I .* grad Ia I .* grad Ia ., ';*D'. +2 . 2 . + + 2 --- " 2a 2a 2a (3.31)
Hence
231
Trace Class Properties
fy2 V; - V; fy2
= i* D' i* D' V; - V;i* D' i* D'
'*D"* grad'a ., 01. 2 'f/
ol,'*D"* grad'a . , - 'f/2 2 (3.32) 2a 2a * grad' a . , 2.* grad' a ., 01. 01.'* grad' a . , 2.* grad 'a , +i 'f/ - 'f/2 2a 2a 2a 2a .* grad' a , '*D' 01. 01.'* grad' a ., 2'*D' +2 . 2 'f/ - 'f/2 2a 2a = i* D'i* grad 'V;.' +i* grad' V; .' i* D' + V;i* D' i* D' +2
-V;i* D'i* D' + i*( grad 'V;.' +V;D')i* grad 'a .' 2a ol,'*D"* grad'a,. +2.* grad'a . , 2.*( grad '01.' - 'f/2 2 'f/' 2a 2a 01.'* grad' a . , 2'*D' -'f/2 2a = i* D'i* grad 'V; .' +i* grad 'V; .' i* D'
+2.* .* +2
' .* grad' a . , grad '01. 'f/' 2 2a grad 'a , .* d '01. ' 2a . 2 gra ' f / ' .
(3.33)
ol'D')
+ 'f/
(3.34) (3.35) (3.36)
The terms in (3.34) are first order operators but grad'v; has compact support and we are done. The terms in (3.35), (3.36) are zero order operators and we are also done since grad' V; has compact support. 2
-,2
2
-,2
Hence (e- tD P-e- tD P')B, B(e- tD P-e- tD P')B are of trace class and the trace norm in uniformly bounded on any compact t-interval lao, al], ao > O. The assertions for (3.11) are done. Next we study the operator (3.37) Denote by Me the multiplication operator with exp( -Edist (m, K)2). We state that for E small enough (3.38)
232
Relative Index Theory, Determinants and Torsion
and
M e- I e -tD2X G,
M-I -tD,2 X e e G
(3.39)
are Hilbert-Schmidt for every compact GeM or G' eM'. Write (d D2 P _ e-~15,2 P')(l - B) =
= [e-~D2 P Me]' [Me-l(e-~D2 P - e-~15,2 P')(l - B)]("*.40) t- 2
t2
t- 2
+[e- 2D P - e- 2D' P')Me]' [Me- I e- 2D' P'(l - B)].(3.41) According to (1.1) - (1.5) and (3.38), each of the factors [... ] in (3.40), (3.41) is Hilbert-Schmidt and we obtain that (3.34) is of trace class and has uniformly bounded trace norm in any t-interval lao, all, ao > O. The same holds for t D2 t 15,2 , B(e 2 P - e- 2 P )(1 - B) (3.42)
(1 - B)(e~D2 P - e-~15,2 P')B (1 - B)(d D2 P - e-~15,2 P')(l - B)
(3.43) (3.44)
by multiplication of (3.37) from the left by B etc., i.e. assertions for (3.13), (3.14), (3.17) are done. Write now t
2
the
t - 2
(e- 2D D - e- 2D' D')B = (e-~D2 D)(e~D2 P _ e-~15,2 P')B +
(3.45)
(e-~D2 D - e-~15,2 D')(e-~D'2 P)B.
(3.46)
(3.45) is done already by (3.23) and le-~D2 Dlop :S pose (3.46) as the sum of
3t-.
Decom-
e-~D2 P(D - [Y) . (e-~15,2 fy) = [e-~D2 P( -'1])] . (e-~15,2 iF)B (3.47) and tD2 tD-,2 tD-,2 (e- 2 P - e- 2 P')(e- 2 D')B (3.48)
[... ] in (3.47) is done. Rewrite e-~D2 P - e-~15,2 P' as (3.49) (3.50)
Trace Class Properties
233
D2
(3.49), (3.50) are done already, hence (3.48) and hence (d pe-~D,2 PI)B, (3.12), (3.15), (3.16), (3.18). This finishes the proof of 3.3. D The proof of theorem 3.2 now follows from 3.3 by adding up the t 2 t -,2 t 2 t -,2 four terms containing e2D p - e-2 D pi or e2D D - e-2 D D', respectively. D
Remark 3.4 We could perform the proof of 3.2,3.3 also along the lines of (2.59) - (2.62), performing first a unitrary transformation, proving the trace class porperty and performing the back transformation, as we indicate in (2.59). This procedure is completely equivalent to the proof of 3.2, 3.3 presented above. D
,2
The operators U*i* D iU and (U*i* D'2iU)2 are distinct in general and we have still to compare e-t(U*i* D,2 i U) pi and e-t(U*i* D,2 iU )2 P'. According to our remark above, it is sufficient to prove the trace class property of (3.51) in
H' = L2((K, E), g, h)tBL2((K', E'), g', h' )tBL 2((M\K, E), g', h). Here we have an embedding i L2 ,K' tB C
1
:
L2((M', E'), g', hi) ~ H'
(i L2 ,K' tB C1)
= i L2 ,K,XK, + i-1XM'\K,, (3.52)
where
c 1 : L 2 ((M' \ K', E'IM'\K'), g', hi) ~ L 2 ((M ' \ K', E'IM'\K'), g'h),
i-1
and
= ,
234
Relative Index Theory, Determinants and Torsion
i* D,2 i similar, all with the canonical domains of definition anal-
ogous to (3.8). P' is here the projection onto im (i L2 ,KI E9 i-I). We define i* D,2 i , (i* D'i)2 to be zero on im p,l... Remark 3.5 Quite similar we could embed L2((M, E), g, h) into H', define P, U DU* and the assertion 3.2 would be equivalent to the assertion for
(3.53) Applying the (extended) U* from the right, U from the left, yields just the expression (3.9).
o Theorem 3.6 Assume the hypotheses of 3.2. Then
(3.54) is of trace class and its trace norm is uniformly bounded on compact t-intervals lao, al]' ao > o.
Proof. We prove this by establishing the assertion for the four cases arising from multiplication by B ,1- B. Start with (3.54). B. Duhamel's principle again yields
(e-t(i* D/2 i) P' _ e-t(i* D' i)2 P')B
J t
=-
/2 e- s(i*D i)((i* D,2 i ) - (i* D'i)2)e-(t-S)(i*D i)2 ds. (3.55) 1
o
An easy calculation yields (i* D,2 i )'lj; - 'lj;(i* D'i)2 = i* D,2'lj; - 'lj;i* D'i* D'
= i* D' grad
''lj; .' +i* grad ''lj; .' D + 'lj;i* D,2
-('lj;i* D,2 + 'lj;i* D'(i* - 1)D')
= i* D' grad ''lj; .' +i* grad ''lj; .' D' -'lj;i* D'(i* - 1)D'.
(3.56) (3.57)
235
Trace Class Properties
The first order operators in (3.56) contain the compact support factor grad ' 'lj; and we are done. Here i* D' (coming from the first term or from grad ''lj;,1 D' = grad ''lj;.'i*-li* D) will be connected 12 with e-s(i* D i) or e-(t-s)(i* Dli)2, depending on the interval [~, t] or [0, The (D')'s of the second order operator (3.57) can be distributed analogous to the proof of 3.9. The remaining main point is 0 ~ 'lj; ~ 1 and i* - 1 Sobolev of order r + 1, i.e. i* - 1 E no,l,r+1(Hom((E'\M'\K', g', h'), (E\M\K' g', h))). The assertion for (3.54)·B is done. Quite analogously (and parallel to the proofs of (3.11), (3.13), (3.14), (3.17)) one discusses the other 3 cases. 0
H
We obtain as a corollary from 3.2 and 3.6
Theorem 3.7 Assume the hypothesis of 3.2. Then for t > 0
(3.58) is of trace class in 'H' and the trace norm is uniformly bounded 0 on compact t-intervals lao, all, ao > O.
The simplest standard example for a Clifford bundle is E (A*T* M ®C, gAo ,vg1\* ) ----t (Mn, g) with Clifford multiplication X ® wE TmM ® A*T* M ®C
----t
X· W = Wx 1\ W - ixw,
where Wx := g(, X). In this case E as a vector bundle remains fixed but the Clifford module structure varies smoothly with g, g' E comp(g). It is well known that in this case D = d + d*, D2 = (d + d*)2 Laplace operator !:1. Theorem 2.12 then yields
Theorem 3.8 Assume (Mn, g) with (I), (B k ), k ~ r + 1 > n + 3, g' E M(I, B k), g' E compl,r+l(g) C M(I, Bk)' Denote by !:1~2(9) = U*i* !:1(g')iU the transformation of!:1' = !:1(g') from L 2(g',g') == L2((M,A*T*M®C),g',g~*) to L2(g) == L 2(g,g) = L 2((M,A*T*M ®C,g,gA*), where i : L 2(g,g') =
236
Relative Index Theory, Determinants and Torsion
L2((M,A*T*M 0C,g,gA*) - - t L 2(g',g') and U : L 2(g,g) - - t L 2(g, g'), U = a!, dq(g) = a(q)· dq(g'), are the canonical maps, i*, U* their adjoints. Then for t > 0
is of trace class and the trace norm is uniformly bounded on compact t-intervalls [aD, al], aD > O.
o Applying 3.7 to the case E = A*T*M 0C, D2 = b., we obtain 2n Theorem 3.9 Let (A*T*M0C,gA* E CLB (1,B k ), k ~ r+ 2 > n + 3, (A*T*M' 0C,g~*) E gencompi~dtlf,rel(A*T*M 0 C, gA*) n CLB 2n ,n(1, B k ), ,0. = b.(g), b.' = b.(g') the graded Laplace operator. Then for t > 0 e-tt. p _ e-t(U*i*t.'iU p'
(3.59)
is of trace class in H' and the trace norm is uniformly bounded 0 on compact t-intervals [aD, al], aD > O. Roughly or more concretely speaking, as one prefers, the means the following. Given an open manifold (Mn, g) satisfying (1) and (B k ), k ~ r + 2 > n + 3. Cut out a compact submanifold K and glue the compact submanifold K' along 8M = 8K, getting thus M', endow M' with a metric g' satisfying (1) and (Bk) and Ig - g'IM\K,g,l,r+1
<
00.
Then for t > 0 has the asserted properties. We can apply a slight modification of 3.8. also to Schrodinger operators.
237
Trace Class Properties
Lemma 3.10 Let (Mn,g) be open with (1) and (Eo), r > n, V E n,l,r (Mn, g) a real-valued function, 0 ~ q ~ n. Then the operator
tlq + (V·) is essentially self-adjoint on C,: (A q).
Proof. According to the Sobolev embedding theorem,
lV(x) I ~ b. Hence (3.60)
o Proposition 3.11 Suppose 9 E M(J, B k ), k 2: r > n V E n,l,r (Mn, g) a real valued function. Then for t > 0
+ 2,
(3.61)
is of trace class and the trace norm is uniformly bounded on compact t-intervals lao, al], ao > O.
Proof. We infer from Duhamel's formula
J t
e- tAq -
e-t(Aq+(V.))
=
e-SAqVe-(t-s)(Aq+(v'))ds,
(3.62)
o t
decompose
~
t
J = J + J and estimate these integrals as in (1.1) -
o 0 ~ (1.42), replacing T} by V.
0
Theorem 3.12 Assume the hypotheses of 3.8 and V E n,l,T (Mn,g), V' E n,l,r(M'ng). Then (3.63)
is of trace class in H' and the trace norm is uniformly bounded on compact t-intervals lao, al], ao > O.
238
Relative Index Theory, Determinants and Torsion
Proof. We write e-t(.t. q +(V'))p.-tU·i·(.t.~+(VI.))iUp' e
= (e-t(.t.q+W))
_ e-t.t.q)p
(3.64)
+e-t.t.qp _ e-tU·i·.t.~iU)p'
(3.65) (3.66)
+(e-tU·i·.t.~iU _ e-tU·i·(.t.~+(VI-)))P'.
(3.64) and (3.66) are of trace class according to proposition 3.10, 0 (3.65) is of trace class according to theorem 3.8.
We proved that after fixing E E CLBN,n(I, B k ), k 2:: r+ 1 > n+ 3, we can attach to any E' E gencompi~t;f,rel(E) two number valued invariants, namely
and
1
E'
--+
tr (-tD2p e - e -t(U'i'D iU)2 p ')
E'
--+
tr (e -tD2 P - e -tU'i' D
12
iUp') .
(3.67)
(3.68)
This is a contribution to the classification inside a component but still unsatisfactory insofar as it 1) could depend on t. 2) will depend on the K C M, K' c M' in question, 3) is not yet clear the meaning of this invariant. The answers to these open question will be the content of chapters V, VI and VII.
V
Relative index theory
1 Relative index theorems, the spectral shift function and the scattering index In many applications the Clifford bundles under consideration are endowed with an involution r : E ---t E, s.t.
[r,X]+
= 0 for
(1.1 ) (1.2) (1.3)
X E TM
[\7, r] = 0 Then L 2 ((M,E),g,h) = L 2 (M,E+) EEl L 2 (M,E-) 0
D= ( D+
and D-
=
D-) 0
(D+)*. If Mn is compact then as usual
indD := indD+ := dim ker D+ - dim ker D-
= tr( re- tD \ (1.4)
where we understand r as
For open Mn indD in general is not defined since re- tD2 is not of trace class. The appropriate approach on open manifolds is relative index theory for pairs of operators D, D'. If D, D' are 2 tD/2 selfadjoint in the same Hilbert space and etD - ewould be of trace class then
makes sense, but at the first glance (1.5) should depend on t. 239
240
Relative Index Theory, Determinants and Torsion
If we restrict to Clifford bundles E E CLBN,n(I, B k ) with involution T then we assume that the maps entering in the definition l,r+l j,rel (E) are T-compatl'ble, I.e. . l,r+l j,F (E) or gen comp L,dij of comp L,dij after identification of EIM\K and fiE'IM'\K holds
(1.6) We call EIM\K and E'IM'\K' T-compatible. Then, according to the preceding theorems,
(1. 7) makes sense.
Theorem 1.1 Let ((E, h, "V h ) ---t (Mn, g), T) E CLBN,n(I, B k ) be a graded Clifford bundle, k ~ r > n + 2. a) If "V'h E compl,r("V) C C~r(Bk)' "V' T-compatible, i.e. ["V', T] = 0 then
is independent of t. b) If E' E gencomptd~;j,rel(E) is T-compatible with E, ~.e. [T, X .']+ = 0 for X E T M and ["V', T] = 0, then
is independent of t. Proof. below.
a) follows from our IV 1.1. b) follows from our 1.2 0
Proposition 1.2 If E' E gencompi~d~;j,rel(E) and
T( e- tD2 P _ e-t(U*i* D'iU)2 P') T( e- tD2 D - e-t(U*i* D'iU)2 (U*i* D'iU» are for t > 0 of trace class and the trace norm of
241
Relative Index Theory
T( e- tD2 D - e-t(U*i* D'iU)2 (U*i* D'iU)) is uniformly bounded on compact t-intervals [aD, al], aD > 0, then
is independent of t.
Proof.
Let ('Pi)i be a sequence of smooth functions E ::s 'Pi ::s 'PHI and t->oo 1. Denote by Mi the multiplication operator with 'Pi on
C~(M \ K), satisfying sup Id'Pil ~ 0,
'Pi
~
°
t->oo
L2((M\K, EIM\K), g, h). We extend Mi by 1 to the complement of L 2 ((M \ K, E), g, h) in H. We have to show
e- tD2 P _ e-t(U*i* D'iU)2 P' is of trace class , hence
trT (e_tD2p - e -t(U*i*D'iU)2 p ') = lim trTMj (e- tD2 p - e- t(U*i*D'iU)2 P')Mj
.
J->OO
M j restricts to compact sets and we can differentiate under the trace and we obtain
Consider
There holds trT ( M j e-tD2 D 2M) j
= tr M j grad 'Pi' T D e-tD2 .
242
Relative Index Theory, Determinants and Torsion
Quite similar trT(Mj(e-t(i* D'i)2 (i* D'i)2)Mj ) = trT
('*D")2 t ('*D")2 , ~
=tr(i*Di)e-2~
2 t('*D")2 = tre- 2t('*D")2 ~ ~ (i* D'i)
= tre-~(i*D'i)2i*(2
= tr2i* M j grad'
.' iT( i* D'i)e-t(i* D'i)2
-trT Mje-t(i* D'i)2 (i* D'i)2 M j , hence trT(Mje-t(i* D'i)2 (i* D'i)2 M j )
= trMji* grad '
and finally d M 'e (-tD2p - e-t(U*i*D'iU)2 p ')M ' -trT dt J J
= trT M j [ grad
= trTMj [( grad
. e-
tD2
+ grad '
Relative Index Theory
243
Q'± = U*i* D!±iU = (U*i* D'iU)±, Q', H' analogous, assuming (1.1) - (1.3) as before and J, '\i" T-compatible. H, H' form by definition a supersymmetric scattering system if the wave operators
W'f(H, H'):= lim
eitHe-itH'.
Pac(H') exist and are complete
t-->'fOO
(1.9) and
QW'f(H, H') = W'f(H, H')H' on V H , n '}-{~c(H').
(1.10)
Here Pac(H') denotes the projection on the absolutely continuous subspace '}-{~c(H') C '}-{ of H'. A well known sufficient criterion for forming a supersymmetric scattering system is given by
Proposition 1.3 Assume for the graded operators Q, Q' (= supercharges) e-tH -e -tH' and
e-tHQ -e -tH'Q are for t > 0 of trace class. Then they form a supersymmetric scattering system. 0
Corollary 1.4 Assume the hypotheses of 1.1. Then D, D' or D, U*i* D'iU form a supersymmetric scattering system, respectively. In particular, the restriction of D, D' or D, U*i* D'iU to their absolutely continuous spectral subspaces are unitarily equiv0 alent, respectively. Unti! now we have seen that under the hypotheses of 1.1 ind(D, fy) = trT(e- tD2 P _
e-
tfy2
P'),
(1.11)
fy = D' or fy = U*i* D'iU, is a well defined number, independent of t > 0 and hence yields an invariant of the pair (E, E'),
Relative Index Theory, Determinants and Torsion
244
still depending on K, K'. Hence we should sometimes better write (1.12) ind(D, fY, K, K'). We want to express in some good cases ind(D, fY, K, K') by other relevant numbers. Consider the abstract setting (1.8). If inf ae(H) > 0 then indD := indD+ is well defined. Lemma 1.5 If e- tH p - e- tH' P' is of trace class for all t > 0 and inf ae(H), inf ae(H') > 0 then
lim trT(e- tH P - e- tH' P') = indQ+ - indQ-.
(1.13)
t--->oo
o We infer from this Theorem 1.6 Assume the hypotheses of 1.1 and inf a e(D2) > O. Then inf a e(D,2), inf ae(U*i* D'iU)2 > 0 and for each t > 0
(1.14) Proof. In the case 1.1 a), inf a e (D,2) > 0 follows from a standard fact and (1.14) then follows from 1.5. Consider the case 1.1 b). We can replace the comparison of a e (D2) and ae((U*i* D'iU)2) by that of ae(U D 2U*) and ae((i* D'i)2). Moreover, for self adjoint A, 0 ~ ae(A) if and only if inf a e(A2) > O. Assume 0 ~ ae(U DU*) and 0 E ae(i* D'i). We must derive a contradiction. Let (1)v)v be a Weyl sequence for 0 E ae(i*D'i) satisfying additionally l1>vlL2 = 1, supp 1>v ~ M \ K = M' \ K' and for any compact L c M \ K = M' \ K'
(1.15) We have lim i* D'i1>v = O. Then also lim D'1>v = O. We use in V--l>OO V--l>OO the sequel the following simple fact. If (3 is an L 2 -function, in particular if (3 is even Sobolev, then (1.16)
Relative Index Theory Now (U DU*)
245
= (U DU* - D')oo ----+ O.
Consider (U DU* - D')
(-
Q
gr:
•
lim v--->oo
1-
grad a .
o.
Q
IE
(1.17)
If a == 1 this term does not appear. Write, according to IV (2.9) - (2.12),
(D - D')
+ r/i
(1.18)
1 ~·I
is bounded (we use a uniformly locally finite cover by normal charts, an associated bounded decomposition of unity etc.). f3 = IV' - V"I is Sobolev hence L2 and by (1.16) g'ik
(1.19)
Now 1V'
lI---l'OO
yields (1.20)
We conclude in the same manner from 1V"
I77i
----+
v--->oo
, Sobolev and
O.
(1.21)
(1.17) - (1.21) yield (U DU*) 0, inf O'e((i* D'i)2) > 0, 0 tf. O'e(i* D'i), 0 tf. O'e(D'), inf O'e(D,2) > O. We infer from 1.2 and 1.5 that for t > 0
trTe- tD2
_
e-t(U*i* D'iU)2 = indD+ - ind(U*i* D'iU)+.
(1.22)
We are done if we can show
. d(U* ~'*D' ~'U)+ = III . dD'+ . In
(1.23)
246
E
Relative Index Theory, Determinants and Torsion
ker(U*i* D'iU)+ means (U*i* D'iU)+
(U*i* D'iU) (XK' + U*C1XM'\K,
O. But this is equivalent to D'+ holds and hence (1.14).
= O. Similar for
D'-.
(1.23) 0
It would be desirable to express ind(D, iY, K, K') by geometric/topological terms. In particular, this would be nice in the case inf (J'e(D2) > O. In the compact case, one sets indaD := indaD+ = dim ker D+ - dim ker(D+)* = dim ker D+dim ker D- = lim trTe- tD2 . On the other hand, for t --t 0+ t->oo
there exists the well known asymptotic expansion for the ker2 nel of Te-tD • Its integral at the diagonal yields the trace. If trTe- tD2 is independent of t (as in the compact case), we get the index theorem where the integrand appearing in the L 2-trace consists only of the t-free term of the asymptotic expansion. Here one would like to express things in the asymptotic expan2 sion of the heat kernel of e- tD ,2 instead of e-t(U*i* D'iU) • For this reason we restrict in the definition of the topological index l,r+l (E) or E' E comPL,dijj,F,rel l,r+l (E) , to th e case E ' E comPL,dijj,F i.e. we admit Sobolev perturbation of g, \1h,. but the fibre metric h should remain fixed. Then for D' = D(g', h, \1,h, .') in L 2 ((M,E),g,h) the heat kernel of e- t (U*D'U)2 = U*e- tD ,2 U equals to a(q)-~W'(t,q,p)a(p)~. At the diagonal this equals to W' (t, m, m), i.e. the asymptotic expansion at the diagonal of the original e- tD ,2 and the transformed to L2((M, E), g, h) coincide. Consider trTW(t,m,m) '" r~b_!!(D,m)+ .. ·+bo(D,m)+··· (1.24) t->o+
2
and trTW'(t, m, m) '" r~b_!! (D', m) t->O+
+ ... + bo(D', m) + ....
2
(1.25)
247
Relative Index Theory
We prove in VI 1.1 and 1.2 Lemma 1.7
n
--2 -< i -< 1.
(1.26) D
indtop(D, D') :=
J
bo(D, m) - bo(D', m).
(1.27)
M
According to (1.26), indtop(D, D') is well defined. l,r+l (E) Theorem. 1 8 A ssume E ' E gencomPL,dijj,F,rel a) Then
ind(D, D', K, K')
J
J
K
K'
bo(D, m) -
+
J
bo(D', m)
+
(1.28)
bo(D, m) - bo(D', m).
M\K=M'\K'
(1.29)
b) If E'
E
gencompi~d~Jj,F(E) then ind(D, D') = indtop(D, D').
(1.30)
c) If E' E gen comptd~Jj,F(E) and inf (Je(D2) > 0 then
indtop(D, D') = indaD - indaD'.
(1.31)
Proof. All this follows from 1.1, the asymptotic expansion, (1.26) and the fact that the L 2-trace of a trace class integral operator equals to the integral over the trace of the kernel. D
Relative Index Theory, Determinants and Torsion
248
Remarks 1.9 1) If E' E gencompi~dt;f,rel(E), g and g', V'h and V',h, . and.' coincide in V = M \ L = M' \ L', L 2 K, L' 2 K', then in IV (2.5) - (2.53) a -1 and the rJ's have compact support and we conclude from IV (3.38), (3.39) and the standard heat kernel estimates that
J
IW(t, m, m) - W'(t, m, m)1 dm::; C . e-~
(1.32)
v and obtain ind(D, D', L, L') =
J
J
£
£'
bo(D, m) -
bo(D', m).
(1.33)
This follows immediately from 1.8 a). 2) The point here is that we admit much more general perturbations than in preceding approaches to prove relative index theorems. 3. inf O"e{D2) > 0 is an invariant of gencompi~dt;f,F(E). If we fix E, D as reference point in gencompi~dt;f,F(E) then 1.8 c) enables us to calculate the analytical index for all other D's in the component from indD and a pure integration. 4) inf O"e{D2) > 0 is satisfied e.g. if in D2 = V'*V' + R the operator R satisfies outside a compact K the condition R
~ "-0 .
id,
"-0
> O.
(1.34)
(1.34) is an invariant of gen compi,~t;f,F(E) (with possibly different K, "-0). 0 It is possible that indD, indD' are defined even if 0 E {Ye. For the corresponding relative index theorem we need the scattering index. To define the scattering index and in the next section relative (functions, we must now use spectral shift functions ~(A) which we introduced in III section 2. According to theorem 2.8 of
Relative Index Theory
249
chapter III, ~(A) == ~(A; A, A') exists if A, A' are self-adjoint and V = A - A' is of trace class. Then, with R'(z) = (A' - z)-l, ~(A) = ~(A,A,A') :=
1T-
1
Iimargdet(1 0:--->0
~(A)
exists for a.e. A E lR.
tr(A - A')
=
+ VR'(A + if))
(1.35)
is real valued, E L1 (lR) and
j ~(A) dA,
1~ILl::;
IA - A'h.
(1.36)
I
If I(A, A') is the smallest interval containing O"(A) U O"(A') then ~(A) = 0 for A ~ I(A, A'). Let
Q={f:lR----+lR
I
fELl
jli(p)l(l+lpl)
and
dp
ffi
Then for rp E Q, rp(A) - rp(A') is of trace class and
tr(rp(A) - rp(A')) =
j rp'(A)~(A) dA.
(1.37)
ffi
We state without proof from [49] Lemma 1.10 Let H, H' 2: 0, selfadjoint in a Hilbert space H,
c tH - e- tH' for t > 0 of trace class. Then there exist a unique function ~ = ~(A) = ~(A, H, H') E L 1 ,loc(lR) e-tA~(A) E L 1 (lR) and the following holds.
such that for t > 0,
00
a) tr(e- tH - e- tH ') = -t J e-tA~(A) dA. b) For every rp
E
o Q, rp(H) - rp(H') is of trace class and
tr(rp(H) - rp(H')) =
j Rrp'(A)~(A) dA. I
c)
~(A)
= 0 for A < O.
o
250
Relative Index Theory, Determinants and Torsion
We apply this to our case E' E gencompi~d~Jf,rel(E). According to corollary 1.4, D and U*i* D'iU form a supersymmetric scattering system, H = D2, H' = (U*i*D'iU)2. In this case
e21ri((>",H,H') = det S('x),
J
where, according to II (2.5) and (2.6), S = (W+)*W- = S('x) dE'('x) and H~c = J,X dE'('x). Let Pd(D), Pd(U*i* D'iU) be the projector on the discrete subspace in 1i, respectively and Pc = 1 - Pd the projector onto the continuous subspace. Moreover we write
(U*i* D'iU)2 =
(H~+
:,_). (1.38)
We make the following additional assumption.
e- tD2 Pd(D), e- t (U*i*D'iU)2 Pd(U*i* D'iU) are for t > 0 of trace class.
(1.39)
Then for t > 0
is of trace class and we can in complete analogy to (1.35) define ~C('x,H±,Hd):=
-1f
lim argdet[l
e-+O+
+ (e- tH ± Pc(H±)
_e- tH'± Pc(H d )) (e-tHt± Pc(H d ) - e->..t - ic)-l]
(1.40)
According to (1.36),
J~C('x, 00
tr(e- tH ± Pc(H±)-e-tH'± Pc(Hd))
= -t
H±, Hd)e- t>.. d'x.
o (1.41) We denote as after (1.11) fy = D' in the case 'il' E compl,r('il) and fy = U*i*D'iU in the case E' E gencomptd~Jf,rel(E). The assumption (1.39) in particular implies that for the restriction
251
Relative Index Theory
of D and fy to their discrete subspace the analytical index is well defined and we write inda,d(D, fy) = inda,d(D) - inda,d(D') for it. Set (1.42) Theorem 1.11 Assume the hypotheses of 1.1 and {1.39}. Then nC(A, D, 15') = nC(D, 15') is constant and
ind(D, 15') - inda,d(D, 15')
= nC(D, 15').
(1.43)
Proof.
ind(D, 15')
2
-,2
tfT(e- tD P - e- tD P')
=
= trTe- tD2 Pd(D)P - trTe- tD ,2 Pd(D')P' + +trT(e- tD2 Pc(D) - e- tD ,2 Pc(D')) =
J 00
inda,d(D,D')
+t
e-tAnC(A, D, 15') dA.
o
According to 1.1, ind(D,D') is independent of _ t. _ 00 holds for inda,d(D, D'). Hence t
The same
J e-tAnC(A, D, D') dA
is inde-
o
00
pendent of t. This is possible only if
J e-tAnC(A, D, D') dA = t o
or nC(A, D, 15') is independent of A.
o
Corollary 1.12 Assume the hypotheses of 1.11 and addition-
ally
o
VI Relative (-functions, 7]-functions, determinants and torsion In this chapter, we apply our preceding considerations and results to the construction of relative zeta functions and related invariants. We will attach to an appropriate pair of Clifford data a relative zeta function, which is essentially defined by the corresponding pair of asumptotic expansions of the heat kernel. Therefore we must first consider such a pair of expansions.
1
Pairs of asymptotic expansions
Assume E' E gencompi~tJf,F(E). Then we have in L 2 ((E,M), g, h) the asymptotic expansion trW(t,m,m)
rv
t-O+
r~b_!!(m) +r~+lb_!!+l 2
2
+...
(1.1)
and analogously for tm-! (m)W'(t, m, m)a! (m) = tr W'(t, m, m) with b_~+l(m)
b~%+l(m)
= b-~+I(D(g, h, \7), m), = b-%+I(D(g', h, \7'), m).
Here we use that the odd coefficients vanish, i.e. terms with r~+!, r%+~ etc. do not appear. The heat kernel coefficients have for l ~ 1 a representation I
b-~+l =
k
LL k=l q=O il +i2+·+ik=2(l-k)
tr (\7i q + 1 RE ... \7ik RE)Ci),···,i k ,
(1.2)
where Cil, ... ,ik stands for a contraction with respect to g, i.e. it is built up by linear combination of products of the gi j , gij' 252
253
Relative (-functions
1
Lemma 1.1 b-~+l - b'-~+l E L (M, g), 0 ~ l ~
nt3.
Proof. First we fix g. Forming the difference b-~+l - b'-~+l' we obtain a sum of terms of the kind 'ViI Rg ... 'V iq Rg tr ['V iq +1 RE ... 'V ik RE
_ 'V /iq + 1 R,E ... 'V /ik R,EJCil, ... ,ik.
(1.3)
The highest derivative of Rg with respect to 'V g occurs if q = k, i1 = ... = i q - 1 = O. Then we have
(1.4)
('V 9 )21-2k Rg.
By assumption, we have bounded geometry of order :2: r > n+2, i. e. of order :2: n + 3. Hence ('V9)i Rg is bounded for i ~ n + 1. To obtain bounded 'V j RLcoefficients of [... J in (1.3), we must assume n +3 (1.5) 2l- 2 ~ n + 1, l < - -2 - . Similarly we see that the highest occuring derivatives of R E , R'E in [... J are of order 2l - 2. The corresponding expression is R E'V 21 - 2RE _ RIE'V/21-2 R'E = (RE _ R'E) ('V 21 - 2RE) + R'E ('V 21 - 2RE _ 'V 21 - 2R ,E ). (1.6) We want to apply the module structure theorem. 'V - 'V' E n1 ,1,r (Q~'l, 'V) = n1 ,1,r (Q~l, 'V') implies RE - R,E E n2,1,r-1. We can apply the module structure theorem (and conclude that all norm products of derivatives of order ~ 2l - 2 are absolutely Hence, integrable) if 2l - 2 ~ r - 1, 2l - 2 ~ n + 1, l ~ E (1.6) E L1 since R , R'E, Cil ...ik bounded. It is now a very simple combinatorial matter to write [... J in (1.3) as a sum of terms each of them is a product of differences ('Vi RE - 'V,i R'E) with bounded terms 'V j R E , 'Vd' R'E. We indicate this for an expression 'Vi RE'V j RE - 'V'i R'E'V /j R 'E , 'Vi RE'V j RE _ 'V'i RIE'V /j R'E = 'ViRE('V j RE _ 'V /j R'E) + 'ViRE'V j RE _ 'V/iRIE'V/j R'E
nt3.
= 'ViRE('VjRE - 'V/jR'E ) + ('ViRE - 'V/iRIE)'V/jR'E. (1.7)
254
Relative Index Theory, Determinants and Torsion
The general case can be treated by simple induction. Remember \7, \7' E CE(Bk). Admit now change of g. We write \7ilR9 . .. \7 iq R9tr(\7i q + 1 RE ... \7i k RE )Ci l, ... ,ik as R}(g)trR2(h, \7)C(g), similarly R 1 (g')trR 2(h, \7')C(g'). Then we have to consider expressions Rl(g)trR2(h, \7)C(g) - R 1 (g')tr'R2(h, \7')C(g')
= [R 1 (g) - R 1 (g')JtrR 2(h, \7)C(g) +R1 (g')trR 2(h, \7)[C(g) - C(g')J +R1 (g)tr[R 2(h, \7) - R2(h, \7')JC(g').
(1.8)
But each term of [R 1 (g) -R 1 (g')J and [R2(h, \7J - R2(h, \7') can be written as a product of terms of type (1.7) composed with bounded terms (bounded morphisms). The terms [C (g) - C (g') J are E Ll since g' E COmpl,r+l (g). Then the module structure theorem for P = PI = P2 yields again that the whole expression (1.8) is ELI. This proves lemma 1.1. 0
Lemma 1.2 There is an expansion tr ( e _W2 - e -t(U' D
1
U)2)
n r"2a_¥-
+ ... + r"2n+[!!H] a_¥-+[~] 2
3 +o(r¥-+[nt ]+I). Proof.
Set
a_¥-+i
=
J
(b_¥-+i(m) -
b~¥-+i(m))
(1.9)
(1.10)
dm
and use tr W(t, m, m)
n = r"2b_~ + ... + r"2n+[n+3] - 2 b_~+[nt3]
3 +O(m, r¥-+[nt ]+I), tr W'(t, m, m) = r¥-b~!! 2
(1.11)
+ ... + O'(m, r¥-+[nt
3
]+1)
(1.12) 1
tr ( e _tD2 - e-t(U' D U)2)
=
J
(tr W(t, m, m) - tr W'(t, m, m)) dm.
255
Relative (-functions
Using lemma 1.1, the only critical point is
j O(m,r~+[~J+l)-O'(m,r~+[~J+l)
dm
= O(r~+[~J+l).
M
(1.13) (1.11) requires a very careful investigation of the concrete representatives for O(m, C~+[~J). We did this step by step, following [38], p. 21/22, 66 - 69. Very roughly speaking, the m-dependence of O(m,·) is given by the parametrix construction, i. e. by differences of corresponding derivatives of the rfa' r'fa, which are integrable by assumption. D If E' E gencomp}~dt;f,F,rel(E) then we immediately derive from IV, theorem 3.2
is of trace class. The heat kernel and its asymptotic expansion split into its restriction to K and M \ K = V or to K' and M' \ K' = V, respectively. 2 (e-tD
+ Wg,h(t, m, m)lv, = W~I,h(t, m, m)lK' + W~I,h(t, m, m)lv 1,
P)(t, m, m) = Wg,h(t, m, m)IK '2
(e-t(u' D UP')(t,
m, m)
hence tr( e- tD2 P _
e-t(u' D '2 U) P')
= j W(t, m, m)dm - j W'(t, m, m)dm K
+ j(W(t,m,m) v
K'
W'(t,m,m))dm
256
Relative Index Theory, Determinants and Torsion
n (K) =r2a_~ ,g
+· .. +r2n +[.!!.±!.] 2 a_~_[~] (K ,g )
+O(r~+[~]+l)
n (K,g ") +r2a_~
, ') + ... + r"2n+[.!!.±!.] a_~_[~] (K,g 2
+O(r~+[~]+l)
n (V ') +r2a_~ ,g,g
') + ... + r2n+[.!!.±!.]a_~_[~] ( V ,g,g 2
+O(r~+[nt3]+1)
= r~(a_~(K, g) - a_~(K', g') + a_~(V, g, g')) () ( , ') +r"2n+[n+3] -2- a_~_[~] K, 9 - a_~_[nt3] K , 9 +a_~_[~](V, g, g')
+ O(r~+[~]+l),
+ ... (1.14)
where the ai (K, g) or ai (K', g') are the integral terms of the asymptotic expansion on K or K', respectively, and
ai(V, g, g') =
J
(bi(m) -
b~(m))dm.
(1.15)
v The existence of the integrals (1.15) follows from the proof of the lemmas 1.1 and 1.2. Hence we proved
Lemma 1.3 Suppose E' E gencomptd~Jf,F,rel(E). Then there exists an asymptotic expansion tr(e-tD2 P _ e- t (u*D /2 u) P') n n+[n+3] = r2L~ + ... + r2 -2- C_~+[nt3] +O(r~+[nt3]+1).
(1.16)
o
2
Relative (-functions
For a closed oriented manifold (Mn, g), a Riemannian vector bundle (E, h) - - - t (Mn, g) and a non-negative self-adjoint elliptic differential operator A : COO(E) - - - t COO(A) there is a
257
Relative C; -functions
well-defined zeta function
((s,A) =
1 ,V
L
(2.1)
)'Eu(A) ).>0
which converges for Re (s) > ~ and which has a meromorphic extension to C with only simple poles. In particualar, s = 0 is not a pole. Hence
is well defined and one defines the (-determinant of A as det (A :=
(2.2)
e-(,(O,A).
This is the first step to define analytic torsion. On open manifolds, (2.1) does not make sense since O"(A) is not nessecarily purely discrete. (2.1) can be rewritten as
rts) Je- (tre00
((s, A)
=
1
tA
-
dim ker A)dt.
(2.3)
o
But (2.3) has a meaningful extension to open manifolds as we will establish in this section. Definition. Assume E' E gencompi~d~}f,F(E). Define
rts) Je1
(1(S, D2, (U* D'U)2) :=
o
1
tr (e- tD2
-
e- t (U*D ' U)2) dt.
(2.4)
We insert the expansion (1.9) into the integrand of (2.4), thus
258
Relative Index Theory, Determinants and Torsion
obtaining 1
tS-lr~tldt =
J
1
o
(2.5)
-!!:2 +l'
8
1
Jo e-lr~+[~l dt
1
= 8
+!!:2 + [n+3] , 2
1
_1_
f(8)
Je-lo(r~+[~l+l)
dt holomorphic for
o
Re (8)
n
n+3
+ (-2") + [-2-] + 1 >
°
(2.6)
and [nt3] ~ ~+1, we obtain a function merom orphic in Re (8) > -1, holomorphic in 8 = with simple poles at 8 = ~ - l, l ~
°
[nt3].
00
Much more troubles causes the integral
J.
Here we must addi-
1
tionally assume
(2.7) (2.7) implies l1e(D,2) = inf o"e((U* D'U)21(ker(U*D1U)2)-L) > 0. Denote by 110(D 2), 110(D,2) = 110((U* D'U)2) the smallest positive eigenvalue of D2, D,2, respectively and set I1(D2) min{l1e(D2), 110(D 2)}, I1(D,2) min {l1e( D,2), 110 (D,2)}, 2 I1(D , D,2) ._ min{I1(D2) , I1(D,2)} > 0.
(2.8)
If there is no such eigenvalue for D2 then set I1(D2) = l1e(D2), analogous for D,2. D2, D,2, (U* D'U)2 have in ]0, I1(D 2, D,2)[ no further spectral values. We assert that the spectral function ~(A) = ~(A, D2, (U* D'U)2) is constant in the interval [0,I1(D 2,D,2)/2[.
259
Relative (-functions
Consider the function We: (X) = { ce:e -
e{~x2 Ixl S E
o
Ixl > E
and choose
Ce: s. t. J We: (x) dx = 1. Let 0 < 3E < % and X[-6-2e:,H2e:] the characteristic function of [-0 - 2E, 0+ 2E]. Then 'P6,e: := X[-6-2e:,H2e:] * We: satifies 0 S 'P6,e: S 1, 'P6,e:(X) = 1 on [-0 - E, + E], 'P6,e(X) = 0 for x ~] 3E, + 3E[, 'P'<> ,e: S K· E- 1 and e:~06~O lim lim 'P6,e: = 0- distribution. Assume + 3E < ~. A regular distribution fED' (] - ~,~ [) equals to zero if and only if (f()..) , we:().. - a) sin(k()" - a))) = 0, (f()..), we:().. - a) cos(k()" - a))) = 0 for all sufficiently small E and all a (s.t. lal + E < ~ ) and for all k. ([71], p. 95). This is equivalent to (f, we:().. - a)) = 0 for all sufficiently small E and a and the latter is equivalent to (f()..) , 'P6,e: - 'P61,e:) = 0 for ~ll 0,0' andallE(s.t. 0+3E,O'+3E< ~). Wegetforf-L=f-L(D 2,D' ),0< 3E < %, 0+3E < ~ that 'P6,e:(D 2) -'P6,e:((U* D'U)2) is independent of 0, E, tr('P6,e(D2) - 'P6,e((U* D'U)2)) is independent of 0, E, 0 = tr( 'P6,e:(D 2) -'P6,e( (U* D'U)2)) -tr( 'P61,e:(D2) -'P61,e:( (U* D'U)2)) =
°
°- °
°
I± 2
J('P6,e: - 'P61,e:)'()..)~()..) d).., i.e. the distributional derivative of ~ o equals to zero, ~()..) is a constant regular distribution. We write ~()..)I[o,~[ = tr('P6,e(D2) - 'P6,e:((U* D'U)2)) == -h. Set quite parallel to [49] ~()..) := ~()..) + h which yields ~()..) = 0 for)" < ~ 00
00
and -t J e-t>'~()..) d)" = h - J e-t>'~()..) d)". The latter integral o ~ converges for t > 0 and can for t ~ 1 estimated by
J1~()..)le-ti 00
e-t~
d)" S
Ce-t~.
I± 2
Hence we proved
Proposition 2.1 Assume E' E compYdtf1fF(E), inf O"e(D2 2 " 2 l(kerD2).L) > 0 and set h = tr('P6,e(D ) - 'P6,e((U* D'U) )) as above. Then there exist C > 0 s. t. tr(e- tD2 _ e-t(u' D U)2) = h + O(e- ct ). (2.9) 1
260
Relative Index Theory, Determinants and Torsion
o Define for Re (s) < 0
rts) J 00
(2(S, D2, D,2) :=
t S - 1 [tr(e- tD2 - e- t (U*D'U)2) - h] ds.
1
(2.10) Then (2 (s, D2, D,2) is holomorphic in Re (s) < 0 and admits a meromorphic extension to C which is holomorphic in s = o. Define finally
rts) Je00
((s,D 2,D,2):=
1
[tr(e- tD2
-
e- t(U*D,U)2) - h] dt
o 1
= (l(S, D2, D,2)
+ (2(S, D2, D,2) -
rts) J
+ (2(S, D2, D,2) -
r(s:
t S - 1 h dt
o
= (l(S, D2, D,2)
1)
(2.11)
We proved
Theorem 2.2 Suppose E' E gencompi~d~;f,F,rel(E), inf O"e(D2 l(kerD2).L) > 0 and set h as above. Then ((s,D 2,D,2) is after meromorphic extension well defined in Re (s) > -1 and holomorphic in s = O. 0 We remark that (U* D'U)2 = U* D,2U, but (U*i* D'iU)2 =I-
U*i* D,2iU. The situation is more difficult if there is no spectral gap above zero. Then tr( e- tD2 - e-t(u* D'U)2) must not - up to a constant - exponentially decrease to zero. But one can ask, whether there exists for t ----+ 00 an asymptotic expansion in negative powers of t. The answer is yes, if the spectral function ~(A) has an asymptotic expansion in postive powers of A near A = 0 (cf. [49]). We perform our considerations at first for the general situation of a supersymmetric scattering system H, H' such that for t > 0 e- tH - e- tH' is of trace class.
261
Relative (-functions
Proposition 2.3 Suppose that for an interval [0, c] there exists u sequence o :S 'Yo < 'Yl < 'Y2 < ... ---+ 00 such that for every N E 1N N
~(,\)
= LCixYi +O(XYN +1 ).
(2.12)
i=O
Then there exists for t ---+ 0 an asymptotic expansion 00
tr( e -tH - e -tHI)
f'V
L
C'Yi b'Yi .
i=O
Proof. According to II (2.17),
J~('\)e-t>'d'\ 00
tHI tr(e- tH - e- )
=
-t
o -t
For t
~
[!
1
';(A)e-tAdH l';(A)e-tAdA
1
J~('\)e-t>'d'\ 00
:S C . e- te / 2
e
and
J,\ 00
o
,
'Y e-t>'d,\
~ t'Y+l
J,\ 00
-
'Y e-t>'d,\
e
+ O(e- te / 2 )
~ + O(e- te / 2 ) t'Y+ 1 o
262
Relative Index Theory, Determinants and Torsion
Looking at proposition 2.3, there arises the natural question, under which conditions an expansion (2.12) is available. We assume Hand H' self-adjoint, 2': 0 and e- tH - e- tH' for t > 0 of trace class. Following [49], we can now establish Proposition 2.4 Suppose ]0, c[e aac(H'), ~(A) continuous in ]0, c[ and that there exists mEN such that for A E]O, c[, det S(A) extends to a holomorphic function on Dc,m, where Dc = {z E Clizi < c} and Dc,m ---t Dc is the ramified covering given by z ---t zm. Then there exist c1, 0 < c1 < c~, and an expansion 00
~(A) =
L CkA~,
A E]O, cd·
(2.13)
k=O
Proof.
We have by assumption that det S(A) is smooth on
]0, cr. I det S(A)I = 1 and ]0, c[e aac(H') imply that there exists cP E Coo(]O, cD such that det S(A) = e- 27ficp (A) for A E]O, cr. We know from II (2.8) det S(A) = e- 27fi((A). Hence there exists k E Z such that ~(A) = cp(A) + k, A E]O, cr. This implies ~(A) smooth on ]0, cr. Differentiation of det S(A) = e- 27fi((A) and the unitarity of S(A) yield
d~(A) = __ 1 dA
27ri tr
(s*( ') dS(A)) /\ dA .
(2.14)
We infer from the assumption Dc,m ---t Dc,m and ( ---t )-extension of det S(A) that lim cp(A) exists, hence lim ~(A) too. Thus we A-+O+
A-+O+
obtain from (2.14) A
~(A)
~(O+) - ~Jtr (s*(a)dS(a)) 27r2
da
o
da
.1.
~(O+) - 2~i ]
o
tr (s*(a
m
m )
dSd:
))
da. (2.15)
Relative <; -functions
263
But tr ( S*(zm) ds~:m») is holomorphic on
Izl <
C1
::;
c~.
We get the expansion (2.13), inserting the power series of tr (s*(zm) ds~:m») into (2.15).
D
Corollary 2.5 a) Under the hypothesis of 2.4 with m exists for t - t 00 an asymptotic expansion
=
1, there
00
tr(e- tH
-
e- tH')
rv
L C{3kb
k,
0=
(30
<
(31
<
(32
< ...
-----t
00.
k=O
(2.16) b) Suppose the hypothesis of 2.4 with m arbitrary and suppose for t - t 00 an asymptotic expansion (2.16). Then the coefficients bk in (2.16) are not local. Proof.
Izl
<
of a) (2.16) follows from 2.3 and 2.4. b) Write for
C1
tr
(S'(zm) ds~:m)) ~
t, dd
(2.17)
Then we see that the Ck in (2.13) and the dk in (2.17) are related by Ck = dk-I!k, k ~ 1. Hence the asymptotic expansion (2.16) determines S(,\) in a neighbourhood of ,\ = 0 up to an additive constant, the bk in (2.16) are nonlocal. D We want to define a relative (-function also in the case ]0, de (Jac(H') , Le. very far from a spectral gap above zero. Lemma 2.6 Suppose the hypothesis of proposition 2.4. Then for t -----t 00
(2.18)
264
Relative Index Theory, Determinants and Torsion
Proof.
00
J~('x)e-tAd'x J0-~'('x)d'x 01
tr(e- tH - e- tH') = -t
o
=
-tA
-t
_e-~('x)I~ [ -t
00
-tA
o
_e -t
J -~(o+) J ~(o+) + ~ J = ~(o+) + ~ J = -~(o+) -
e-tA(('x)d'x
o
00
J
E:
=
-
(('x)d'x -
e-tA(('x)d'x
o
E:
E:
=
2~z
o
(e-tAtrs*(,X) dS('x)) d,X d,X
+ O(e-t~
E:1
2~z
o
(e-tAffitrs*(,X) ds(,Xm)) d,X d,X
+ O(e-t~) (2.19)
o Now we would like to define again
((s, H, H') .- (l(S, H, H')
Je00 J
+ (2(S, H, H')
E:
.- _1_ f(s)
1
tr(e- tH - e-tH')dt
0
+ _1_ r(s)
t s - 1tr(e- tH - e-tH')dt
(2.20)
E:
and thereafter perform meromorphic extension.
Theorem 2.7 Suppose the hypotheses of 2.4 and for t
---> 00
an
Relative (-functions
265
asymptotic expansion 00
tr(e- tH - e- tH') '"
L ajt
CXj
,
-00 < ao < al < ... -+ 00.
j=O
(2.21) Then (2.20) defines a relative (-function which is meromorphic inC and holomorphic in s = O.
Proof. We start with
J 00
(l(S, H, H') =
ts-1tr(e- tH - e-tH'dt.
(2.22)
o
Inserting (2.21) into (2.22) yields as in (2.5) a meromorphic extension with simple poles at s = -aj which is holomorphic at s = 0 (since we multiply with f(s))' According to (2.19),
J 00
/" (S,' H H') =
':,2
tS-1tr(e- tH - e-tH'dt
JtS-l~(O+)dt + J 00
__1_
f(s)
00
_1_
tS-10(C-ln)dt.
f(s)
c
c
(2.23) Here the first integral is absolutely convergent in the half-plane Re( s) < 0 and equals there to s . f(s)
f -,-;-----=
s(~+'Y+s.h(s))
1 + s . 'Y . s . h( s) ,
which admits a meromorphic extension toC. The second integral yields a holomorphic function in the half-plane Re( s) < ~. The meromorphic extension to C is given by integration of (2.16) D with simple poles at s = i3k, k > O.
266
Relative Index Theory, Determinants and Torsion
Remark 2.8 We see from the proof of theorem 2.7 that we only need (2.18) to have a well-established (2, i.e. the stronger condition (2.21) is not necessary for that. Moreover, using tr(e- tH - e- tH')
1
00
= -t
e-tAe(,\)d,\,
we see that
tr(e- tH - e- tH') = bo + O(r{!) for t
-----t
00
(2.24)
is equivalent to
e(,\) = -bo + O('\{!) for ,\
~ 0+.
(2.25) D
Corollary 2.9 Suppose e- tH -e-tH' of trace class, (2.21) and (2.18) or (2.24). Then (2.20) defines a relative ( -function which is meromorphic in the half-plane Re( s) < {! and holomorphic in s = 0, which is explicitely given by
J 00
I bo 1 (2(s,H,H)=-qs+1)+qs)
(-tH -e -tH') -bodt. ) t S-I( tre
1
We apply this to our Clifford bundle situation.
Theorem 2.10 Let E E CLBN,n(I, B k ), k ~ r + 1 > n + 3, EI E gen comp~4t;f,F,rel(E) n CLBN,n(I, Bk)' Suppose additionally 10, E [c (J ac (D' ) and det S (,\) extends to a holomorphic function on De,I' Then (2.20) defines a relative (-function ((s, D2, (U* D ' U)2) which is meromorphic in C and holomorphic in s = O.
Proof. According to IV, theorem 3.2 for t > 0 e-tD2 P e-t(u* DU)2 pI is of trace class. Hence the wave operators are defined, complete and Sand are well-defined. Lemma 1.3 yields an asymptotic expansion of type 2.21 and we get (1 (s, D2, (U* D* U)2) as in (2.22) and thereafter. The assumptions provide (2.18) and we obtain (2(8, D2, (U* D ' U)2) as above. D
e
Relative (-functions
3
267
Relative determinants and QFT
It is well-known that the evolution of a quantum system is described by the S-matrix of QFT. For the elements of this matrix there exist well-known formulas, given by the Feynman path integrals. The exact mathematical understanding and meaning has occupied up to now many mathematicians. There are several approaches. One essential part of these path integrals is the so-called partition function which can be written (perhaps after a so-called Wick-rotation) as
z:=
J
e-S(A) dA.
(3.1)
Here S(A) is an action functional and A runs through an infinite dimensional space, e.g. a space of connections (in gauge theory), a space of Riemannian metrics and embed dings (in string theory). In many cases S(A) is of the kind (HA, Ah 2 , where H is an elliptic self-adjoint non-negative differential operator. The model, how to calculate (3.1) is now the Gaufi integral (3.2)
This yields a hint, how to attack, better to define the integral in the infinite dimensional case. One simply replaces the determinant in (3.2) by the zeta function determinant, if the latter is defined. Suppose the underlying space M n to be compact. Then (J"(H) is purely discrete and one defines
det(H)
:=
det(H)
:= e-/.-((s,H)ls=o = e-('(O) ,
(3.3)
(
((8, H)
=
L
A-s
>.eCT(H)
>'>0
with meromorphic extension.
(3.4)
268
Relative Index Theory, Determinants and Torsion
In the case of string theory,
00
Z =
L
Zp, I:p closed surface of genus p.
p=O
If the underlying manifold Mn is open then (J(H) is not purely discrete and (3.3), (3.4) do not make sense. We are now able to rescue this situation by considering relative determinants, det(H, H') := e-('(O,H,H'). (3.5) 1 +1
- 2
If E', E" E gencomp/diff,F(E) then we denote as above D' = - 2 (U* D'U)2, D" = (V* D"V)2 for the transformed operators acting in L2((M, E), g, h). Theorem 3.1 Suppose E', E" E gen compt~tf1fF(E) and inf (J 2 ' , (D l(kerD2)J.) > O. -2 -2 -2 -2 a) Then ((s, D2, D' ), ((8, D2, D" ), ((8, D' ,D") are after meromorphic extension in Re (8) > -1 well defined and holo2 morphic in 8 = O. In particular det(D2, D,2) = e-('(O,D ,D'\ 2 det(D 2,D,,2) = e-('(O,D 2,I5 det(D,2,D,,2) = e-('(O,D,2, I5 11 ) are well defined. b) There holds
I1\
(3.6) etc. and
Proof. a) follows from 2.1 and the fact that E, E" E gencomp(E) implies E" E gencomp(E')(= gencomp(E)). tD2 b) immediately follows from the definitions and tr( ee- tJ5112 ) = tr(e- tD2 _ e- tD ,2) + tr(e- tD ,2 _ e- tI5l1 \ D
Relative (; -functions
4
269
Relative analytic torsion
If we now restrict to the case E = (A*T*M 0C,gA) then g' E gencompl,r+l(g) does not imply E' = (A*T*M 0C,g~) E gen comp};~dtJf,F(E) since the fibre metric changes, gA ----t g~. Hence the above considerations for constructing the relative (function are not immediately applicable, since they assume the invariance of the fibre metric. Fortunately we can define relative (-functions also in this case. We recall from [38], p. 65 - 74 the following well known fact which we used in (1.1), (1.2) already. Let P be a self adjoint elliptic partial differential operator of order 2 such that the leading symbol of P is positive definite, acting on sections of a vector bundle (V, h) ----t (Mn, g). Let Wp(t,p,m) be the heat kernel of e- tP , t > O. Then for t ----t 0+ trWp(t,m,m)
rv
r%_!j(m)
+ r!j+lb_!j+l(m) + ...
and the bv(m) can be locally calculated as certain derivatives of the symbol of P according to fixed rules. As established by Gilkey, for P = D. or P = D2 the b's can be expressed by curvature expressions (including derivatives). This is (1.1), (1.2), (1.3). We apply this to e-tL:;. and e-t(U*i*L:;.'iU) but we want to compare the asymptotic expansions of WL:;.(t, m, m) and WL:;.,(t, m, m). The expansion of (4.1)
and
(4.2) coincide since
The point is to compare the expansions of (4.4) and
(4.5)
270
Relative Index Theory, Determinants and Torsion
i.e. we have to compare the symbol of i* b.'i = i* b.' and b.'. For q = 0 they coincide. Let q = 1, m E M, WI, ... ,Wn a basis in T:nM, E nl(M), b.'lm = + ... + ~nwn' Then, according to IV (2.55) i*(b.'i)lm = gklg~k~iwl' i.e.
eWl
((i* - l)b.~
(4.6)
Hence for the (local) coefficients of i* b.'i as differential operator holds (4.7) coeff of (i* b.~ i) = (gkl g~k) coeff of (b.~). Quite similar for 0 < q < n, e.g. coeff of (i* b.'2 i)
= (gkIhgk212g'tIkI g't2k2 ) coeff of (b.') 2 .
nt
3, Proposition 4.1 Let r > n + 2, g' E compl,r+l(g), l ::; b_~+l(b.(g, gA*), g, gA*, m) and b_~+I(U*i* b.'(g', g~* )iU, g, gA*) the coefficients of the asymptotic expansion of tr gAo W ~ (t, m, m) and tr gA * WU*i*~/iU(t, m, m) in L 2 (M, g), repectively. Then
b-~+I(b.,g,gA*,m)
Proof.
- b_~+I(U*i*b.'iU,g,gA*,m) E L1(M,g). (4.8)
Write
b-~+l(b.,
g, gAo, m) - b-~+I(U*i* b.'iU, g, gAo, m) = b-~+l(b., g, gAo, m) - b_~+l(b.', g', g~*, m) + +b-~+l(b.', g', g~*, m) - b-~+l(i* b.'i, g', gAo, m) + +b-~+l(i* b.'i, g', gAo, m) - b-~+I(U*i* b.'iU, g, gAo, m)
(4.9) (4.10)
(4.11)
where b-'!!:.+l(b.', g', g~*, m), b_'!!:.+l(i* b.'i, g', gA*) are explained in 2 2 (4.5), (4.4), respectively. (4.11) vanishes according to (4.3). (4.9) E L1(M,g) according to the expressions (1.3) and g' E compl,r (g). We conclude this as in the proof of 1.1. Finally (4.10) ELI according to i*b.'i = (i* -l)b.' + b.', coeff (i*b.'i) = coeff (i* - 1) + coeff (b.'), (i* - 1) E no,r(End (A*)), the rules for calculating the heat kernel expansion and according to the module structure theorem. 0
Relative (-functions
271
Theorem 4.2 Let (Mn,g) be open, satisfying (1), (B k ), k ~ r+1 > n+3, g' E compl,r+1(g),.6. = .6.(g,gA'),.6.' = .6.(g',g~,) the graded Laplace operators, U, i as in (2.54), and assume inf O"e (.6.1 (kerA).L) > O. a) Then for t > 0 e-tA -e -tU'i' A'iU is of trace class.
b) Denote h = tr(ip,5,c(.6.) - ip,5,e(U*i*.6.'iU)) for 0 < 3E < ~,
8 + 38
<
~, f.1
= inf{nonzero spectrum of .6.,i*.6.'i}. Then
J 00
(q(s,.6., .6.') := rts)
e-1[tr(e- tAq
-
e-t(U'i'A~iU)) -
h] dt
o
has a well defined meromorphic extension to Re (s) > -1 which is holomorphic in s = O.
c) The relative analytic torsion r a(Mn , g, g') , 1
log ra(Mn, g, g') :=
n
2 2)
-l)qq . (~(O,.6., .6.')
(4.12)
q=O
is well-defined.
Proof. a) is just IV theorem 3.8. b) immediately follows from theorem 2.2 and the proof of theorem 2.2. c) is a consequence of b). 0 We defined in II 4 gencompz~t}f,rel(g) which induces at the level of A*T*M gencompZ~dt}f,rel(A*T*,gA)' According IV 3.9, for t > 0 e-tA P _ et(U'i' A'iU) p' is of trace class. Then we can apply the asymptotic expansion (1.16) and obtain as above relative (-functions
(q(s,.6., .6.')
= (q(x, (M, g, .6. 9 ), (M', g', .6.~)),
which are holomorphic at s = O. Hence we got
Relative Index Theory, Determinants and Torsion
272
Theorem 4.3 Let (Mn,g) be open, satisfying (1), (B k ), k 2: l,r+lJ,reZ (Mn) r > n + 3 ,g' E gen comp L,diJ , 9 an d suppose inf (Te(~I(ker~)l.) > O. Then there is a well-defined relative analytic torsion Ta((M,g), (M',g')),
logTa((M,g), (M',g')):=
~ t(-l)qq. (~(O,~,~').
(4.13)
q=O
o The last step ist to give up assumption inf (Te(~I(ker~)l.) > O. Theorem 4.4 Let (Mn,g) be open, satisfying (1), (B k ), k 2: r > n + 3, g' E gencomp;;4:fJ,rez(Mn,g) and suppose additionally ]0, E[e (Tac(~') and that det S(.\) extends to a holomorphic function on De,ffi' Then there is a well defined relative analytic torsion Ta((M,g), (M',g')), logTa((M, g), (M', g')) :=
~ t ( -l)qq(~(O,~, ~'),
(4.14)
q=O
where
(q(s,~,~')
is defined by (2.20) and theorem 2.10.
0
Remark 4.5 The assertion of theorem 4.4 remains valid if we replace the assumption det S(.\) extends to a holomorphic func0 tion on De,m by (2.18).
5
Relative 1]-invariants
Finally we turn to the relative 7]-invariant. On a closed manifold (Mn, g) and for a generalized Dirac operator the 7]-function is defined as
J 00
7]D(S)
:=
'" Lt
sign .\
1
~ = r (81 1 )
0
t
8-1 2
tr(De
-tD2
) dt.
A E cr(D)
AiO
(5.1)
273
Relative <: -functions
is defined for Re (s) > n, it has a meromorphic extension to C with isolated simple poles and the residues at all poles are locally computable. r (Btl) . 'rJD( s) has its poles at n+~-v for /J E IN. One cannot conclude directly 'rJ is regular at s = 0 since r(u) is regular at u = ~,i.e. r (stl) is regular at s = O. But one can show in fact using methods of algebraic topology that 'rJ( s) is regular at s = O. A purely analytical proof for this is presently not known (cf. [38], p. 114/115). (5.1) does not make sense on open manifolds. But we are able to define a relative 'rJ-function and under an additional assumption the relative 'rJ-invariant. De- tD2 is an integral operator with heat kernel DpWD(t,m,p) which has at the diagonal a well defined asymptotic expansion (cf. [16], p. 75, lemma 1.9.1 for the compact case) 'rJD(S)
(5.2) In [3] has been proved that the heat kernel expansion on closed manifolds also holds on open manifolds with the same coefficients (it is a local matter) independent of the trace class tD2 property. The (simple) proof there is carried out for e- , trW(t, m, m), but can be word by word repeated for De- tD2 , DW(t, m, m). The rules for calculating the b-n±l (D2, D, m) are 2 quite similar to them for b-n±l (D, m) (cf. [38], Lemma 1.9.1). 2 We sum up these considerations in Proposition 5.1 Let E' E gencompi~dt;f,F(E), r Then for t > 0
+ 1 > n + 3. (5.3)
is of trace class, for t
---t
0+ there exists an asymptotic expansion
tr( e- tD2 D - e-t(u* D'U)2 (U* D'U))
=
I: Ja-~±l /=0 M
(m) dvolm(g)t
-~±l + O(d).
(5.4)
274
Relative Index Theory, Determinants and Torsion
Proof. The first assertion is just IV theorem 2.9. We recall from [38] the existence of an asymptotic expansion for the diagonal of the heat kernel
2D 1 D- , = (-tD e - e -t1512 15 ) (m,m )
=
~[2 ~ b_!!fl (D ,D, m) -
-2 - ]
b_!!fl (D' ,D', m) C
ill 2
+ O(t2). 4
1=0
(5.5) It can be proved, absolutely parallel to lemma 1.1, that [ ] E L 1 , O(d) ELI. Integration of (5.5) yields (5.4). D We recall from [49] the following
Proposition 5.2 Assume that D and fy = U*i* D'iU satisfy
(5.3), (5.4) and that the spectra of D and D' have a common gap [a, b], (CJ(D) U (J(fY)) n [a, b] = 0. Then there exists a spectral shift function ~(A) = ~ (A, D, D') having the following properties. 1) ~ E L 1,loc(lR) and ~(A) = 0 for A E [a, b]. (5.6) 2) For all
2 3) tr (e-tD D - e- t1512 D') =
J d~ (Ae-t.~2)~(A) dA.
(5.8)
lR
D
Pr~position 5.3 Assume E' E gencompZ~dtJf(E) and inf (Je (D l(kerD2).L) > O. Then there exists c > 0 s. t.
1 tr(e- tD2 D - e- t(U*i*D iU)2(U*i*D'iU)) = O(e- ct ).
(5.9)
Proof. We conclude as in the proof of 2.1 that there exists /1 > 0 s. t. (J(D) U (J(i* D'i) n ([-/1, -~] U [~, /1]) = 0 for all
275
Relative C; -functions
1/
~
•
I/O.
Hence, accordmg to (5.5),
J ft(>.e- tA )~(>.) d>' = 0 and J1,
2
-J1,
-00
J 00
+
e-
tA22
11_ 2t>'II~(>')1 d>.]
J1,
= C· e -t!!c"2. o
Theorem 5.4 Assume E' E gencompi~d~}f,F(E), k ~ r + 1 > n + 3 and inf o"e( D21 (ker D2).l.) > O. Then there is a well defined relative ",-function 00
'1l(s D D'):= 'I
"
r
1
Jt
(S~l) 0
S
;l
tr(De- tD2 _U*D' Ue- t (U*D ' U)2) dt
(5.10) which is defined for Re (s) > ~ and admits a meromorphic extension to Re (s) > -5. It is holomorphic at s = 0 if the coefficient J a_!(m) dvolm(g) of t-! equals to zero. Then there is a well 2 defined relative ",-invariant of the pair (E, E') ,
Proof.
We write again U* D'U
= fy. Then according to
276
Relative Index Theory, Determinants and Torsion
proposition 5.1, 00
~(s, D, fy) ~ r (~) [
I
[1',' tr( e~w' D -
1
e- HY ' 15') dt
00
+r (~) 1
t ',' tr(e-
W
'
1
n+3
D - e-H'''fj')dt
J
= r (s+1) s n I 1 a_!!±l dvolm(g) - 2 '" ~---+-+2 1=0 2 2 22M
J 1
+r
1
(s~1)
0
8-1
4
t-2 O(t2)dt
(5.11)
(5.12)
We infer from (5.9) that (5.13) is holomorphic in C. (5.12) is holomorphic in Re (s) > -5. (10.45) admits a meromorphic extension toC. T/(s, D, D') is holomorphic at s = 0 if the coefficient J a_! (m) dvolm(g) equals to zero. 0
M
2
Theorem 5.4 immediately generalizes to the case of additional compact perturbations. Theorem 5.5 Assume E' E gencompi~dtJf,F,rel(E), k ~ r+1 > n+3
Relative r:;, -functions
277
Then there is a well defined relative 'T/-function
J 00
D fy) :=
71(S '1
"
r
e! 1
1
(;1 tr(De- tD2 P - fYe- tfy2 Pl)dt
) 0
'
which is defined for Re (s) > ~ and admits a meromorphic extension to Re (s) > -5. Here fy = U* D'U as above. 'T/(s, D, fy is holomorphic at s = 0 if the integrated coefficient a_!2 =
-J
J
b_!2 (D2, D, m) dvolm(g)
K
b_ frac12 (fy2, fy, m/) dvolm, (g')
K'
+
J
M\K=M'\K'
equals to zero. We repeat for the proof the single arguments from the proof of 5.4 which remain valid in the case of 5.5. 0
6
Examples and applications
In this section, we present examples of pairs of generalized Dirac operators which satisfy the assumptions of sections 1-5 and present applications of some theorems of these sections. Let (Mn,g) be open with finitely many collared ends Ci, the collar [0, oo[ XN;-l of Ci endowed with a warped product metric, i.e. glC:i ~ dr2 + fi(r)2da}vi' Ni closed, hi = da7vi, i = 1, ... ,m. We consider one end C with collar [0, oo[xN with the warped product metric ds 21c: = dr2 + f(r)2da 2 and we first calculate the curvature. Let Uo, U1 , ... , Un - 1 be an orthogonal basis in T(r,u) ([0, oo[xN) with respect to ds 2, U = :" U1 , ... , Un - 1 orthonormal in TuN
278
Relative Index Theory, Determinants and Torsion
with respect to d(J"2 = h. Then, in coordinates (r,u\ ... ,un-I), we get for the Christoffel symbols r~,,8(g), Ct, (3, '"Y = 0, ... , n -1, the following expressions
rgo = 0,
rg
c k > 0, r k00 = 0 lor
r OJk
r?j = -1' fh ij for i, j
> 0,
j
= 0 for --
j
> 0,
!Ls:k c . k 2 f' U j lor J,
(6.1)
> 0,
r~j = r~j(h) for i, j, k
>
o.
For the curvature tensor and the sectional curvature holds
1"
R(Uo, Ui)UO = jUi
(6.2)
R(Uo, Ui)Uj = -1" fhijUO (6.3) R(Ui , Uj)Uo = 0 (6.4) R(Ui , Uj)Uk = - f'2(h jk Ui - hikUj ) + RN(Ui , Uj)Uk , (6.5) which implies immediately
(6.6) (6.7) Here i, j, k = 1, ... ,n - 1. The easy calcualtions are performed in [28]. It is now easy from (6.2) - (6.7) to calculate the general curvature K(V, W).
Examples 6.1 1) Take f(r) = e- r , N fiat, then K == -1, E satisfies (Bo) but rinj(E) = O. 2) Choose f(r) = e- r , KN =1= 0, then E does not satisfy (Bo) and again rinj(E) = O. 3) If f(r) = er , N fiat, then E satisfies (Bo) and rinj(E) > O. 4) Finally take f(r) = er2 , then E does not satisfy (Bo) but (I). Hence all good and bad combinations of properties are possible. D
279
Relative (-functions
Proposition 6.2 Suppose f(r) such that
inf f(r) > 0 or f monotone increasing
(6.8)
If(lI)1 ::; cllf, v = 1,2, ....
(6.9)
r
and Then 9 Ie satisfies (/) and (Boo).
Proof. rinj(C:)
inf f (r) > 0 and
h) > 0 immediately imply > O. (6.6) and (6.7) immediately imply (Bo). (Bd is r
rinj (N,
equivalent with (B o) and
IV'eJR(eiJ,e),)e,,)lx::; C IV'e",eiJlx ::; c,
(6.10) (6.11)
eo, ... ,en-l tangential vector fields, orthonormal in TxM. We apply this to x = (v,y) E [O,oo[xN, Uo = Ui = a~i' eo = Uo, Un-I Then accor d'mg t 0 (6 . 1) , el -- UI f ' ... , en-l -- -1-'
tr,
V'ei eO
We see, each term on the r.h.s. of (6.12) - (6.15) is - up to a constant or bounded function - a sum of terms
l'
l'
1
jei, jeo, 7el
(6.16)
with pointwise norms (w.r.t. gle)
I'll'7'
Ij
(6.17)
280
Relative Index Theory, Determinants and Torsion
i.e. (6.11) is satisfied. Next we establish (6.10)
f 1'" - ~f' 1"
P
(6.18)
ei,
1
P "Vuj(R(Uo, Ui)Uo) =
1" 1 p "VUjjUi
f"
k J3 "VUjUi = f J3" (-,f fhijUO+ rij(h)Uk)
1" f' 1" k -yhijeO + p rij(h)ek,
(6.19)
1
"V Ua pR(Uo, Ui)Uj ) -
~' R(Uo, Ui)Uj ) + )2 "V Ua ( - 1"fhijUO)
-
~' (-1"fhijUO) + )2 (-1"'f -
2f'1" (fill phijeO- T (
1"f')hijUO
1"f') +Y hijeo
f'1" Y - Tfill) hijeo,
(6.20)
)3 "Vuk(R(Uo,Ui)Uj) = )3 "VUk (-f"fhijUO) 1"
I
- phijrkOUl I1"f'
1"
If'
I
= - phij '2j[)kUl
-'2phij Uk ,
(6.21)
Relative (-functions
1
J2 "VUo(R(Ui , Uj)Uo) =
281
0
1
J3 "Vuk(R(Ui, Uj)Uo) "Vek(R(ei,ej)eO),
(6.22)
r1 "Vul(R(Ui, Uj ), Uk) =
1 "V Ul [/2( f4 - f hjkUi - hikUj
+
RN(Ui , Uj)UkJ
1'2
-]4(hjk "VUIUi - hik "VUIUj )
+
1 f4 "Vul(RN(Ui, Uj)Uk)
1'2
-]4(hjkrr:(h)Um - hikrlj(h)Um)
+
r1 "Vul(RN(Ui , Uj)Uk).
(6.23)
If we take the pointwise norm of the r.h.s. of (6.18) - (6.23) and apply the triangle inequality, then we obtain on the r.h. sides a finite number of terms, each of which is - up to a constant or a bounded function - of the type
1
If(Vl)I·lf(V2)1
fa
J2
a 2: O.
(6.24)
But according to (6.9), each term of the kind (6.24) is bounded on c, i.e. we established (Bl)' To establish (B2)' we have at the end to estimate expressions of the kind
(j).(j)',
( f').~ f f' (6.25)
282
Relative Index Theory, Determinants and Torsion
Again, according to (6.9), each term of the kind (6.25) is bounded. A very easy induction now proves (Bk) for all k, i.e. (Boo). 0 Collared ends are isolated ends. Hence, if all ends of an open manifold Mn are collared, then Mn can have only a finite number of ends. If an open manifold has an infinite number of ends, then at least one end is not isolated.
Theorem 6.3 Let (Mn,g) be open. If each end E of Mn is collared then M has only a finite number of ends, El,.··, Em. Suppose glc; ~ dr2 + Jl(r)dO"Jv; such that each fi satisfies {6.8} and {6.9}. Then (Mn, g) satifies (1) and (Bo). This follows immediately from proposition 6.2. 0 Interesting examples for the fs are f(r) = e9 (r) , g(r) > 0 and g(v)(r) bounded for all v. We consider here the special case g(r)
= b· r, b> 0 i.e. (6.26)
In the sequel, we need the knowledge of the essential spectrum O"e of such manifolds.
Theorem 6.4 Suppose (Mn,g) has only collared ends Ei, i = 1, ... , m, each of them endowed with a metric of type {6.26}. Then there holds
O"e(t:.q(Mn , g)) \ {O}
~ [m,m ( min { (n - ~q 00 [ \
{O} for q =I ~
1)\~, (n - ~q + 1) \; }) , (6.27)
and (6.28)
Relative (-functions
283
We refer to [3J for the proof which essentially relies on [28J.
0
Corollary 6.5 Suppose the hypotheses of 6.4, n even and minb~ ~
> O.
(6.29)
Then the graded Laplace operator D. = (D.o, ... ,D.n ) has a spec0 tral gap above zero.
A special case of theorem 6.4 is the case of a rotationally symmetric metric at infinity, i.e. (Mn \ K M, gIM\K) ~ (lRn \ KlRn, dr2 + e2br dO'~n-l)' Then for b > 0
for q -:F
n
2'
If we replace e2br by (sinh 1')2 and set K real hyperbolic space H:!:.l and 0'
e
(D. (Hn )) = q
-1
(6.30)
= 0, then we get the
(~)2b2} [ [{( ~)2 2' 2 ' 00 } [1 [ { {O U 4,00
for q -:F % for q = % (6.31)
Corollary 6.6 In the case (Mn,g) ~ (lR 2k ,dr2 + e2brd~n_l) or (Mn, g) = H~1, the graded Laplace operator D. = (D.o, ... , D. n ) has a spectral gap above zero. 0 Corollary 6.7 In the following cases, the graded Laplace operator D. = (D.o, ... ,D.n ) has a spectral gap above zero. a) (Mn,g) is a finite connected sum of manifolds with collared ends, warped product metrics {6.26} satisfying {6.29} and n is even, b) any compact perturbation of manifolds of a}, c) any finite connected sum of manifolds of type (Mn \ K M, gIM\K) ~ (lRn \ KlRn,dr2e2brdO'~n_l)' b> 0 and n even,
284 d) e) f) g)
Relative Index Theory, Determinants and Torsion any compact perturbation of c), any finite connected sum of the hyperbolic space H'!.-1, any compact perturbation of e), any (M2k, g'), g' E comp~~+1(g), 9 of type a) - f).
0
Remark 6.8 Compact perturbations and connected sums of collared manifolds with (6.26) are again of this type. We introduced 6.6. a), b) to indicate how to enlarge step by step a given set of such warped product metrics at infinity by forming connected sums and compact perturbations. 0 We apply the facts above to the case E
=
(A*T* M ®C, gA*), V 9A *), D
= d + d*, D2 = D. = (D.o, ... ,D. n ).
Theorem 6.9 Let (M2k,g) be one of the manifolds 6.6 a) - f), g' E comp~~+l(g), g' smooth, r + 1 > 2k + 3. Then the relative (-function (q(s, Do, Do') as in section 4 and the relative analytic torsion, Ta((M, g), M', g')),
2k
log Ta((M, g), (M', g')) =
2:) -l)qq· (~(O, Do, Do') q=O
are well defined.
Proof. According to proposition 6.2, (Mn,g) and (M',g') satisfy (1) and (Boo), and the general Laplace operator has a spectral gap above zero. The assertion then follows from theorem 4.4. 0
Corollary 6.10 Let (M,g) be as is 6.6 a) - g). Then the attachment (M,2k, g') ----t Ta((M, g), (M', g')) yields a contribution to the classification of the elements of gen compi~~Jf,rel (A*T*,gA*)'
285
Relative (, -functions
Remark 6.11 If n = 2k + 1 and (Mn, g) belongs to one of the classes 6.6 a) - g) then the relative (-functions (( s, !::l.q, !::l.~) are for (Min, g') E gencompi~dtJf,rel(Mn, g) n C r +l and q 1= k, k + 1 well defined. D
Another very special case is given by b = 0 in (6.26), i.e. cylindrical ends E, (6.32) ds 2 = dr 2 + da'iv. Suppose, we have m cylindrical ends
Ei,
i
= 1, ... , m,
and let {Xl (i)} k be the (purely discrete) spectrum of !::l.q (Nr- l , hi), i = 1, ... , m. Proposition 6.12 Then
ae(!::l.q(M, g» = U U([Ak(i), OO[U[Arl(i), oo[). i
(6.33)
k
We refer to [3] and [28] for the proof.
D
Corollary 6.13 a) If Hq(Ni ) = Hq-l(Ni ) = (0), i = 1, ... , m, then a(!::l.q(M)) has a spectral gap above zero. b) If for at least one i Hq(Ni ) 1= 0, then a(!::l.q+l(M)) = a (!::l.q(M) = [0,00[. e) In the case of cylindrical ends, the graded Laplace operator never has a spectral gap above zero. Proof. a) and b) immediately follow from (6.34). c) follows from HO(Ni ) 1= 0 for all i, hence a(!::l.o(M)) = a(!::l.l(M)) [0,00[= a(!::l.o, ... ,!::l.n). D
Corollary 6.14 Suppose (Mn, g) with cylindrical ends El,'" ,Em and let (M,n,g') E gencompL,diff,rel(Mn, g). If Hq (Ni ) = Hq-l(Ni ) = (0), i = 1, ... , m, then ((s, !::l.q, !::l.~) and det(!::l.q, !::l.~) are well defined. D
286
Relative Index Theory, Determinants and Torsion
A special case is given by the pair
(Mn,g) and
(Q
N, x [0,
Q
oo[~ N x [0,00[' dr' + INr )dO"~,) ~ (M'", g').
Here M'n is a manifold with boundary 8M'n = N = UN i , and the latter falls out from our considerations. But if we consider the case q = 0 and b.. o(Min, g') with Dirichlet boundary conditions at 8M' = N = U N i , then we get an essentially selfadjoint operator b..~, O"(b..~) = [0,00[= [O,oo[U U [Aj, 00[, and Aj>O
e- tLlo -
e-tLl~pl is of trace class.
The latter fact is an immediate consequence of the proof of theorem 3.9. Hence the wave operator W±(b.. o, b..~) exist, are complete and the absolutely continuous parts of b.. o and b..~ are unitarily equivalent. We intend to present an explicit representation of the scattering matrix S(A), of tr(e- tLlo - e-tLl~) and of the relative (-function. Here we essentially follow [49]. Then
and O"p(b.. o) consists of eigenvalues
o < /-ll
::; /-l2 ::; /-l3 ::; . . . --+
00
of finite multiplicity without finite accumulation point and (6.34) (6.35) immediately implies for t > 0 (6.35) and (6.36)
287
Relative C. -functions
where ~OIL2,O"p is the restriction of ~o to the subspace spanned by the L 2-eigenfunctions. If we apply (6.34) to the case q = 0 then we obtain (6.37) i
k
j
where the AjS are the eigenvalues of ~o(N) and simultaneously the thresholds of the (absolutely) continuous spectrum. We describe the continuous spectrum in terms of generalized eigenfunction. Consider ~o(N)(= (~O(Nl)"'" ~o(Nm)), an eigenvalue Aj E O'(~o(N)) and let E(Aj) be the corresponding eigenspace. For f../, > Aj and different from all thresholds and for <.p E E(Aj) there exists a unique eigenfunction E(<.p, f../,) E COO(M) of ~o for the eigenvalue A, the restriction to lR+ x N of which has the representation E(<.p,f../,)(r,y) = eirVI1-->'j<.p(y)
+L
e- ir ,/J1.->'I(Tjl (f../,)<.p)(y)
+'l/J,
>'1<11(6.38) 'l/J E L2(lR+ x N) and Tjz(f../,) : E(\) ----t E(AZ) a linear operator. Let ~ be the minimal Riemann surface to which all functions f../, ----t f../, - Aj, j E 1N extend as holomorphic functions.
J
Lemma 6.15 Each generalized eigenfunction E(Q, f../,) extends to a meromorphic function on ~. D
See [49] for a proof. Define now
and set S(f../,) =
EB
Sjz(f../,).
>'j,>'I<11S(f../,) is the scattering matrix of S(f../" ~o, ~~). The E(Q, f../,) are meromorphic functions on ~, which implies the same for Tjl(f../,) and Sjz(f../,). This yields (cf. [49])
288
Relative Index Theory, Determinants and Torsion
Proposition 6.16 The scattering matrix S(f.L) is a unitary operator of finite rank, smoothly depending on f.L E] Aj -1, Aj [. The rank changes when f.L crosses a threshold. The coefficients of S(f.L) extend to meromorphic functions on ~. D Proposition 6.17 For t > 0 there holds
tr(e- tllo - e-tll~pl) = Le-t/Lj
1
00
+"2 LdimE(Ak)e- tAk k=l
j
1
+ 4trS (0)
J 00
+
1 27ri
tA d e- dA log det S(A)dA. (6.39)
o D We refer to [49] for the proof. Next we want to establish an explicite expression for the relative (-function ((8, ~o+z, ~~+z), which is well defined for ~(z) > O.
Proposition 6.18 Under the above assumptions, the relative (-function ((8, ~o + z, ~~ + z) for ~(z) > 0 is given by
j
k
(6.40)
Proof. At first we remark that for 0 < A < AI, S(A) is a scalar, S (A 2 ) extends to a meromorphic function on {z E ell z I < AI}, which is holomorphic at A = 0 and, in particular, S(A) is holomorphic in Uc(O), E small. Using this and equation (6.39), we see that for t ---> 00 we have a representation tr( e- tllo e-tll~PI) = bo + O(re), (2 > 0, which is equivalent to ~(A) = -bo - O(Ae) for A ---> 0+. Hence (2.24) and both assumptions
289
Relative (, -functions
of proposition 2.4 are satisfied. We obtain from corollary 2.5 a) and (6.39) an asymptotic expansion
1 tr(e-t~o-e-t~~ PI) "" dim ker ~+4trS(0)+
z=
00.
Cjr~,
t --+ 00.
j=l
Moreover, for t --+ 0+, we have the standard expansion 00
tr(e-t~o - e-t~~pl)
"" Z=ajr~+j,
t --+ 0+ ,
j=O
and we get the existence of the relative (-function ((8, ~O, ~~) as a meromorphic function. According to [49]
J:z A
log det S(A)dA = O(A n) as A --+ 00.
(6.41)
o
Using this and inserting (6.39) into (,), we obtain finally (6.40). D
It is clear that the product geometry of cylindrical ends is an extremly special case of possible geometries on E = N x [0,00[, and one should admit much more general bounded geometries on M, e.g. bounded geometries of the type gc = gINX[O,oo[ = dr 2 + (e 9 (rl)2dc/iv, g(r) > 0, g(l/l(r) bounded for all 1/. Again we get a pair (~o = ~o(g), ~~ = ~~(gc))' where (E, gc) E l,r J,rel (M) If gen comp L,diJ , g, e -t~o - e _t~10 pI 0 f t race c1ass. inf O"e(~O) > 0 or for t --+ 00 (6.42) then the relative (-function ((8, ~O, ~~) and relative determinant are defined. Explicit g(r) leads to explicit calculations. Similarly for q > O. If e.g. n = 2m, 9 on M is such that glc = dr2+e rdO"'iv and 9 = m then inf O"e(~ml(ker~mlJ.) > O. Here we take for ~~ the Friedrichs' extension of ~m on M x [O,oo[
290
Relative Index Theory, Determinants and Torsion
with zero boundary condition on M x {O}. Hence ((8, .6. m , .6.~) and det (.6. m , .6.~) are well defined. In the case of a manifold M = M'UN x [0,00[,9) with cylindrical ends, the main and interesting part of the geometry is contained in the compact part M', 8M' = N. At the boundary we assume product geometry. X = (M'UM', 9x = 9M,U9M' is then a closed manifold. It is now N a natural and interesting question, how are the .6.-determinant for X and the relative .6.-determinant for M related? The answer would also give a meaning, an interpretation of the relative determinant. A certain answer is contained in [53], and we give an outline of the corresponding result. Consider the following situation. Given (Mn, 9) closed, oriented, connected, Y c M a hypersurface, separating M into two components MI , M 2 . Set Mi = Mi. i = 1,2. Mi are compact with boundary Y, M = MI U M 2 , Y = 8MI = 8M2 . Morey over, let E ---+ M be a Hermitean vector bundle and .6. = .6. M : COO(M, E) ---+ COO(M, E) a Laplace type operator, i.e . .6. is symmetric, non-negative with principal symbol 0' O by setting Mr = MI U N r U M 2 ,
::2
Y
y
where N r = Y x [-r, r] and 9rlMi as before, 9rlNr = dr2 + 9y. Similarly, we extend the bundle E ---+ M to Er ---+ Mr. Then .6. Mr , ((8, .6. Mr ) and det .6. Mr = e-('(O,
::2
;22
291
Relative (-functions
gen COmPL,diff,rel(Ei,oo ----t (Mi,oo, 9i,oo)) and, according to the above considerations the relative (-function (( s, .0.i,oo, .0. 0 ) and the relative determinant det(.0. i,oo, .0. 0 ) = e-('(O,Ll.i,oo,Ll.O) are well defined. Denote r(s-!) 1 ~y(s) := J7ff(:) ((s - 2' .0. y). Theorem 6.19 Assume ker.0. y 1,2. Then .0. Mr is invertible for
= {O} and ker .0. i,oo = {O}, i = l' ~ 1'0
and
2
lim er~~(O) det .0. Mr
r--+oo
= (det .0. y)-! II det(.0. i,oo, .0. 0 ),
(6.43)
i=l
We refer to [53] for the proof.
0
Remark 6.20 (6.43) establishes a connection of the desired kind. We cannot expect a "simple formula" which relates det(.0. M ), det(.0. Mll det(.0. M2 ) since this would contradict the global character of ('(0) and det(·). 0
For E' E gencompi:~d~Jf,rel(E), inf oAD21(kerD2).L) > 0, we have the asymptotic expansion (1.16),
c¥ c_!!2
+
+ ... + c¥+[~]c -2+ [!!H] 2 n
O(C¥+[n;3])
and can express the logarithm of the determinant more explicitely. First we remark that in U(s = 0) 1 f(s) = - +1" s· h(s), s where h( s) is a holomorphic function near s = 0 and I' denotes the Euler constant. Hence
~I
_1_
ds s=of(s)
~I
ds 8=0 1
=1
,
(_1_~) r(s) S
r(s) 18=0 = O.
_~I
1
_
- ds 8=of(s + 1) -
-f'(l)
,
292
Relative Index Theory, Determinants and Torsion
(1.16), (2.4) and (2.11) then imply -10gdet(D2, D,2) = :8 Is=0(1(8, D2, D'2) - h· f'(l)
J 00
+
r1[tr(e-
tD2
P - e-
tIJl2
P') - h]dt
1
[~]
L
J 1
C-~+l -
cof'(l)
+
r1[tr(e-
tD2
P - e-
tD'2
P')
0
1=0
[~]
-L
C_~+lr~+l]dt - hf'(l)
1=0
J 00
+
r1[tr(e-
tD2
P - e-
tDl2
P' - h]dt.
(6.44)
1
Let as in section 1 Pd(D) be the projector on the discrete subspace and Pc = 1 - Pd the projector on the continuous subspace and denote by D d and Dc the corresponding restrictions of. These subspaces are orthogonal and remain invariant under Dd or Dc, respectively. Suppose that e-tD~ is of trace class. Then e-tD~ - e- tDl2 is of trace class too. If for e-tD~ there exist an asymptotic expansion of the type (1.1) then there exists a well defined zeta function ((8, D d ) and det DJ can be defined by det D~ := e-('(O,D~).
(6.45)
Then (3.7), (6.45) and
+ e-tD~p _ e- tfy2 P') tre-tDJp + tr(e-tD~ P _ e- tfy2 P') tr(e-tDJP
yield
(6.46) If a > 0, then inf a"e(D2 + a) > 0 then h = 0 in (2.9), (2.24) for the pair D2 + a, iy + a is satisfied and the corresponding relative
Relative (-functions
293
(-function is given by
rts) J 00
«(s, D2 + a, fy2
+ a) =
tS-le-tatr(e-tD2 - e-
tfy2
)dt,
o (6.47) Re( s) > -1. The r.h.s. of (6.47) is also well defined if we replace a by any z E C with Re(z) > O. Then the corresponding function «(s, z, D2, fy2 admits as function of s a merom orphic extension to C which is holomorphic at s = 0 and we finally define
det(D2
+ z, fy2 + z)
:=
e-/s ls =o((s,z,D 2,D/2).
An important property concerning the z-dependence is expressed by l,r+l (E) an d Proposition 6.21 Suppose E' E gen comp L,diJ J,rel {2.24}. Then det(D 2 + z, fy2 + z is a holomorphic function of z E C\]O, 00[.
Proof. Here we essentially follow [49]. According to V, lemma 1.10 a),
Je-tA~(A)dA Je-tA~(A)dA. 2c
tr(e-
tD2
Pe- tD/2 PI)
=
-t
00
- t
o
2c
The second integral on the r.h.s. is O(e- tC ) for t e-tA~(A) E L1IR for t > O. Moreover.
Je-tA~(A)dA
----+
(6.48) 00 since
N
2c
=
o
I:
Ck tk
+ O(t N +1 )
for t
----+
0,
(6.49)
k=O
N E IN arbitrary. (1.16) and (6.49) imply (by taking the dif00
ference) that
J e-tA~(A) dA has a similar asymptotic expansion as 2c
(1.16). We infer from this that the integral
rts) J Je-tA~(A)dAdt 00
F(s, z) =
00
tSe- tz
o
2c
294
Relative Index Theory, Determinants and Torsion
in the half planes Re(s) > -~ and Re(z) > -c absolutely converges and, as function of s, it admits a meromorphic extension to C which is holomorphic at s = o. The first integral on the r.h.s. of (6.48) can be discussed as follows. We obtain for Re(z) > 0
J Je-t>'~()")d)"dt J~()..) J J + )..)-(s+l)~()")d)". 2c
00
tSe- tz
__1_
f(s)
o
0
2c
00
= _1_ f(s)
ee-t(zH)dtd)..
o
0
2c
= -s
(z
(6.50)
o
Hence, for Re(z) > 0, 2c
2
J (zH)-l~(>')d>'+ ~F (O,z).
- 2
det(D +z,D' +z)=e o
s
(6.51)
The r.h.s. of (6.51) has an obvious extension to an analytic (0, z) is holomorphic in Re(z) > function of z E C\] - 00,0], -c. c > 0 was arbitrary, hence we get an analytic extension to
c::
C\]- 00,0].
0
It is an interesting question, how the relative determinant and the relative torsion change under I-parameter change of the metric. We could consider e.g. the most natural evolution of the metrix which is given by the Ricci flow,
o
07 g(7)
.
= -2RIC (g(7)),
g(O)
=
go·
(6.52)
If (Mn, go) is complete and has bounded curvature then, according to [69], [25], there exists for 0 :::; 7 :::; T in the class of metrics with bounded curvature a unique solution of (6.51).
295
Relative (, -functions
Denote ~(T) diagram
~q(g(T)).
-
A(T)
is defined as before by the
~(O) ----t
L2(g(0)) :) 'D{),.(O)
~
L2(g())
~(T) • dvol(O) dvol(r)
i . dvol(r)
1
dvol(O)
~(T)
L2(g(T)) :) 'D{),.(r)
----t
L2(g( T))
It is not yet clear, whether g(T) E compl,r+l(g(O)). We proved in [25] that g(T) E bcomp2(g(0)) , but g(T) E compl,r+1(g(O)) is still open. Therefore we make the following · E' A ssumpt Ion.
l,r+lf (E) C gen comp L,dif l,r+lf (E) . Here comp L,dif comp(·) denotes the arc component. Hence there exists an arc {E(T)}-e~r~e connecting E and E', we assume in the sequel to the arc to be at least C l . E
Then we get a Cl-arc {D(T)}rl and we will study the behaviour of the relative determinant det(D2(0), D2(T)) under variation of T. Additionally, we suppose again (2.24). As before and in the sequel, D2 (T) denotes the transformed to the Hilbert space for
T = 0 D2(T). Denote D2(T)
= trD2(T). By Duhamels principle,
d~ e- tjj2 (r)
J t
=
e- sjj2 (r)
D (T)e-(t-s)jj2(r)ds. 2
(6.53)
o
If D2(T)e- tD2 (r) is for i > 0 of trace class with trace norm uniformly bounded on compact i-intervals lao, al], ao > 0, then according to (6.53), d~e-tjj2(r) is also of trace class for i > 0 and (6.54)
296
Relative Index Theory, Determinants and Torsion
To establish in the sequel substantial results, we must make two additional assumptions.
Assumption 1. D2(T) is invertible for T E [-s,s]. Assumption 2. There exists for t pansion 00
tr(D 2 (T)b- 2(T)e-
tjj2
(T))
<"V
--t
(6.55)
0+ an asymptotic ex-
k(j)
L L>jk (T)(Yj logk t + C (T) + C2(T) log t 1
j=Ok=O (6.56) with -00 < 11 < 12 < ... , Ij =1= 0, j E IN and Ij implies the existence of c > 0 such that
tr(D 2(T)[)-2(T)e- tJ52 (r)) = O(e- ct ) for t We infer from (6.54)-(6.57) that for Re(s)
»
--t
--t
00.
00.
(6.55) (6.57)
0
d~ ((s, D2(0), [)2(T))
Tn
Proposition 6.22 Suppose that the arc {D( -o
-
dT logdet(D2(0), D2(T)) = C 1 (T)
+ r'(1)C2(T),
C1(T), C2(T) as in {6.56}.
Proof. Consider
J 00
t S - 1 tr(D 2 ( T)[)-2( T)e- tJ52 (r))dt.
o
(6.59)
Relative (, -functions
297
This integral has an analytic extension to a meromorphic function of sEC with poles determined by the expansion (6.56). The pole at s = 0 has order ~ 2 with Laurent expansion
Inserting this into (6.58) and differentiating at s (6.59).
o yields o
Corollary 6.23 Suppose the hypothesis of 6.22. Then (6.60)
o Until now we defined the relative (-determinant only for pairs D2, D'2. But there is a reasonable way also to introduce det(D, D'). Let (Mn, g) be closed. Then Singer already defined in [70] by very natural considerations (6.61)
Consider now the open case, E' E comptdtJf,F,rel(E) , and assume for a moment inf 0"(D2), inf 0"(D,2) > O. Then D and D' are invertible and inf O"e(D'21(ker DI2).L) > O. We can apply 2.2 and 5.4 with h = 0 and obtain
((s, D2, D,2) = tr((D 2)-S _ (fy2)-S) = tr(IDI- 2S _ Re (s) > -5.
ID'I- 2S), (6.62)
(6.62) yields (6.63) and (6.63) has a meromorphic extension to C which is holomorphic at s = O. Hence
Relative Index Theory, Determinants and Torsion
298
is well defined and we set in analogy to (6.61) det(D, D') := det(IDI, ID'I) . e¥(1](O,D,D')-((O,IDI,I1J 'I)).
(6.64)
But to define (6.64) we don't need D, D' invertible. The former assumption inf a"e(D2 I(ker D2).L) > 0 is sufficient to define (6.64) since this assumption is sufficient for the 1.h.s. of (6.63) and D hence for the Lh.s. of (6.64). Finally we draw some conclusions for the Schrodinger operator fl. q + V, V as in IV, proposition 3.11. Theorem 6.24 Suppose 9 E M(I, B k ), k ~ r > n + 2, V E n1,r (Mn, 9) a real-valued function. Then the absolutely continuous parts of fl. q and fl. q + (V-) are unitarily equivalent. Proof. This immediately follows from IV 3.11 and the completeness of the wave operators. D
Corollary 6.25 Suppose V E n1,r (IRn, 9standard) , real-valued, r >n
+ 2,
fl.
of fl. and fl.
=-
+ (V-)
n
2
i=l
'
L~' Then the absolutely continuous parts
are unitarily equivalent.
D
VII Scattering theory for manifolds with injectivity radius zero In chapters II - VI we always assumed rinj(M, g) > O. The background for this assumption was the fact that to establish our uniform structures, we used the module structure theorem for Sobolev spaces and this theorem has been proved until now under the assumption rinj > O. An extension to the case rinj = 0 for weighted Sobolev spaces is in preparation. In [54] W. Muller and G. Salomonsen introduced a scattering theory for manifolds with bounded curvature, admitting Tinj(M, g) = O. They restrict themselves to the case of the scalar Laplacian ~o = (\79)*\79. We partially follow here an extended to arbitrary Riemannian vector bundles version of their approach but using our language of components of uniform spaces and our procedures of chapter IV to establish trace class properties.
1 Uniform structures defined by decay functions We apply the procedure of [27], [32], our chapter II and reformulate and extend the approach of [54]. Let V E CO ([1 , oo[) be non-increasing. It is called of moderate decay if (1.1 ) 1) sup x· V(x) < 00 XE[l,oo[
and 2) there exists C = C(V) such that V(x + y) ~ C(V) . V(x) . V(y). 299
(1.2)
300
Relative Index Theory, Determinants and Torsion
It is called of sub exponential decay if for any c > 0
eCXV(x)
---t
(1.3)
00.
x---?oo
Examples 1.1 1) V(x) = e- tx is for t > 0 of moderate decay. 2) If V, Vi, V2 are of moderate decay then this holds for V", 6 > 0, Vi . V2 too. The same holds for subexponential decay. 3) V(x) = X-I and e- X <> , 0 < a < 1, are of sub-exponential ~~ D Denote by M = M(M) the set of all complete (smooth) metrics on M and let g, hEM. Then we define
m-I
mig - hlg(x) := Ig - hlg(x)
+ L I(\7 9 )i(\79 -
\7 h)lg(x) (1.4)
i=O
and
b,ml g _ hl g := sup mig - hlg(x). xEM
Remark 1.2 The conditions b,Olg - hl g == big - hl g < 00 and big - hlh < 00 are equivalent to g and h quasi-isometric. D Lemma 1.3 Suppose big - hl g, big - hl g < for every m ~ 0 a polynomial
00.
Then there exists
with non-negative coefficients and without constant term such that mig - hlh(X) ::; Pm(lg - hlg(x), ... , 1(\7g )m-I(\7 g
-
\7h)lg(x))' (1.5)
Proof. The proof is essentially contained in that of II, proposition 2.16. There we work with b,ill, but all steps remain valid without "b". D
The Case Injectivity Radius Zero
301
Lemma 1.4 Suppose gl, g2, g3 quasi-isometric. Then there exists for every m 2: 0 a polynomial
with non-negative coefficients and without constant term such that
o Proof. This proof is also contained in that of II, 2.16, replacing there b,ill by ill(x). 0
Lemma 1.5 Let V be a function of moderate decay, g, hEM, Xo E M, c > 0, and suppose
(1. 7) Then there holds a) 9 and hare quasiisometric,
b) there exist constants
C1, C2
> 0 such that
and C1 V(I+d g(x,
xo)) :::; V(I+d h (x, xo)) :::; c2V(I+dg(x, xo)), x E M.
(1.9) We refer to [54] for the simple proof.
o
Remark 1.6 If V(x) is of moderate decay then there exist constants c, C1, C 2 such that
(1.10)
302
Relative Index Theory, Determinants and Torsion
o We refer to [54] for the proof. Define now for m ~ 0, V a function of moderate decay, g, hEM m,vlg _ hl9 = inf{c> 01 mig - hI 9 (x) ~ c· V(1 for all x E M}
+ d9 (x, xo))
if { ... } i= 0 and m,vlg - hl9 = 00 otherwise. Let m ~ 0, 6 > 0, C(n,6) = 1 + 6 + 6y''-C" 2n---C('--n-_-l::-;-) function of moderate decay and set
M 2 IC(n, 6)-1g ~ h ~ C(n,6)g and m,vlg - hl9 < 6}. (1.11)
V8 = {(g, h)
E
Proposition 1. 7 ~ = {V8} 8>0 is a basis for a metrizable uniform structure m,v,U( M) . Proof.
Certainly holds
nV8 =
diagonal = {(g, g) Ig EM}.
8
For the symmetry, we have to show m,vlg - hl9 < 6 implies m,vlg - hlh < 6'(6) such that 6'(6) ----t O. But (1.5), (1.10), (1.11) and C(n, 6)-1g ~ h ~ C(n,6)g immediately imply
mig - hlh(X)
~
cV(1
+ dh(x, xo))
for all x E M,
c = c(C(n,6),c1(6),c2(6),6), limC(n,6) = 1, limC1(6),C2(6) = 8->0 8->0 1 and This yields the symmetry. Similarly we get the transitivity from 0 (1.6). Denote by ~M the pair (M, m,v'u(M)) and by m,V M the com--u
pletion ~M . The elements ofm,v Mare Cm-metrics (cf. [67]). There is an equivalent metrizable uniform structure, based on Ig - hi, 1\79 (g - h)I,· .. , 1(\7 9 )m(g - h)1 instead of Ig - hi, 1\79 \7 hl, 1\79(\7 9 - \7h)I, ... , I(\7 9 )m-1 (\7 9 - \7 h )1, i.e. a uniform structure giving the same (metrizable) topology.
The Case Injectivity Radius Zero
303
Suppose Ig - hI 9(x) :::; C9 . V(1 + d9(x, xo)), hence Ig - hlh(X) :::; ChV(1 + dh(x, xo)). According to lemma 1.5, for any (p, q) E (Z+)2, there exist constants C 1 (p, q), C 2 (p, q) > 0, such that for any (p, q)-tensor t
The equivalence of the uniform structures would follow from an inequality
Cd (\79)i(g -
h)19(X) < 1(\79)i-1(\7 9 - \7 h)19(X) < C2,il(\79)i(g - h)19(X), (1.13)
since then
if and only if
1(\7 9)i-1(\7 9 - \7 h)19(X):::; E2V(1 +d9(x,xo)). We prove (1.13) and set B
= h - g,
D
= \7 h
-
\7 9 . Then
1(\79)iBI9(X) = 1(\79)ihI9(X), 1(\7 h)iBlh(X) = 1(\7 h)iglh(X), h(D(X, Y), Z) = -
(1.15)
(1.16)
~{\7~B(Y, Z) + \7~B(X, Z)
\7~B(X, Y)}
(1.17) \71B(Y, Z) = g(D(X, Y), Z) + g(Y, D(X, Z)). (1.18) If we insert into (1.18) for X, Y, Z local (with respect to h) orthonormal vector fields ei, ej, ek, square, sum up and take the square root, then we get
l\7 h(h - g)lh(X) < v'2 (?=g((\7Zi -
\7~Jej, ek)2)
1
2"
t,),k
v'2lg((\7h - \79)00, Olh(X),
(1.19)
304
Relative Index Theory, Determinants and Torsion
According to (1.12)
Ig((\7 h - \79)0(-), (·))lh(X)
:S
C1(~' 0) Ig((\7 h -
C1(~' 0) l\7
h -
\79)0(-), ('))19(X)
\7919(X)
< C1 (2, 1) l\7h _ \791 (x) - C1 (3,0) h,
(1.20)
together with (1.19)
l\7 h(h - g)lh(X) :S d~,ol\7h - \79Ih(X), or, exchanging the role of 9 and h,
i.e. we have the first inequality (1.13) for i same manner from (1.17)
= 1. We get in the
together with
finally
1\79 - \7hI 9(x) :S C2,il\79(h - g)19(X) for i = 1, (1.13) is done. For i > 1, the proof follows by a simple but extensive induction, differentiating (1.17), (1.18) and applying (1.16). 0 We now set m
m,metl g _ hI 9(x) :=
L 1(\79)i(g -
h)19(X),
i=O
m,met,Vlg _ hl9 := inf{E > 0lm,metl g - hI 9(x) :S EV(1 + d9 (x, xo)) for all x E M}
305
The Case Injectivity Radius Zero
if {- .. }
I- 0 and m,met,vlg -
hl g =
00
otherwise and define
M 2 IC(1, <5)-1g :::; h:::; C(n, <5)g and m,met,vlg - hi < <5}.
Vo = {(g, h)
E
Proposition 1.8 ~ = {v,,}o>o is a basis for a metrizable uniform structure m,met,vit(M). D
Denote by m,met,v M the corresponding completion. Then, as metrizable topological spaces
m,V M = m,met,v M.
(1.21)
Proposition 1.9 m,v M is locally contractible and hence locally arcwise connected. Proof. According to (1.21), we would be done if we could prove the assertion for m,met,v M. Locally a neighbourhood basis is given by
Uo(g) = {h E m,met,v MIC(n, c)-1g :::; h :::; C(n, c) . 9 and m,met,vlg - hl g < c}. (1.22) Let h E Uo(g), gt = th + (1 - t)g, Ig - gtlg(x) = tlg - hlg(x), m,met,vlg _ gtl = t . m,met,vlh - gig < t . c :::; c, i.e. all of the arc is contained in Uo (g) and this arc deforms to g. D
Remark 1.10 Quite analogous to (1.22) on defines
Uo(g) = {g'IC(n, c)-1g :::; g' :::; C(n, c)g and m,vlg - g'lg < c}. (1.23) D
Theorem 1.11 a) In m,v M coincide components with arc components.
306
Relative Index Theory, Determinants and Torsion
b) There exists a topological sum representation m,V M = Lcompm,V(gi).
(1.24)
iEI
c) For 9 E m,v M, compm,v(g) = {g'
E
m,V Mlm,vlg - g'lg < oo} == m,vU(g).
Proof. a) and b) immediately follow from proposition 1.9. We remark further that
g'
E
m,vU(g) if and only if 9
E
m,vU(g').
(1.25)
This follows from the symmetry in proposition 1.7. In particular, 9 and g' are quasi-isometric. Moreover,
m,vU(g) n m,VU(g')
=1=
0 implies m,vU(g) = m,vU(g').
(1.26)
n m,VU(g') and + (g" _ g) + (g _
Let g" E m,VU(g)
gill E m,VU(g). Write g' gill = (g' _ g") gill) and apply (1.. 6) to get m,Vlg' _ gllllgi < 00. This yields m,vU(g) ~ m,vU(g'). From symmetry considerations m,VU(g') ~ m,vU(g). Suppose now g' E compm,v (g) and let {gt }o::;t::;1 be an arc between 9 and g', go = g, gl = g'. Then for even be covered by a finite number of
E:
> 0, the arc can neighbourhoods
m,vUc:(go), m,vUc:(gtJ, . .. ,m,VUc:(gtr) = m,VUc:(g'), m,VUc:(gti_J m,vUc:(gtJ =1= 0. If gi,i-1 E m,VUc:(gti_l) n m,VUc:(gtJ, then m,vU(gti_l) :J m,vUc:(gti_l) :7 gi,i-1
E
m,VUc:(gtJ
C
n
m,vU(gtJ,
i.e., according to (1.26)
m,VU(gti_l) = m,vU(gtJ. Repeating this conclusion, we obtain
m,VU(g) = m,VU(g') , g' E m,vU(g) , compm,v (g) ~ m,VU(g). We have to show that m,vU(g) ~ compm,v (g). Suppose now g' E m,VU(g). Set gt =
The Case Injectivity Radius Zero
307
tg' + (1 - t)g. Then we conclude immediately from the proof of proposition 1.9 that {gdO::;t9 is an arc in m,VU(g) which connects 9 and g', i.e. m,vU(g) ~ compm,v (g). 0
Proposition 1.12 The following properties are invariants of compm,v(g), m 2:: 2. a) IVi R I :::; Ci , 0 :::; i :::; m - 2, b) sectional curvature bounded from below or bounded from above, respectively,
c) Ricci curvature bounded from below or from above, respectively.
0
2 The injectivity radius and weighted So boley spaces We assumed in chapter II - VI the positivity of the injectivity radius, rinj (M, g) > o. The background for this assumption was the fact that we often used the Sobolev embedding theorem II 1.6, II 1.7 and the module structure theorem II 1.17. In the proof of these theorems, the assumption rinj > 0 plays an essential role. Unfortunately, open complete manifolds with finite volume fall out from these considerations. Nevertheless, a big part of the necessary Sobolev analysis can be rescued if we work with weighted Sobolev spaces. We present in a forthcoming paper a comparatively comprehensive investigation on weighted Sobolev spaces for open complete manifolds with rinj = o. Brought all discussions to a point, one has to work with the weight (rinj(x))-~ or a weight function with at least this growth. For the trace class investigations in this chapter, we need much less of the corresponding Sobolev analysis and we essentially follow [18] and [54]. Lemma 2.1 Let (Mn,g) be open, complete, 9 and h quasi isometric, IKgl, IKhl :::; K. Then there exist constants c, c' > 0 such
308
Relative Index Theory, Determinants and Torsion
that rinj(h, x) 2: min{ c . rinj(g, x), c'},
x E M.
We refer to [54] for the proof. D A special case is given by h E comp°,v (g), V of moderate decay. Set for x E M
hnj(x) := min { 12JK, rinj(X) } . Then, under the assumptions of lemma 2.1,
hnj(h, x) 2: C2i\nj(g, x). Using the standard volume comparison theorem
27r~
r(~)
<
Jr (sinh tv'K) ° V(K)
n-l
27r~ JT (sinh tv'K)
- f(!!:) 2
0
v'K
dt S
n-l
vol(Br(xo))
dt
(2.1)
and theorem 4.7 from [18], _
rinj(X) 2:
r
1
"2 1 + (v;.~+s/vol(BT(y)))(Vdfx,y)+r/~K)'
(2.2)
ro + 2s < 7r / v'K, ro S 7r / 4v'K, VgK = volume of a (?-ball in hyperbolic space Hr:. K , then one immediately gets
(2.3) C = C(K). (2.3) implies the existence of a constant C = C(K, y) such that
r-·InJ·(x)
> _ Ce-(n-l)vKd(x,y) ,
X
E M.
(2.4)
Lemma 2.2 There exists a constant C = C(K), such that
(2.5)
309
The Case Injectivity Radius Zero
D We refer to [54] for the proof. We recall from II, lemma 1.10 the existence of appropriate uniformly locally finite covers of (Mn, g) if Ric (g) ~ k. Set for s > E ~ 0, /'i,c:(M, g, s) E IN U {(X)} = smallest number such that there exists a sequence {Xi}~l such that {Bs-c:(Xi)}~l is an open cover of M satisfying
sup #{i E INlx E B 3s+c:(Xi)} ~ /'i,c:(M,g,s)
(2.6)
xEM
and set /'i,(M, g, s) := /'i,o(M, g, s), /'i,(M, g, 0) := 1. We need for norm estimates of cos V75. in the forthcoming section 3 the following Lemma 2.3 /'i,c:(M, g, s) is finite for all s > E and there exist constants C, c > 0 depending on K such that for s > ,(R + E
/'i,c:(M, g, s) ~
c· eCs •
(2.7)
D We refer to [54] for the proof. Next we collect some facts concerning weighted Sobolev spaces. Let (E, hE, '\7 E) ---+ (Mn, g) be a Riemannian vector bundle, ~ a positive, measurable function on M which is finite a.e .. Define the associated weighted Sobolev spaces as follows
n~,r(E, '\7)
= {
E
Lp,loc(E)
(J ~
11
r 1
({l IV'
< oo},
where '\7 i is iteratively applied in the distributional sense. In the case p = 2 we write n~,r(E, '\7)
Set
~
= W[(E, '\7) == (W[(E, '\7), Ilw[).
= '\7*'\7,
n~,2r(E,~) =
{
E
Lp,loc(E) 11,~
(J ~({llt>'
r 1
dvolx(q)
< oo}
310
and for p
Relative Index Theory, Determinants and Torsion
=2
We refer to [27], [54] for further general information on weighted Sobolev spaces. For ~ = 1 we get back standard Sobolev spaces. Lemma 2.4 The natural inclusion Wt(E) bounded.
wt,
Proof. Let rp E IrpI2,2r,'V,~ < 00. b;,rp \i'trg = 0 imply for the pointwise norms
=
'----7
Ht(E) is
-trg(\i'2rp) and
hence and
o In the sequel, we will establish some Sobolev estimates by locally finite covers. We proved in [29] the following
Proposition 2.5 a) If (Mn,g) satisfies (B k ) and it {(Ua, cI>a)}a is a locally finite cover by normal charts, then there exist constants C(3, C~, C~, multiindexed by (3, "y such that
and
(2.9) all constants are independent of a.
b) If (E,hE' \i'E)
- - - t (Mn,g) is a Riemannian vector bundle satisfying (Bk(M,g)), (Bk(E, \i'E)), then additionally to (2.8), there holds for the connection coefficients r;J.t defined by
The Case Injectivity Radius Zero
V -LC{J).. = 8u'
rf).. C{JJL ,
311
a local orthonormal frame defined by
{C{JJL}JL
radial parallel translation,
(2.10)
o Consider {] > 0, B{} = B{}(O) c IRn, m, k E IN, K, A > 0 and denote by Ell~({], K, A) the set of elliptic operators
P
=
L
aa(x)Da,
aa(x)
= (aa,i,jh:5,i,j:5,k
lal:5,m
satisfying 1) aa,i,j E Cm(B{}), 2)
L
lal<m
3)
laalco(Bu):::; K,
L
laaIc 1 (Bu):::; K,
lal<m
A-ll~lm:::; lal=m L ( L aa'i,j(X)) ~a :::; AI~lm, l:5,i,j:5,k
where laaict(B u) = sup sup sup ID.6aa,i,j(X) I. i,j 1.61:5,1 xEBe We recall a standard elliptic inequality for Euclidean balls B{} and refer to [27], II (1.52), p. 75. Lemma 2.6 For given k, K, A, there exist {] = {](K, A) > 0 and C(A) > 0 such that for all {] :::; (]o, P E Ell~({], K, A)
lulwm(Bu~k :::; C(IPuIL2(Bg,Ck) for all u E C':(B{}).
+ luI L2 (B e,C k)) o
Using 2.5 and 2.6, we immediately get a generalization to balls in Riemannian manifolds and bundles satisfying (B k ) (as in [27], II (1.52)). Lemma 2.7 Suppose (E,hE,VE) - - t (Mn,g) being a Riemannian vector bundle satisfying B 2k (M, g), B 2k (E, VE), IVi Rgl,
312
Relative Index Theory, Determinants and Torsion
l\7j REI :s; K, i = 0, ... ,2k. Then there exist constants Qo(K) > 0, C(K) > 0 such that for all Xo E M and Q :s; min{Qo,i\nj(xo)} there holds
o We refer to [27] or to [54]. If V : [1,00[---t IR+ is of moderate decay, then we define the associated V : Mn ---t IR+ of moderate decay by V(x) = V(1 + dg(x, Xo)). Lemma 2.8 Let (E, hE, \7 E ) ---t (Mn,g) beaRiemannianvector bundle satisfying (Bk(Mn,g)), B 2k (E, \7 E ), k even, and let f3 : M ---t IR+ be of moderate decay. Then there exist bounded inclusions (2.12) and
(2.13)
Proof.
We apply (2.6) to get a cover {Brin;kXi) (Xi)
}:1
by
balls satisfying for all x E M (2.14) Set for u E GOO(IR) with u = 1 on [0,1] and = 0 on [2, oo[ and l:S;j:S;k . d(x,y)
Uj,x(Y)
=
u(2 {
o
J' f· ·(x) InJ
,
Y E Bi'i~j(X)(x), otherWIse.
Hence Uj,x E Cgo(M) and for «J E Hk(E, b..), Uj,x«J E Hk(Bi'inj(X)' E).
313
The Case Injectivity Radius Zero
We infer from (2.11)
Uj,x'P
E W k (B finj (x) (x), E and
Moreover, we infer from (2.11) IUk,x'PIWk(E)
~ C 1 1'Plwk(B
+ c,
f.
2k~ll (x)
t G)
(x),E)
(fi"j(x)t'
< C1 1'PIHk(B f . .
2k'~ll (x)
·I U k-l,.l"lw'-'(E)
(x))
k
+
C3
~ (~) (rinj(x)t 1 u
I k-1,x'PIHk-I(E)'
(2.15) We conclude from (2.14) by an easy induction
o
Lemma 2.9 Let V : [1,00[----+ IR+ be a function of moderate decay. Then for all x, y, q E M there holds C v . V(1
V(1 + d(x, q)) + d(x, y)) ~ V(1 + d(y, q)) ~
1 C v . V(1
+ d(x, y))'
and for every q' E M there exists C = C(q, q') > 0' such that
C- 1 . V(1
+ d(x, q'))
~
V(1
+ d(x, q))
We refer to [54J for the simple proof.
~ C·
V(1
+ d(x, q')). o
314
Relative Index Theory, Determinants and Torsion
Let now c.p E Hk(E, ~). Then lemma 2.9, lemma 2.7, (2.14) and (2.16) imply 00
Ic.pl Wt(E,V')
< C
I: V! (xi)luk,Xic.plwk(E,V') . hnj(Xi)k i=1 00
< C
I: V! (Xi) IUk,Xic.pIHk(E,~) i=1 00
c'" ~ V! (xi)hnj(Xi)-k ·1c.pIHk(B-. .( .) E)'
<
T1nJX'J,'
i=1
(2.17) According to (2.4), there exists Dl > 0 such that
f\nj(Xi)-ki\nj(x)kn ::; Db hence 00
'" V! (Xi) . hnj(Xi)-klc.pIHk(B_. .(x,.)(xi),E) ::; D21c.pIHkf:-2knV(E) , ~ rinJ
bl
~
which yields together with (2.17) the inclusion (2.12). The proof of (2.13) is quite analoguous.
o be continuous. Then W[(E, \7) n cOO(E) n COO (E) are dense in W[(E, \7) or Hr(E, ~), re-
Lemma 2.10 Let
or Hf(E,~) spectively.
~
We refer to [54], lemma 3.1 for the simple proof. We conclude this section with the invariance of weighted Sobolev spaces W[(E, \7 E , g), Hr(E, ~E, g) under change of the metric 9 inside compr,V(g). Proposition 2.11 Let V be of moderate decay, (E, hE, \7 E )
----t
(Mn, g) be a Riemannian vector bundle satisfying (Bo (M, g)) and suppose h E compr,v (g). Then W€1!(E, \7 E, g) ~ Wt(E, \7 E, h), 0::; (] ::; r, (2.18) as equivalent Sobolev spaces.
The Case Injectivity Radius Zero
315
For r = 0, the assertion is clear since, according to (1.8), 9 and h are quasi-isometric. The case r = 1 is also clear, since into the derivative \IE the derivatives \19 or \l h do not enter. In the case r = 2, (M, h) satisfies (Bo(M, h)) too, and h is a C r -metric. We set
Proof.
(2.19) (2.20) !?-1
\I!?
= (\IE ®
® \19)
0 •.. 0
(\IE ® \19)
0
\IE, (2.21)
1
\1 /2
= (\IE Q9 \l h) 0 \IE,
(2.22) (2.23)
!?-1
\II!?
= (\IE ®
® \l h)
0 . '.0
(\IE ® \l h) 0 \IE. (2.24)
1
According to II (1.34), for t.p E wt(E, \IE, g)
n COO (E)
!?
\l1!?t.p
=
L \li-1(\l1 - \I)\lI!?-it.p + \I!?t.p.
(2.25)
i=1
By assumption ~~\\I!?t.p\ E L2(M,g). There remains to consider the terms (2.26) Iterating the procedure, i.e. applying it to \l1!?-i and so on, we have to estimate expressions of the kind (2.27) with
i1
+ ... + i!? = {l,
i!?
<
{l.
Consider a typical expression
In a local bundle chart
\lit.p\U
=
L 'ljJ ® t1 ® ... ® ti · finite
316
Relative Index Theory, Determinants and Torsion
We calculate ('V' - 'V)'Virp at a typical term 'l/J ® t1 ® ... ® ti:
i
= ('V
E
®
0 1
'V h - 'V E ®
= 'l/J ® ('V h - 'V g)t1 ® t2
i
0
g 'V ) ('l/J ® t1 ® ... ® t i )
1
® ... ® ti
+ ...
+'l/J ® t1 ® ... ® t i-1 ® ('V h - 'Vg)t i .
(2.28)
Linear extension of (2.26) and Schwartz' inequality for the pointwise norm yield
I('V' - 'V)'VirplhE,9,X:S i 'l'VirplhE,9,X' 11g - hlg(x), together with (1.8)
I('V' - 'V)'VirplhE,h,x :S C· i . l'VirplhE,9,X . 11g - hlg(x). Iterating the calculation and estimate, we obtain
I('V' - 'V)2'V irplx :S C(i, 2) 'l'VirpWlg - hl;(x) +21g - hlg(x)), I('V' - 'V)j'Virplx < C(i,j) 'l'Virpielg - hl~(x) +jlg - hlg(x)), I'V('V' - 'V)j'Virplx < ~,j,lelg - hlg(x), ... , }+llg - hlg(x)) . 1'V}+lrplx,
+ ...
l'Vk('V' - 'V)j'Virplx < ~,j,kelg - hlg(x), ... , j+kl g _ hlg(x)) . I'V}+krplx , where ~,j,k is a polynomial in the indicated variables without constant terms. Continuing this way, we finally get
I'ViI ('V' - 'V)i2'Vi3'Vig-2('V' - 'V)ig-I'VigrplhE,h,x TJ < _ FiI,i2, ... ,ig (...j,l 9 - hi 9() x ,... )lniI+i3+·+ig v rp IhE,g,X
:S PiIh ... ,ig(V(X), ... , V(x)) . l'ViI+i3+··igrplhE,9,X' (2.29)
317
The Case Injectivity Radius Zero
By assumption,
L 2 (M , g) ~ - L 2 (M , h) .
<"dlnil+ia+.+iecpl v hE,g,x E
Pil,i2, ...ie(V(X), ... , V(x)) is bounded, even Pil.i2,,,.ie(V(X), ... , V(x)) ---t 0 uniformly, x-oo
hence
~L IVil (V'- v)i2v ia ... V i e- 2 (V'- V)i e - I viecplhE,h,x E L2(M, h),
cp E Wt(E, V, h), Wt(E, V, g) ~ Wt(E, V, h), 0 :::; {} :::; r. Exchanging the role of 9 and h, we obtain the other inclusion. o
Corollary 2.12 Suppose the hypotheses of 2.11 and Br (M, g), Br(M, h), Br(E, VE). Then HQ(E, il, q)
~
WQ(E, V, g)
~
WQ(E, V, h)
~
HQ(E, V, h)
as equivalent Hilbert spaces.
Proof.
This immediately follows from 2.11 with
[27], [66].
3
~
=
1 and 0
Mapping properties e- til
In the decisive next section, we need mapping properties of the operator e-tt::. which we establish now. Here we densely follow section 4 of [54]. On a complete manifold the operator il = V*V : Cgo(E) ---t L2(M, E) is essentially self-adjoint and for an admissible function fO the operator f(il) is defined by the spectral theorem. The same holds for V/S., i.e.
J 00
f( V/S.) =
f()..)dE>.,
o
Relative Index Theory, Determinants and Torsion
318
where {E.d A is the spectral family for ..J/S. For fELl (lR.) we denote by j the cosine transformation of f
J +00
jp..)
=
f(x) cos(Ax)dx
-00
and f(..J/S) has a representation as
J +00
f(~) = ~ 21f
j(A) cos(A~)dA.
(3.1)
-00
Theorem 3.1 Let (E, hE, \7 E) - - t (Mn, g) be a Riemannian vector bundle satisfying (Bo( Mn, g)) and let V : M - - t lR.+ be a function of moderate decay. Then cos( s..J/S) extends to a bounded operator in L 2,v(M, E) = {'P I V~ I'PI E L2(M, g)} for all s E lR. and there exist C, c > 0 such that
I coS(S~)IL2,V,L2,V and cos(s..J/S) : L2,v(M, E) uous ~n s.
~ C· eels I,
--t
s E lR.,
L2,v(M, E) is strongly contin-
Proof. Recall K,(M,g,s) from (2.6), let s > 0, {Xd~l be a sequence in M which minimizes K,( M, g, s) and let Pk (.) = X k-l . Pk is an orthonormal projection in L 2 (M, E) B.(Xk)\ U B.(Xi) i=l
00
and L 2 ,v(M, E) satisfying PkPk, = 0 for k =F k' and
2: Pk =
1
k=l
strongly. cos( t..J/S) has unit propagation speed which implies for 'P E L 2 (M, E) supp cOS(S~)Pk'P C B 2s (Xk) and
The Case Injectivity Radius Zero
319
This yields
00
k=1 00
= 2:= (cos(sVK)Pk
As operator cos(sVK) : L2(M, E) the operator norm :S 1, hence
---t
L2(M, E), cos(sVK) has
I(cos(sVKPk
sup
yE B38(Xk)
V(y)· IPk
The firs two factors in (3.2) can be expressed as sup
yE B38(Xk)
=
V(y)IPk
J
IPk
M
:S
Cv:
1
sup
yE B38(Xk)
(Vv((y))) V(x) dvolx(g) x
1 V(l
+ 4s) IPk
where we used supp Pk
:S
Cv:
1
V(l
~ 6s) IPk
According to lemma 2.3, K(M, g, s) < 00
00,
and we obtain
2:= IXB38 (Xk)
320
Relative Index Theory, Determinants and Torsion
We conclude from (3.1) and (3.3) 1
cos(8~)'PIL,v(M,E)
1 :::; GV V(l -1
:::; GV V(l
1
£; IX 3s(Xk)'P1 00
+ 68) l'PIL,v(M,E) 1
B
+ 68) f'l,(M, g, 8) ! 1'P 12L2,v(M,v)' 2
(3.5)
We have L2(M, E) c L2,v(M, E) as a dense subspace since, according to (1.11), V(x) :::; G(l + d(x,p))-1, x E M. Hence COS(8J~) extends to a bounded operator L2,v(M, E) ----t L2,v(M, E). (1.11) and (2.7) imply 1
cos(8~IL2,V,L2,v :::; Ge cs ,
which extends to 8 E IR since cos( 8~) 1
8
E [0,00[,
= cos( -8~),
cos(8~)IL2,v,L2,V :::; Ge isi .
The local bound of the norm and the strong continuity on the dense subspace L 2(M, V) c L2,v(M, E) yield the strong continutity of COS(8~) : L 2,v(M, E) ----t L 2,v(M, E). D Define for given c
~
0
J +00
F1(C) = {J
E L 1 (IR)1
li().)lecl>'ld)' < oo}.
-00
Proposition 3.2 Let (E,hE,'\i'E) ----t (Mn,g) be a Riemannian vector bundle satisfying (Bo (M, g)) and let V be a function of moderate decay. Then there exists a constant c = c(M, g, V) such that for all even f E F1(C) the operator f(~) extends to a bounded operator L 2,v(M, E) ----t L 2,v(M, E), and there exists a constant G1 = G1(M,g, V) such that
If(~) IL 2,v,L2,v :::; G11ilL 1 eel.1 (JR)
(3.6)
for all even f E F1(C). If f'l,(M, g, 8) is at most subexponentially increasing, then c( M, g, V) > 0 can be choosen arbitrarily.
321
The Case Injectivity Radius Zero
Proof. According to theorem 3.1, there exist constants G, c >
o depending on M, g, V such that
Icos(s~)IL2,v(M,E),L2,v(M,E) ~
Ge clsl
for all s E JR. Let
J1cos(A~)dA
~ 27r
L2,V(M,E)
-00 +00
<
~ Jill' G· ecl-XldA
-00
(3.7) from which we infer the bounded extension to L 2 ,v (M, E). The last statement follows immediately from the meaning of /'i,(M, g, s) and the proof of theorem 3.1. D
Corollary 3.3 Suppose (E, hE, '\7 E) -----t (Mn, g) and V as above. Then for every t > 0, the operator e- t /). extends to a bounded operator on L 2 ,v(M, E), and its norm is uniformly bounded in t on compact intervalls lao, all, ao > O. Proof. This immediately follows from proposition 3.2 and
-00 +00
J
_1_ cos(xy)dx
A+X2
=
J
7r e-v>'hl.
....;"X
-00
D
Relative Index Theory, Determinants and Torsion
322
4
Proof of the trace class property
In this section, we will prove the trace class property of e-tD. g E e- tiih , h E comp2,V(g) , Ll9 = _gij"VE"V t J' For this , we essentially reduce the task to the case of the scalar Laplacian, using the Semi-group domination principle. Let (E, hE, "VE) ----t (Mn, g) be a Riemannian vector bundle, P an operator of the kind P = LlE + R = ("VE)*"V E + R, R a O-order Weitzenboeck term. Set b = ~ffL b(x), b(x) = min r1/>l~l hE('l/J, R'l/J). Let
Wp(t,x,y) == e-tP(x,y) or WD.o(g)(t,x,y) = e-tD.o(g)(x,y) the heat kernels of e- tP or e-tD.o(g) repectively. Then there holds for the pointwise norms
(4.1) for aUt> 0, x,y E M. We refer to [23] for details. 0 In this section we must distinguish between Llg = ("VE)*"V E = - gij "V f"V * = *(g), Llg acting on sections of E, and Llo (g) = "V*"V = -gij"V i"V j , where "Vi = "Vf is the Levi-Civita connection of 9 and Llo (g) acts on a function on M. Suppose (Mn, g) with (Bo(M, g)), Jsectional curvature KgJ < K. Let WD.o(g)(t, x, y) = e-tD.o(g) (x, y) be the heat kernel of Llo(g), 0< ao < a1.
f,
Lemma 4.1 There exist C1, C1 > 0 such that
e-tD.o(g) (x, y) :::; C1i\nj (x) - ~ i\nj (y) -~ e- Cld2 (x,y) ,
t E lao, a1]' (4.2)
We refer to propostion 1.3 of [18]. For c < C1 there exists according to (4.2) and (2.4) a C > 0 such that n(n+1) cd 2 ( X,y ) e- tD. 0 (9 ) (x , y) < CfInJ . .(x)--2-e'
t E lao, ad·
(4.3)
The Case Injectivity Radius Zero
323
Lemma 4.2 Let V be a function of moderate decay. Suppose that a, b E IR are such that a) a + b = 2, b) Vb E Ll (M, g), _
n(n+l)
c) Varinj(xt-2-
Loo(M, g).
E
Let Mv the operator V" p E IN. Then the operator Mvl::!,.o (g )Pe-tAo(g) is Hilbert-Schmidt, and for any compact intervall lao, all, ao > 0, the Hilbert-Schmidt norm is uniformly bounded. Proof. Write
Mvl::!,.o(g)Pe- tA
(Mve-~Ao(g»)(l::!,.o(g)Pe-~Ao(g».
=
(4.4)
The second factor on the r.h.s. of (4.4) has bounded operator norm on any intervall lao, al], ao > O. Hence we can restrict to the case p = O. According to corollary 3.3, e-tll.o(g) extends to a bounded operator in L 2 ,vb (M, g) with uniformly bounded norm in 0 < a :::; t :::; b. We infer from Vb E Ll(M,g) that 1 E L 2,vb(M, g) and e- tAo (g)1 E L2,vb(M, g). Hence
b=
(1, e-tll.o(g) 1) L 2,V
JJ
V(X)be-tAo(g) (x, y)dydy
MM
is well defined, and we obtain for the Hilbert-Schmidt norm of Mve-tAo(g)
IMve-tAo(g) Its = =
JJ
lV(x)e-tAO(g) (x, y)1 2dydx
MM
JJ
V(x)2e- 2tll. O(g) (x, y)dydx
MM
:::; sup lV(ute-tll.o(g)(u, v)I' u,vEM
JJ J
V(X)be-tll.O(g) (x, y)dydx
MM
n(n+l)
:::; C· sup lV(u)ali:;~j-2-(u)1
lV(x)b(e- tAo (g)I)(x)dx
uEM M
:::; C l
.
le-
tAo
(g)liL
2
Vb (M,g)
<
00.
0
324
Relative Index Theory, Determinants and Torsion
Lemma 4.3 Let (Mn, g) and V be as above and suppose that a, b are real numbers such that a) b ~ 1, a + b = 2, b
b) Va E Ll(M,g),
n(n+1)
c) V~f\~j
Loo(M, g). Then the operator M r :-2nMv D.. o(g)Pe- t6.o(g) is for p E 1N of trace InJ class, and the trace norm is bounded on any compact t-intervall lao, all, ao > O. E
2
Proof. Write
M r :-2n MvD..o(g )P e-t6. o(g) InJ
= [MTInJ __ 2nMve-~6.o(g) M _1]' [M 1D.. o(g)pe- t6. o(g)].(4.5) v J VJ The properties of V and b ~ 1 immediately imply V~ ~ C· V~, hence according to assumption b) V ~ E Ll (M, g). According to c) and the proof of lemma 4.2, the second factor on the r.h.s. of (4.5) is Hilbert-Schmidt, and the HS-norm is bounded for t E lao, al]' ao > O. We have to show this for the first facn(n+1) 2 tor. c) immediately implies V af:;j-2-- n E Loo(M, g). Hence, together with (4.3),
JJ
Ifinj(xt2nV(x)e-t6.o(g) (x, y)V(y)-~ 2 dxdy 1
MM
~
n(n+1) 2 C sup ifinj(ut-2-- nV(u)al uEM
.JJV(x)be-t6.o(g)(x,y)V(y)-~dxdy. MM b
•
2
2
b+4
2
By assumption b), Va ELI, l.e. V-aV-aV-3 ELI, V-a E b+4 b L ,,¥(M,g). Moreover, we get from V-3 ~ C· Va and
2,v
325
The Case Injectivity Radius Zero
b+4 • ~ ( ) b) that V-3 ELI. Accordmg to corollary 3.3, e- t 0 9 has an extension to a bounded operator in L 2¥ (M, g), and we
get
J e-t~o(g)(x,y)V(y)-~dy AI
2,V
E
2¥(M,g) with uniformly
L 2,V
bounded norm in t E lao, all. ao > O. Write Vb . VbV-£¥ = V~ . V~(b-1), from which follows Vb E L2 v-2¥ (M, g). Between
L2,v(M, g) and L 2,v-l (M, g) there is a st~ndard pairing (,) (cp,7jJ) =
J
cp(x)7jJ(x)dx,
cp E L 2 ,v, 7jJ E L 2 ,v-1.
(4.6)
(4.6) allows to rewrite
JJV(x)be-t~o(g)(x, y)V(y)-~dxdy
= (Vb, e-t~o(g)V-~) <
00.
AI AI
(4.7)
o Now we prepare our main result. Let (E,h E , 'VE) ----t (Mn,g) be a Riemannian vector bundle, (Mn, g) complete, satisfying (B2(M,g)), V a function of moderate decay, h E comp2,v(g) satisfying (B 2 (M, h)). We denote 'V = 'V E , 6. g = 'V*'V = -gij'Vi'V j , 6.h = 'V*(h)'V = -hij'Vi'V j , 6.o(g), 6. o(h) are the scalar Laplace operators. We see, hE remains unchanged and we consider only perturbations of the metric on M, 9 and h are quasi-isometric. We denote by dvolxg(x), dvolxh(x) the corresponding volume forms. It is really very easy and quite natural to define a metrizable uniform structure m,v U( vol) quite analogous to m,vU(M) and to establish the following Lemma 4.4 h ( dvol(g)).
E
comp2,v (g) implies dvol( h)
E
comp2,V
o
The metrics hE, g, h define Hilbert spaces L2((M, g), (E, hE)) == L 2(g, hE)' L2((M, h), (E, hE)) L 2(g, hE) and L 2(g, hE) ~ L 2(h, hE) as equivalent Hilbert spaces, but 6.h must not be self-adjoint in L 2 (g, hE)' even not symmetric.
326
Relative Index Theory, Determinants and Torsion
But we are exactly in the situation which we described in section IV, 2, between (2.1) and (2.3). We have to graft b.h to L 2 (g, hE)' We write dvolAg) == dx(g) = a(x)dx(h) == a(x) dvolx(h) and define U : L 2 (g, hE) ---+ L 2 (h, hE) by U'P := a~'P. Then U is a unitary equivalence between L 2(g, hE) and L 2(h, hE)' li h = U*b.hU acts in L 2(g, h), is self-adjoint on U-l(VD.h) since U is a unitary equivalence. Lemma 4.5
(I (\i'9)2(a - 1)19 + 1\i'9(a - 1)19 + la - 119)(X) :S c· V(x). Proof. Write (a -1) dvol x(h) lemma 4.4.
= dvol x(g) - dvol x(h) and apply 0
Proposition 4.6 Suppose (E, hE, \i'E) ---+ (Mn, g), V and hE comp2,v (g) as above. Then for t > 0, e-tD. g(a - 1) is of trace class and the trace norm is uniformly bounded on compact tintervalls lao, all, a>O. Proof. We write
e-tD.g(a -1) = (e-~D.gMvk)· (Mv-ke-tD. g(a -1))
(4.8)
and estimate, using (4.1), the proofs of 4.2, 4.3., lemma 4.5,
(4.6)
le-~D.g Mvk I~s =
JJ
l(e-tD.g(x, y),
.)V~ (xWdxdy
MM
:S Cl
JJle-tD.O(9)(x,YWV~(x)dxdy
<
00,
MM
IMv_~e-~D.g(a - 1)I~s
=
JJlV(y)-~(e-~D.g(x, JJV(x)2e-tD.O(9)(x,y)V(y)-~dxdy
y), ·)(a - 1)(xWdxdy
MM
:S C2
MM
<
00.
327
The Case Injectivity Radius Zero
Now we are able to state and to prove our
Main Theorem 4.7 Let (E, hE, \7 E)
-----t (Mn,g) be a Riemannian vector bundle, (Mn, g) being complete and satisfying B 2 (Mn, g). Let V be of moderate decay and hE comp2,v(g)nC 4 , satisfying (B 2 (M, h)) and set a(x) = dvolx(g)/ dvolx(h), U L 2 (g, hE) -----t L 2 (h, hE) defined by U
is for t > 0 of trace class and the trace norm bounded on compact t-intervalls lao, all, al > O.
2S
uniformly
Proof. According to Duhamels principle IV (2.5), 1
1
e- tLlg _ e-tU*LlhU = e-tLlg _ e- t(a- 2Llh a2 )
J J t
= e-tLlg(a - 1) -
e- sLlg (.6. g
-
a-~.6.ha~)e-(t-S)a-~Llha~ ds
e- sLlg (.6. g
-
a-~.6.ha~)a-~e-(t-s)Llha~ds.
o
t
= e-tLlg(a - 1) -
o
According to proposition 4.6, the term e- tLlg (a - 1) is done. We write
e- SLlg [.6. g (l - a~)
+ (.6. g -
.6.h)a~
+(1 - a-~).6.ha~la-~e-(t-s)Llha~ .6. g e- SLlg (l _ a~)a-~e-(t-s)Llha~ +e- sLlg (.6. _ .6.h)e-(t-s)Llha~
(4.10)
+e- SLlg (l - a-~).6.he-(t-S)Llha~
(4.11)
=
g
(4.9)
328
Relative Index Theory, Determinants and Torsion
~
t
and decompose J = J o 0
t
+ J.
This yields altogether 6 integrals,
&
~ ~ ~ t t t J(4.9)ds, J(4.10)ds, J(4.11)ds, J(4.9)ds, J(4.1O)ds, J(4.11)ds.
o
0
0
i
The definition of a implies 0 < lemma 4.5
1 1
-a
-II_II 2
-
1
1_
fa -
-
i
2
yu
C1 ::;
2
a ::;
1- ~ 1+_1 -
C2,
i 2
and we obtain from
~(1a) <_ C 2 · V. 1+_1
Vii
Vii
t
Now we decompose J(4.11)ds as follows, t
2"
t
J J[{ e-~t!.g Mv~}{ Mv_~ e-~t!.g(1 (4.11)ds
t
2"
t
=
-
a-~)}](~he-(t-S)t!.ha~)ds.
t
2"
But { ... }, { ... } in [... J are Hilbert-Schmidt operators: I(e-~t!.g, .)Mv~ I
IMv_~(e-~t!.g, ·)(1- a-~)I
< C3Ie-~t!.o(g)(x, y)Mv~ I, (4.12)
< C41V-~(y)e-~t!.o(g)(x,y)V(x)l· (4.13)
According to lemma 4.2 and (4.7), the right hand sides of (4.12), (4.13) are square integrable, and the value of the HS-norm is uniformly bounded on the interval [~, tJ. Finally t
We conclude as in IV (1.29)-(1.34) that J(4.11)ds is a trace class t
2"
operator and its trace norm is bounded in any compact t-interval
The Case Injectivity Radius Zero
329
lao, all, ao > O. Quite analogous, we handle the other 5 integrals and indicate this by the decompositionn of the integrands and their estimate: t
"2
J(4.11)ds: o
e- s 6. g (l _ a-~)~he-(t-S)6.ha~
= e- s 6. [{(l_ a-~)e-t"4S6.hMv_~}{Mv~e-t"4S6.h}J g
t-8
1
'~he-2""" ~ha"2,
1(1 - a-~)e- t"4 6.h Mv_~ I s
:s;
c ·1V(x)e-t"4S6.o(g)(x,y)V(Y)-~1
IMv~e-t4S6.hl
:s; C
·1V(x)~e-t"4S6.o(g)(x, Y)I,
t
J(4.10)ds: t
"2
t
"2
J(4.10)ds: o
t
everything is done, J(4.9)ds : t
"2
330
Relative Index Theory, Determinants and Torsion
t
2
everything is done, J(4.9)ds : o
everything is done. This finishes the proof of 4.7.
o
Remarks 4.8 1) Theorem 4.7 and the theorems in chapter IV are neither disjoint nor there is an inclusion. 2) In 4.7 we only permitted a perturbation of the metric of Mn. It would be also interesting to consider additionally appropriate perburbations of the fibre metric and the fibre connection. The advantage of this chapter is that we do not restrict to the case nnj(M, g) > O. Hence we admit e.g. locally symmetric spaces of finite volume. 0 As an immediate consequence of theorem 4.7 we have Theorem 4.9 Suppose the hypotheses of theorem 4.7. there exist the wave operators
Then
and they are complete. Hence the absolutely continuous parts of and /j.h are unitarily equivalent. 0
/j.g
We have in mind still other admitted perturbations to establish a scattering theory. This will be the topic of a forthcoming treatise.
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List of notations page b ,2(M) .................... 30 d!i,~iff,F,rel, (El , E2) ....... 136 il Q ,2(M) ................... 31 det(D2, D,2) .............. 268 q
(Bk) ...................... 65 DP,r(M, N) ............... 123 BZ~ff(I, B k) ............. 131 DP,r(M) .................. 123 Bz~r/(J, B k) ............ 131 D~{(El' E 2) .............. 124 BZ'~r[,F(J, B k) ........... 135 ~b,rel(El' E 2) ............ 124 BZ~r[,rel(I, B k) .......... 134 TJ(s,D,D') ............... 275 compp,r(g) ............... 109 gen compj;~diff,rel(M, g) ... 129 b,mcomp(g) ............... 105 gen compj;~diff,rel(E) ...... 134 b,2 com pp,2(g) .............. 16 gencompj;~diff,F,rel(E) .... 136 compp,r (
List of Notations
339
9J1L(nc) , 9J1L,h(nC) ........ 116 O,p,r(M, N) ............... 122 9J1L ,top,rel . . . . . . . . . . . . . . . . . 120 O,~C ...................... 150 9J1L ,h,rel . . . . . . . . . . . . . . . . . . 119 0,~C(C8) ................... 151 nC(A, D, IY), nC(D, IY) ... 251 O,~c(I, Bk ) . . . . . . . . . . . . . . . . 150 JV(e) , JVK(e) " ............ 57 O,~C(ne) .................. 162 O,~'P(E), O,q,p,r(E), O,p,r(E) .73 Pm(g) ..................... 98 oq,p,r(E), op,r(E) .......... 73 P!:((g) .................... 98 O,q,p,r(E),O,p,r(E) .......... 73 S(A) ..................... 188 ~o'(E), b,mo'(E) ........... 74 O"(A) ..................... 184 O"p(A),O"c,R(A) ........... 184 O,~,r(E), o'V(E) , O"e(A), O"c(A), O"p,J(A) ..... 184 o~,r (E), o';,~ (E) .......... 309 O"pp(A), O"ac(A), O"sc(A) .... 184 O,Pd!(E, \1) ................. 73 O"pd(A) ................... 184 o'~s(E,~) ................. 85 'Ta(M, M') ................ 272 02,2S(E,~) ................ 85 ~(A) ...................... 188 0,2,2S(E,~) ................ 85 W±(A, B) ................ 185 0,2,2S(E,~) ................ 85 wp,r(E) ................... 64
o';(E, D), 02,r(E, D) ...... 87 W(t, m,p) ................ 178 0,2,r(E, D), 0,2,r(E, D) ..... 87 ((8, D2, Iy2) ............. 266 O,p,r(coo(E)) .............. 94 b,mO,(coo(E)) .............. 97
Index page analytic torsion ............. 272 asymptotic expansion .. 246, 252 bordism group ......... 150, 151 bounded geometry ........... 65 characteristic classes ...... 12, 13 characteristic numbers .... 13, 14 Clifford bundles .............. 63 Clifford multiplication ........ 63 coarse cohomology .......... 143 coarse structure ............. 143 coarse map ................. 142 collared end ................. 277 cusp manifold ................ 55 Duhamel's principle ......... 181 end ......................... 159 Gaffneys theorem ............ 15 generalized component ............... 129, 134, 136, 140 generalized Dirac operator ... 63 geodesic ray ................ 159 GH-cohomology ............ 145 GH-component ............. 112 GH-distance ................ 111 GH-uniform structure ...... 112 heat kernel .................. 178 hyperbolic space ............ 284 index theorems .............. 49, 50, 51, 54, 56 K-groups .................... 61 Lipschitz uniform structure ............................ 115 340
- cohomology .......... 145 - component ...... 117, 118 - distance .............. 114 - map .................. 114 Lp-cohomology ............. 10 mapping manifold ......... 122 non-expanding end ........ 160 non-parabolic .............. 56 relative characteristic number ............................ 155 - determinant .......... 268 - eta function .......... 275 - index ................. 243 - index theorem ........ 247 - zeta function ......... 266 scattering matrix .......... 188 spectrum .................. 184 - absolutely continuous . 184 - components of ........ 184 - continuous ........... 184 - essential .............. 184 - purely discrete ....... 184 - resolvent continuous .. 184 - singular continuous ... 184 spectral shift function ..... 188 Sobolev spaces ............. 73 - embedding theorems ......................... 68,75 - module structure theorems ......................... 67,75 spaces of connections ..... 5, 7
Index
- metrics ............ 104, 108 supersymmetric scattering system ................... 243 uniform structure ............ 90
341
warped product metric .... 277 wave operator ............. 185 weighted Sobolev space ....................... 309,310
