Harmonic Mappings and Minimal Immersion

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: Fondazione C.I.M.E., Firenze Adviser: RobertoCo...

Author: Edited by E. Giusti

166 downloads 1045 Views 10MB Size Report

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

Report copyright / DMCA form

DOWNLOAD PDF

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: Fondazione C.I.M.E., Firenze Adviser: RobertoConti

1161 Harmoniic Mappings and Minimal Immersions Lectures given at the 1st 1984 Session of the Centro Intemationale Matematico Estivo (C.I.M.E.) held at Montecatini, Italy, June 24 - July 3, 1984

Edited by E. Giusti

Springer-Verlag Berlin Heidelberg New York Tokyo

Editor Enrico Giusti lstituto Matematico "U.DINI", Universit& di Firenze Viale Morgagni 67/a, 5 0 1 3 4 Firenze, Italy

Mathematics Subject Classification ( t 9 8 0 ) : 4 9 F 10, 5 3 A 10, 5 3 C 4 2 , 5 8 E 2 0 ISBN 3 - 5 4 0 - 1 6 0 4 0 - X Springer-Verlag Berlin Heidelberg N e w York Tokyo ISBN 0 - 3 8 7 - 1 6 0 4 0 - X Springer-Verlag N e w York Heidelberg Berlin Tokyo

Library of Congress Cataloging in Publication Data. Main entry under title: Harmonic mappings and minimal immersions. (Lecture notes in mathematics; 1161) 1. Harmonic maps-Congresses. 2. Immersions.(Mathematics)-Congresses. t. Giusti, Enrico. IL Centro internazionale mathematico estivo. III. Series: Lecture notes in mathematics (Springer-Verlag); t 161. QA3.L28 no. 1161 [QA614.73] 5t0 s [514'.74] 85-27644 ISBN 0-387-16040-X (U.S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re~use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210

PREFACE

The

international course on Harmonic mappings and Minimal

Immersions was held

at Monteeatini, Italy, June 24-July 3, 1984, organized by the Fondazione CIME. The

purpose

of the course was to describe recent results and to introduce to

the various methods that have been used in the study of harmonic maps between Riemannian manifolds. The

theory

recent years, surfaces, survey

of

and

topology,

of

all

harmonic

maps

has

received

several

important

contributions

in

has been successfully applied in various fields such as minimal complex analysis.

recent

advances,

but

The course was not aimed at giving a complete rather

at

describing

in

detail

some

specific

topics. The lecturers,

S. Hildebrandt,

J. Jost and L. Simon, were given complete free-

dom in the choice of the subject within the theme of the conference, and have provided

an

image

of

a

rather broad area. The texts of their lectures form the main

body of the present volume. During

the

course,

there

have

been

two

additional

lectures

by

J.

Sampson,

whose text is included, and by M. Seppala. I wish

to

express

my

gratitude

to

the lecturers and to all the participants

for their contribution to the success of the course.

E. Giusti

C.I.M.E.

Session

on

"Harmonic

Mappings

and

Minimal

Immersiqns"

List of Participants

G. Arca,

Istituto Matematico

L. Boccardo, Dipartimento 00173 Roma R. Caddeo,

UniversitY,

di Matematica,

Istituto Matematico

I.M. Costa Salavessa, G. D'Ambra,

Via Ospedale 72, 09100 Cagliari 2 ^ Universit~

UniversitY,

Department

Istituto Matematico

S. Hildebrandt, P.A. Ivert,

Mathematisches

Department

UniversitY,

Abteilung fur Mathematik,

T. Karlsson,

Department

Istituto Matematico

Istituto

Matematico

I.H.E.S.,

G.

Modica,

Istituto di Matematica 50139 Firenze

F.

Nicolosi,

K.

Ottarsson,

M.P.

Piu,

Y.S.

Poon,

35 Route

2 ^ Universit~

Ospedale

91440

Applicata,

72,

FacoltA

di

UniversitA,

Viale

A.

Institute,

University

des

Lumi6re,

Institute,

24-29

Sampson,

The Johns Hopkins Piazza dei Cavalieri

University 7, 56100

A.

Sanini,

Dipartimento Abruzzi 24,

M.

Seppala,

Department of Mathematics, SF-O0100 Helsinki

di Matematica, 10129 Torino

L. Sibner,

Polytechnic

Institute

R.

Mathematics

Department, N.Y.

11201,

N.Y.,

Doria

of Warwick, 68093 Giles,

Oxford

and Pisa

Seuola

Normale

OXI

of Helsinki,

College,

Brooklyn, City

95125

S. Maria

Catania CV4

7AL,

U.K.

3LB,

U.K.

Superiore,

University

St.,

Via

France

di Torino,

Jay

France

Coventry

Politecnico

333

Cagliari

6,

Mulhouse,

St.

Brooklyn USA

09100

Ingegneria,

Matematico,

Fr6res

S-581 83 Linkoping

Bures-sur-Yvette,

Trieste

4 rue

i0, Bonn

di Roma, Via O. Raimondo,

34127

Mathematical

S-881 83 Linkoping

Via Ospedale 72, 09100 Cagliari

Via

de Chartres,

i0, Bonn

D-4630 Bochum

UniversitY,

Mathematics

Brooklyn,

Fine Hall, Princeton,

Linkoping University,

UniversitA,

J.

Sibner,

Univ.,

di Matematica,

Seminario

I.S.E.A.,

72, 09100 Cagliari

Ruhr Univ. Bochum,

UniversitY,

Meeks,

Istituto

Princeton

di Matematica,

W.

R. Musina,

Via Ospedale

Linkoping University,

of Mathematics,

P. Marcellini, Dipartimento 00173 Roma P. Matzeu,

Av. 24 de Julho 194 3 ^ , 1300 Lisboa

Institut der Univ. Bonn, Wegelerstrasse

J. Kampmann,

I. Leurini,

72, 09100 Cagliari

Institut der Univ. Bonn, Wegelerstrasse

of Mathematics,

J. Jost, Mathematisches

Via Ospedale

of Mathematics,

C.L. Epstein, Department of Mathematics, N.J. 08544, U.S.A.

di Roma, Via O. Raimondo,

Univ.

Corso

Duca

Hallituskatu

N.Y.

i1201,

N.Y.,

degli

15,

USA

3,

V

L. Simon,

Department of Mathematics, Canberra, ACT, Australia

Australian

P. Smith,

Department of Mathematics, Texas Y78~3, USA

Texas A. & M. University,

V. Vespri,

Scuola Normale Superiore,

G. Well, Polytechnic P. Wong,

Piazza dei Cavalieri

University,

P.O. Box 4,

College Station,

7, 56100 Pisa

Institute of N.Y., 300 E 33rd 3t., New York, N.Y. 10016, USA

Department of Mathematics, Indiana 46556, USA

R. Ye, Beringstrasse

National

University

of Notre Dame, Notre Dame,

4, D-53 Bonn

J. Zyskowski, Institute of Mathematics, 90-238 Lodz, Poland

Lodz University,

ul. Stefana Banaeha 22,

TABLE

S.

HILDEBRANDT,

Harmonic Manifolds

J.

JOST,

J.

H.

L.

SIMON,

Lectures

SAMPSON,

OF

CONTENTS

Mappings

on H a r m o n i c

Harmonic

of

Riemannian

................................

Maps

Maps

.......................

in K ~ h l e r

Isolated

Singularities

Geometric

Variational

of

Geometry

Extrema

Problems

..........

118

193

of

.................

206

H A R M O N I C M A P P I N G S OF R I E M A N N I A N M A N I F O L D S

Stefan

I.

The D i r i c h l e t

2.

Harmonic

3.

Liouville

problem

mappings

4.

Estimates

5.

Riemann

6.

Existence

Removable

to m i n i m a l

for Jacobi

normal

7.

A priori

8.

Proof

9.

Miscellaneous

fields

maximum

harmonic

estimates

maps

Bibliography

2

...........................

11

in ~ n + m

25 ...............

................................

44

.................................

62

principle,

and r e g u l a r i t y

69

...................................

theorem

maps

.......................

92

............................

103

results "'"

10.

................

singularities.

for h a r m o n i c

of the u n i q u e n e s s

mappings

submanifolds

coordinates

proof,

for w e a k l y

for h a r m o n i c

into s p h e r o i d s

theorems.

Applications

Hildebrandt

. . . . . . . .

"

. . . .

" ° ' "

. . . .

° ' ° ° ' ° ' - . . . - ° - I O 8

...............................................

112

I. T h e

Let N

Dirichlet

X

and

problem

M

be complete

, respectively.

associate

for harmonic

mappings

Riemannian

To every mapping

manifolds

U : X ÷ M

of d i m e n s i o n

of c l a s s

C]

n

and

we can

energy

an

E(~) = fx ~Jd~!i Z' '2dv°l which

is o b t a i n e d

by i n t e g r a t i n g

e(U)

of

U

with

respect

o f the m e t r i c respect

to

tensor

to the m e t r i c

functional

U :X ÷ ~

= int X

Here of

and

2e(U) M

. The

is s a i d

satisfies

is the t r a c e

under

of t h e p u l l - b a c k

the mapping

functional

E(U)

of

local

u N)

M

we can write

harmonic

to be

the E u l e r

In t e r m s on

X

]

~trx

U

taken with

is c a l l e d

energy

Dirichlet integral.

or

A mapping

X

of

=

= ~IldU ]i2

<',->

energy density

the

coordinates

x =

these

if it is o f c l a s s

equations

of the Dirichlet

(x I, .... x n)

Euler

on

equations

Axul + Flik(u)Y~8~aui$suk

X

and

in t h e

form

C2

on

integral. u =

(ul,...,

= 0

where

£X = Y-1/2~{Y1/2ya~3 B] is t h e L a p l a c e - B e l t r a m i the f o l l o w i n g

operator

on

elements

on

coordinates

x

and

summed

I

to

from

. Here

and

in t h e

sequel we use

notations:

d o2 = ya~ ( x ) d x a d x ~

the l i n e

X

X u

and

and

M

= gik(U)duiduk

, respectively,

. Repeated

n = dimX

ds2

Greek

, Latin

with

indices

indices

are

respect

a,8,..,

to the

are

i,k,..,

from

is the

local

local

to be I

to

N =

w

dimM

. Moreover

u = u(x)

: (u I (x) ..... u N ( x ) ]

on

X

and

M

U :X ÷ M

; (y~B)

with

is the

respect

inverse

of

to the c h o s e n (X~B)

local

the

Christoffel

symbols

pl = ik gl3Fijk 2

J

on

M

'

Fijk

~ •

~u 3

r

and "

,

a

coordinates

y = d e t (y~8) '

denote

representa-

/

k

tion of a mapping

_ 2 I ( $ i g j k - 3j g i k + 3kgij) (glj)

~X a

=

(gik)-1

F1 ik

In l o c a l

coordinates,

e(U)

c a n be w r i t t e n

I e(U)

as

ui~ u k

= ~ g i k ( U ) XaS~ a

Dirichlet's problem for harmonic maps is o n e of t h e i n t e r e s t i n g in d i f f e r e n t i a l

geometry.

mappings

U : X + M

manifold

M

map

case

n = I

since

the

case

n = 2

boundary

fundamental

references For

reduces

has

that

introduce

Let

BR(p)

and denote

in s o m e some

treated

in

with

~X

8X

M

lectures

boundary with

into a

a preassigned

~ . and has

and Morse.

been well

The much

A survey

studied

more

and

intricate

further

by J o s t .

completely

is o p t i m a l .

understood.

In o r d e r

We

shall prove

to d e s c r i b e

this

result,

R

center

notations.

C(p)

ball

the c u t

BR(p)

a regular ball in

M

in

locus

M

of radius

of

its c e n t e r .

= {q 6 M : d i s t M ( q , p)

and with We call

p

the ball

_< R

if

(i)

/ ~ R < ~/2

(ii)

Here we have

the h a r m o n i c

on

to

questions

to i n v e s t i g a t e

by M o r r e y .

is n o t y e t

sense

be a geodesic by

of Hilbert

in the

X

agree

to g e o d e s i c s

been

are presented

manifold

which

are homotopic

work

first

n ~ 3 , the problem

a result

is the p r o b l e m

of a compact

without

~ : X ÷ M , and which

The

we

This

C(p)

~ BR(p)

is e m p t y

set < = max{O,sup_ (p)K sR

where

is the sectional

K

curvature of

M

, and

similarly

we define

~ = min{O,infBR(p)K} L e t us c o n s i d e r I. If

M

regular, 2. If

is s i m p l y by virtue

M

hemisphere 3.

If

~n+m

M

a f e w examples connected

SN

sphere

SN

K ~ 0 , then each

ball

in

M

is

theorem. , then

each ball

contained

in an o p e n

is r e g u l a r .

is the G r a s s m a n n

, equipped

and

of Hadamard's

is the u n i t of

of regular balls.

with

in t h e N - d i m e n s i o n a l

the

manifold

invariant

manifold

G(n,m) Cartan

G(n,m)

of o r i e n t e d

metric,

, N = nm

n-planes

then each ball

, with

/-~R < ~/2

in BR(p)

,

is

regular.

Moreover,

s =

I

and

O

K =

2

for

s ~ 2

g K

if w e

g 2

set

for

s = min{n,m}

s ~ 2

(employing

. That

the

, we

is,

standard

< =

obtain I

K

for

m I

s =

normalization

of

for

I , and

the

Cartan

metric). Now

we

these

can

state

the

main

results

such

Suppose

I.~.

MoreoVer,

let

that

exists

that

X

~ : ~X ÷ M

a continuous

map

UI~XL = ~

problem

U :X ÷ M

as well

U :X + M

for

C°(X,M) ball

~

and

the

Theorem

small

is

if U

U

~X

=

following

the

course

of

said is

~

which

that

~X

ball

is non-void.

mapping BR(p)

is harmonic

o f class

CI

. Then

there

int X

and

in

as

c BR(P)

to

a small

be

a harmonic

such

that

map

solution

of

int X

condition

fact,

2

be the

(1.1)

same

hemisphere H

sphere

whereas by H

Theorem

the

of the D i r i c h l e t into

holds

M

with

for

some

U

6

regular

each

can

pair

the

1.2

sense

an

of

int X

connected

of

antipodal

of

geodesic

one of

example

points

exists

that

= 7/2

is o p t i m a l . of

be

verified.

at most

of

into For

boundary

one

M

on

the

arcs

contained

theorem

1.2,

7

if

the

, that

n =

open

geodesic

equator in

ball.

•

in one

admit

values

n = N ~

instance,

exactly

points

cannot

a regular

, and

contained

by

can

~ .

definition

, /
map

solutions

there

,

for

the

exhibit

BR(P)

pair be

small

,

contained

~H

the

I

northern

can

closed

be

hemi-

. to p r o v e

following

1.2'.

into

c

Theorem

a continuum

is a regular int X

shall

, every SN

in in

a harmonic

sense,

of

problem

< ~/2

~(~X)

to

M = SN H

In order

blish

2, w e

for

~ 6 CI(~x,M)

is o p t i m a l /~R

with

extended

, and

connected

of

section

: ~X ÷ M = S N

In

1.1

instead in

result

of the D i r i c h l e t

I. T h e o r e m

~ z/2

cannot

uniqueness

For each m a p p i n g

1.2.

solution

Remarks.

3.

in

BR(p)

Then

in

proved

boundary

in a r e g u l a r

U(X)

A mapping

2.

be

and

be a p r e s c r i b e d

(1.1)

N =

is compact,

is c o n t a i n e d

t(~X)

satisfies

In

will

lectures.

Theorem

/~R

which

For ball

M

the

uniqueness

seemingly

each in

with

weaker

~ 6 C I (~X,M)

M , there U 6 C°(X,M)

exists and

it

suffices

to

esta-

result:

with

¢(~X)

at most U I ~X =

c BR(P)

one ~

, where

harmonic

such

that

BR(p)

map

U

U(X)

c BR(P).

of

This

observation

The

uniqueness

follows theorem

Dirichlet

problem

solution.

For

in any

ball.

As an e x a m p l e ,

They

If

We

there

and a l a r g e

between

solutions

of the

the

it is e a s y T o this

end,

circular $2

let

< i~

~3

can

are

holds

B

i . Then

whence

through

of M o r s e

which S2

surfaces

yields

- ~ u = u I~u I 2

on

that

in c a s e

E

Jost

~x £ ~ n [

that

be a small

that

~(~B)

exist

large

: Ixl

~ 17 )

data

be a

in

~3

that

u x.uy

=o

I-I

H

i . Both

way

. Thus

onto

that

B

,

a large

solution

2 for the d e r i v a t i o n

problem.

On a c c o u n t

solution,

for

have

proved

~ : ~B ÷ S 2 , t h e r e

rotationally Sn

cap

such

in a

6

F

to the

of the Euler

M = S N ).

is c o n t a i n e d

into

B U C

let

hemisphere

I

,

Palais-

Nevertheless

B = {(x,y)

curvature

of

is e n t i r e -

do exist.

a n d a large

furnishes

and B r ~ z i s - C o r o n [ 5 ]

boundary

of the D i r i c h l e t

o n e of t h e m m u s t

there

(see s e c t i o n

[48,49]

nonconstant

maps

u :B ~ S 2

the

and set

lUxl 2 = lUyl 2

computation

Ulc

out

,

an e l e m e n t a r y

for

assume

mean

a

yield

. Moreover,

S

"

before.

a n d the n u m b e r

in t h e n o r t h e r n

cap

q2

that form

solutions

=

are not

and

indicated

not s a t i s f y

= I!i y2

a small

M

u = "'Cu1(x,y),u2(x,y),u3(x,y)l

values

solutions

~u~ :

u = u(x,y)

infer

q2

contained

which ql

2 , the s i t u a t i o n

n = 2 , large

of c o n s t a n t

2ux^Uy

S2

out by L u c L e m a i r e .

is c o n t a i n e d

into

and

of

n ~ does

pointed

in case

boundary

trary

For

functional

been

on

in the s e n s e

properties

problem.

energy

are not

joining

ql

the

than one

and Ljusternik-Schnirelman

topological

equation Recently,

respectively,

which

q2

passing

, and that

we may

and

arcs

find a m a p p i n g

on

ql

two g e o d e s i c s

M = $2 = {u 6 ~ 3

arc o~

caps

two p o i n t s

, C = ~B = ~(x,y)

As=

we

large solutions.

it has

. It d i v i d e s

of t h e s e we

solutions

[46].

that

have more

call

them

see

possibility

might

exactly

to see that,

~2 : x 2 +y2

of

as

¢

can exist

Dirichlet

property,

the

data

shall

geodesic,

ly d i f f e r e n t s i n c e the Smale

exclude

principle;

there

n = I , the t h e o r i e s

relations

maximum

exist

are c o m p l e m e n t a r y

small

not

fixed b o u n d a r y

we consider

Then

a suitable

1.2 does

for

instance,

regular

antipodal.

from

the

other

one

3 < n ~ 6,

1.1

a large

ball

of

for arbi-

at l e a s t

of Theorems

in a r e g u l a r symmetric,

that,

exist

and

1.2,

solution,

if

S 2 . It turns

harmonic

maps

of

but

for

n ~ 7

none

two

Bn = (el.

[43],

and

the r o l e results

4.

[2],

It is

fairly

Theorem

two

1.3.

with

there

37-38).

of a c r i t i c a l described

following

CI

pp.

dimension

in s e c t i o n

easy

to see

~(~X)

that

c BR(P)

a solution

(I .2)

E(U)

where

the class

(1.3)

be

that

Theorem

C

, where of the

÷ min

~E(U,~)

direction o f a n y

tangential

and satisfies

Theorem

BR(p)

among

1.4.

for

that

if

all

: U -~

but

is s o l v e d

convergence

bypass

this

variational

of the

in

M

Then

,

and '

harmonic,

~

that is,

E

at

along

is a weakly

~

U

U

the first

vanishes

that

and,

stated

< I}

of

at

harmonic

is c o n t a i n e d

in

is s u f f i c i e n t l y

, then

~n

w.(x) into

(of ball

U 6 C2(~,M)

U 6 C°(X,M)

in T h e o r e m

map

map

in a regular

in particular,

~ 6 CI(~x,M)

1.4

=

is o p t i m a l .

(~,O)

Sn

As

, x 6 B

is w e a k l y

, of

harmonic

x = O

theorem

method

semicontinuity

H I'2 N L ~(~,F) will

field

result

H I'2

ball

of class

.

on

existence

in

U 6 C

c BR(p) I

2, the equator

by the d i r e c t

lower

mapping

6 ~1,2(~,~{)

o f which

discontinuous

to w e a k

class

other

problem

integral

U :X ÷ M

{ x £ ~ n : Ix[

to the b a s i c

space

= 0

U(~)

regularity

the

this

vector

is h a r m o n i c

ploying

the

that

in s e c t i o n

ball

n ~ 3

a consequence

is a regular

is weakly

UI~ X = ~ , and

see

unit

The key

U

interior

shall

the

Suppose

Then

5. The we

Various

plays

by

U

~[~I

M ) , the range

Moreover,

is

n = 7

direction.

is a b o u n d a r y

of the Dirichlet

regular

BR(p)

1.1

variational

is defined

~ = int X . Moreover,

into

mappings.

in the s a m e

~ : DX ÷ M

C : {U 6 H I'2 n L~(~,~4)

variation

the d i m e n s i o n

for h a r m o n i c

2 point

U(~) and

that

results.

Suppose

exists

It c o u l d

1.1

is the v a r i a t i o n a l

of the

calculus

of the D i r i c h l e t Although

there

for m a n i f o l d s

not be c o n t i n u o u s ,

there

difficulty.

A solution

of

inequality.

It, h o w e v e r ,

~

of v a r i a t i o n s ,

integral

turns

with

em-

way

and

mappings

A~

will

since

out that

to d e f i n e

possibilities

first

only

satisfy

solutions

(1.2)

respect

is no o b v i o u s

are v a r i o u s

(1.2)

problem

in

to a

of such

v a r i a t i o n a l i n e q u a l i t i e s satisfy a kind of m a x i m u m p r i n c i p l e which, turn,

implies that every s o l u t i o n of

the assumptions of Theorem

(1.2) is weakly continuous

in

~ , and T h e o r e m

are carried out in section

if we m a k e

1.3. The r e g u l a r i t y t h e o r e m then yields that

each s o l u t i o n of the v a r i a t i o n a l p r o b l e m harmonic

in

(1.2)

is continuous on

X

and

1.1 is proved. The details of this approach

6. The proof of regularity

is a c h i e v e d by a

careful analysis of weak solutions of the s e m i l i n e a r system I (u) ~ ui~Bu k = 0 Axul + F i k Y~8

(1.3)

that is a special case of the general q u a s i l i n e a r s y s t e m -~B{a aB (x,u) 8 ul} = fl(x,u,~u)

(I .4)

the right hand side of w h i c h derivatives

an estimate

that a c u r v a t u r e a s s u m p t i o n of type for the q u a d r a t i c m a t r i x

normal coordinates, other words, hand side

is of quadratic growth w i t h respect to the

~u .

One first observes yields

,

/kR < w/2

ulF~k(U)-

but not for the Christoffel symbols

aet only a s o - c a l l e d

only

if we use Riemann itself. We, in

"one-sided" c o n d i t i o n for the right

fl , and it has b e e n proved by Struwe

[80] and Meier

[61]

that, in general, we can neither expect r e g u l a r i t y of w e a k solutions nor can w e hope to o b t a i n a priori estimates d i t i o n holds. system

(1.3)

if only such a o n e - s i d e d con-

F o r t u n a t e l y the s i t u a t i o n is much better for the special than for the general s y s t e m

is a g e o m e t r i c a l

(i.e., invariantly

(1.4)

formulated)

since c o n d i t i o n condition,

and

/KR < 7/2 (1.3)

are the Euler equations of a functional that is covariant with respect to d i f f e o m o r p h i s m s of

M . These p a r t i c u l a r properties of our geometrical

p r o b l e m lead to r e g u l a r i t y results and a priori estimates h a r m o n i c mappings.

stoffel

symbols with respect to Riemann normal coordinates.

tes, w h i c h

are d e r i v e d in section

5, follow from bounds

along geodesics that are e s t a b l i s h e d in section needed seen,

for w e a k l y

The analysis rests on c e r t a i n estimates

for the proof of the uniqueness

theorem

of the ChriThese estima-

for Jacobi

fields

4. Related results are 1.2' which,

as we have

implies the stronger result of T h e o r e m 1.2. The proof of T h e o r e m

1.2' is given in section

8.

By s h a r p e n i n g the methods that lead to the regularity also o b t a i n a priori

estimates

theorem 1.4, we

for h a r m o n i c or weakly h a r m o n i c mappings.

These results are d e s c r i b e d and derived in section

7. A first but de-

cisive step for the d e r i v a t i o n of global a priori estimates is formed by the following local estimate that will also be proved *) in section 7.

Theorem

1.5. Let

regular

ball

subset

~o

u = (ul,u 2,...,u N)

BR(p) of

X

of

~ , and

that maps

let

~o

be R i e m a n n

normal

X : n O + B2d

Suppose

also

respect

to the

that

coordinates

x =

of

onto

,

0 < I ~ ~ , for all there

exist

~ 6 ~n

numbers

and

•

all

the m e t r i c

x 6 B2d

on

n , N , I , V , e = min{O,infBR(p,K} not / on

d , such

that \

u(x)

[U I (X) ..... uN(x))

BR(P)

is e s t i m a t e d

the H ~ l d e r of every

harmonic

ball

tensor

of

X

with

satisfy

,

k > 0

, and

seminorm

on an o p e n

the o p e n

•

, 0 < • O

(x 1,...,x n)

I i ~ I 2 _< yc~B(x)~a~ B _< ~ I ~ i 2

(1.5)

Then

yaB(x)

the c o m p o n e n t s

local

be a c h a r t

homeomorphically

B2d = {x 6 ]Rn : Ixl ~ 2d}

coordinates

that d e p e n d

only

< = m a x { O , S U P B R(p)K} of

map

the

local

U :X + M

,

representation with

U(~ O) c

by

(1.6)

[u]

~ kd -T C ~ (Bd)

This result can be used to obtain L i o u v i l l ~ t y p e sults on the r e m o v a b i l i t y

of singularities

theorems

this to the Gauss map of m i n i m a l s u b m a n i f o l d s in Bernstein- and B e t s - t y p e theorems. ted in ~ e c t i o n

as well as re-

of h a r m o n i c maps. ~n+m

Applying

we can derive

Some theorems of this kind are presen-

3.

It should be m e n t i o n e d that the a priori estimates of section 7 lead to another a p p r o a c h to solve D i r i c h l e t ' s p r o b l e m for h a r m o n i c mappings,

if

we combine them with S c h a u d e r - L e r a y degree theory.

C O M M E N T S and R E F E R E N C E S to the LITERATURE: An important example of h a r m o n i c mappings, submanifolds

the Gauss map of m i n i m a l

in E u c l i d e a n space, will be studied in section 3. Other im-

portant examples are p r o v i d e d by conformal m a p p i n g s

and by m i n i m a l

sur-

faces w i t h t w o - d i m e n s i o n a l p a r a m e t e r domains. A survey of p o s s i b l e a p p l i c a t i o n s of h a r m o n i c maps is given in the paper [12] by Eells-Sampson,

in the surveys

and in the lecture notes

[13] and [14] by Eells and Lemaire,

[48],[49] by Jost. We also refer to the lectures

*)Except for S e c t i o n 9 , w e

shall only treat the case n>2

(cf. JOST for n=2).

of Jost p r e s e n t e d For

n k 3 , the

established proved

(see

stence that

proof

are c o m p a c t

of

M

20-21).

Hamilton

result,

using

to m a n i f o l d s

[33] was not optimal.

in 1975,

in case

suitable

a priori

with positive Such a r e s u l t

Theorem

Details

1.1

is in some

sting results. (cf. Jost

of this p r o o f

For

[47])

that

M

the p r o p e r t y

and

that

there

be an o p e n boundary

M0

that

exists bounded

mapping

subset

suppose

out

M

is

simpler

which

proof

could be

K , but the result formulated

as T h e o r e m

the D i r i c h l e t

in section

it covers

6. A l t h o u g h

by no means

all intere-

theorem:

of a complete

convex

o f some

convex

function

Riemannian

Riemannian

of class

on

to

M° ,

M 0 . Moreover,

manifold

H I '2 ( ~ , M O)

manifold

with respect

t 6 H I ' 2 ( ~ , M o ) . Then

that

U : ~ + M0

estimates,

[34], m i n i m i z i n g

is s t r i c t l y

a strictly

that

the w o r k of S c h o e n - U h l e n b e c k

subset

9M o

the first exi-

In case

that we have

the f o l l o w i n g

is a c o m p a c t

%~ , a n d

harmonic

from

ode can d e d u c e

Suppose with

sense optimal,

instance,

gave

found a much

curvature

are carried

Hart-

for such h a r m o n i c

n ~ 3 , also a s s u m i n g

[32]

1.1 was p r o v e d by H i l d e b r a n d t - K a u l - W i d m a n integral.

U : X ~ M . In 1967, theorem

flow method.

Hildebrandt-Jost-Widman

and if the

then any s m o o t h m a p

map

[27],

in 1964

flow method,

and w i t h o u t boundary,

problem

the heat

maps was

by the heat

is n o n p o s i t i v e ,

for the D i r i c h l e t

over

on h a r m o n i c

into a h a r m o n i c

and e m p l o y i n g

of H a m i l t o n ' s

theorem

[12] who,

this r e s u l t by an u n i q u e n e s s

connected,

carried

M

K

[13], pp.

K ~ 0

simply

and

can be d e f o r m e d

supplemented

maps

of

X

existence

and S a m p s o n

curvature

:X ÷ M man

first

by Lells

that if

sectional

at this meeting.

with

there

with

let

smooth exists

boundary

a

values

.

If

M

admits

noncompact sets

f (P) = c~

preceding M

convex

c}

sectional

f , such as c o m p l e t e do,

have a s t r i c t l y

Thus we o b t a i n

o

function curvature convex

the f o l l o w i n g

the sublevel

boundary

corollary

~Mo =

of the

theorem:

admits

c a n be s o l v e d

a strictly f o r all

It should be noted the corollary,

convex

boundary

that u n d e r

an u n i q u e n e s s

p. 492,

remark

3).

exhaustion values

function,

the D i r i c h l e t

~ 6 H 1'2 n L~(~,M)

the c o n d i t i o n s theorem

not true as is seen by g e o d e s i c [47],

exhaustion

of p o s i t i v e

Mo = ~p 6 ]M : f ( p ) <

~p 6 M

If

a strictly

manifolds

of this

analogous

theorem

to Theorems

arcs on a p a r a b o l o i d

problem

.

or even of

1.2 or

of r e v o l u t i o n

1.2' (see

is

10

The p r e c e d i n g theorem is based on an interior r e g u l a r i t y energy m i n i m i z i n g harmonic maps due to G i a q u i n t a - G i u s t i Uhlenbeck

[76]. Moreover,

theorem for [18] and Schoen-

b o u n d a r y r e g u l a r i t y for m i n i m i z e r s was

e s t a b l i s h e d by S c h o e n - U h l e n b e c k

[77] and by J o s t - M e i e r

The fundamental uniqueness t h e o r e m 1.2' was

[51].

found by J ~ e r

and Kaul

the slightly s h a r p e n e d version of Theorem 1.2 was derived by Jost who e m p l o y e d his heat flow t h e o r e m of principle

[42];

[46],

[45] as well as Hamilton's m a x i m u m

for solutions of the h e a t equation.

The Theorems

1.3 and 1.4 were

the first a priori estimates

(essentially)

n o n p o s i t i v e sectional curvature of result w e r e

proved in

for h a r m o n i c mappings

[34]. Furthermore,

(for not necessarily

M ) w e r e derived in

first derived by G i a q u i n t a and H i l d e b r a n d t

[33], but optimal [20]. These com-

plete results w e r e p r e c e d e d by the interior C a - e s t i m a t e s s t a t e d in T h e o r e m 1.5 that were proved by H i l d e b r a n d t - J o s t - W i d m a n

[35]. Very i n t e r e s t i n g

a priori estimates w e r e later on e s t a b l i s h e d by Jost and K a r c h e r and by Sperner

[79]. Very recently,

Schoen

[50],

[75] has s k e t c h e d a simple,

direct approach to a priori estimates that is based upon the B o c h n e r formula for maps.

Ae(U)

and, therefore,

cannot be applied to w e a k l y h a r m o n i c

11

2. H a r m o n i c

In

this

mappings

chapter,

into

we

shall

spheroids

mainly w:B

of

the

unit

the

harmonic

mappings

÷ Sn a

ball B = {x

of

consider

n-dimensional

sn{ =

6 ]Rn : Ix[

Euclidean

w

:

(u,z)

< IfI

space

~n

6mnxm

n_>3

, into

the

n-dimensional

: [Url2+(z/a) 2

a

Such

a mapping

(u ( x ) , z ( x ) )

=

Aw =

Iv]-2(Aw

satisfies

• v)v

sna

£W±Tw

,

A =

spheroid

11 f

=

w(x)

,

or

~ +~2{...+~

where =

(u,a-2z)

1-a 2 2 ]ml 2 = I + - - - ~ z a

and

From

lul 2 + (z/a) 2 = 1 we

obtain

• Aw = -{',~ul 2 +a-21~zL 21] Thus

a harmonic

map

w : B + Sna ' g i v e n

by

: (u(x) , z ( x ) )

w(x)

, x 6 B

,

%

is c h a r a c t e r i z e d

by

the

elliptic

system

(2.1) -Az

where

h(a,z)

I~zl 2 = ZI~ If

= [1 + ( 1 - a 2) ( z / a 2 ) 2]

the

spheroid

system

(2.1)

S na

=

-£w

w =

C~(B'~n+I)c for

, 3 = (~1,~2 .....

(u,z)

be

a harmonic

a tangential

x 6 B . Then

the

coincides

{w 6 ] R n+]

reduces

(2.2)

Let

+a-2]DzI

2

3 n)

,

I3ul

2 = [L~uL2,

zL2

a = I , the

Sn

and

= a-2h(a,z)]I~u]2

with

unit

sphere

: jwl = I }

to

=

l~wl2w

map

B ÷ S na

, and

Vector field along

second

the

variation

w

~2E(w,~)

denote , that of

by is, the

~ : ~(x) energy

(v,q)

6

£ Tw(x) S an functi-

12

onal

E

Here

D* =

which

isa g l' v~xa e n by

tion R

at

in d i r e c t i o n

D ....

1Rn + l

of

denotes

holds

~1,2

for

positive

Moreover, along

w

w

and

~

is g i v e n

derivative

denotes

<w I , w 2 >

is

the

we

respectively.

> O

vector

field

zero

~

(that

to be u n s t a b l e

is s a i d

can d e f i n e

of

s na

'

and

along

is,

w

~(x)

which

~ 0

is of c l a s s

on a s u b s e t

of

if t h e r e

is s u c h

a vector

field

= O

fields

a weakly

W,(X)

=

but weakly

< O

to be weakly harmonic if

vector

map

is a s i n g u l a r ,

metric

strictly stable if

is c a l l e d

~2E(w,~)

is n o t

tangential

unstable,

equator

Riemann

w ,

projec-

tensor.

SBdx

as b e f o r e

The

mapping

measure).

w : B + Sa n

for all

the

the o r t h o g o n a l

satisfies

(2.3)

holds

along

g

~2E(w,~)

A mapping

by

, where

w : B + Sa n

is s a i d

that

of

covariant

;

tangential

N L ~ ( B , ~ n+l)

with

T w S an

curvature

mapping

every

the

D~ = ~

onto

is it R i e m a n n

A harmonic

B

w

~

along

harmonic

( u,(x),O ) harmonic

map

,

map,

w

. In the

same w a y

to be

strictly

stable

x

U,(X)

since

= T~

,

x6B

u, £ H I ' 2 ( B , ~ n )

or

,

for

n ~ 3 , and Aug + 13u, 12u~ = O

A well note

known

reasoning

by De G i o r g i

in

now

B-

implies

{O}

(2.3);

cf.

[11].

We also

that ~u~(x) I 2 =

(2.4)

where In the B ÷ S na

(n- I)Ix1-2

d e n o t e s the a r e a of S n-1 n following we shall particularly

and

E(w~)

= ~I ~n-1

~n

'

e

which

is the p r o t o t y p e

investigate

of a s i n g u l a r

the

equator

but weakly

map

harmonic

w, :

map.

13

We

first

note

In general, tric

that

We

if t h e r e

~ -= ~/2

same

compare

boundary

function

= I~

, (2.5)

values

with on

the

such

equator

as

w~

to be r o t a t i o n a l l y

symme-

that

, a cos~ (Ixl)>

all o t h e r

~B

is said

~

sin~ (,xl)

yields

w~

symmetric.

HI'2(B,Sa n)

w 6

is a s c a l a r

w(x)

shall

is r o t a t i o n a l l y

a mapping

(2.5)

For

w.

map

w..

weakly

harmonic

. To this

end,

maps

we

that

have

introduce

the

the two

classes C ~ = {w 6 H I ' 2 ( B , S n )

: w - w,

6 HI'2(B,]Rn+I)}

and rot = It w i l l

turn

w 6

out

that

:w

a crucial

Theorem w ~ w~

2.1. . On

role.

> I

n

2/~L-~ (n-2)

We n o t e

for

If

that

3 < n -< 6

,

E(w)

a ~ a n , then

the

symmetric

the n u m b e r

anplays

is r o t a t i o n a l l y

other

hand,

if

a

n

 E(w.)

n -> 7

for

a~l

w~

is

a < a n , then

w 6 Cro t

with

unstable

(even

Crot).

within

It is u n k n o w n m u m of

E

as to w h e t h e r

within

w~

even

class

C . The

weakly

harmonic

furnishes

following

the u n i q u e

result

shows

absolute

that

mini-

this m i g h t

be true.

Theorem that

2.2.

a ~ a

The n e x t mapping mappings

Theorem w # w~

The

mapping

w~

is

strictly

stable

provided

n

result within

shows

of c l a s s

2.3.

If

C

C

, then

w~

is e s s e n t i a l l y

whenever

w~

the o n l y

is a m i n i m i z e r

weakly

of

E

harmonic

among

all

.

E(W~)

. Moreover,

H2"2(B,3Rn+I)

that

class

if

= infcE w

w

, then

is a w e a k l y

= w~

E(w)

> E(w~)

harmonic

. In p a r t i c u l a r ,

for

mapping

there

is

of

all

w

class

no r e g u l a r

6 C

with

C 0 harmonic

14 map

within

We,

however,

Theorem

C

.

have

2.4.

If

the f o l l o w i n g

a ~

I

and

partial

n ~

The T h e o r e m s

2.3 and 2.4 imply a

NONEXISTENCE

RESULT

There

is

w(x)

=

no

regular

(x,O)

for

In o r d e r

7 ,

w : B ÷ S an

map

Theorem

= infcE

-

PROBLEM :

Ix[ = I , p r o v i d e d

to e s t a b l i s h

E(w~)

then

for the D I R I C H L E T harmonic

result:

that

2. I-2.4,

with a ~

the

I

we shall

boundary

and

n ~

values

7 .

first derive

some a u x i l i -

ary results. Lemma

2.1.

For

L~(B,]R n+1)

any

tangential

along

the

(2.5) ~2E(w~,~g)= Proof.

Note

~I +<°2

w~

<01 =

into a c o m p o n e n t a component

<s2

vector

w~

, we

field

maps

(v,O) <01

B-{0}

, <02 =

into

(0,~)

of

n-1 2 2 q 2} dx a r

2

H 1'2 n

class

, where

the e q u a t o r

of S na . Thus a d e c o m p o s i t i o n of

, yields

that is t a n g e n t

perpendicular

(v,~)

~ =

have

]B{,~vI2 - -n-1 - ~ Ivl 2} d x + ~ l a q i r B

that

with

map

to the e q u a t o r

to the equator.

sphere

A simple

of

w~> = O

,

w~> = <~ w~,~ w~>Ivl 2 - ~ < ~ u ~ , v

n-1 = ~ a r

2 ~

>2

Since

= <[[a ~ 1,a ~2> = O we infer

that ~2E(w~,~)

Moreover,

= ~2E(w~,~1)

+ @2E(w~,~2)

<~a~02,~ %02> = I ~ ] 2

62E(w~,~2 ) = SB{I~I2

0 = <~1,w~>

=

that

---~--~n-12 } d x a

Finally,

implies

yields

r

<~au~,v>

= -<~ v , u ~ >

Since D~I we then o b t a i n

= ~a~l

= ~I

- < ~ a ~ 1 'w*>w*

= ~I

-<~

<0 =

sna ' and

computation

yields

r = Ixl-

v,u.>w.

15 2 < D ~ I ' D ~ ~I> = <~e~1'~a~1 > - [<~a v ' u * > i2 - ~ 2 and

JE~w,,~ t) : SB{l~vi2-~-~IvI2}dx

.

r

if w e take

(2.4)

Lemma

If

2.2.

into a c c o u n t .

~ 6 HI'2(B,~)

S~2r-2dx <

(2.6) Let

i.e.,

x = re

r,m

~B~2r-2dx

~ ~ 0 ,

then

~ -" ~12ax

B Proof.

and

(n-2) 2]B ~-~

be polar coordinates

,

of

r=Ix[

x 6 ~n

centered

a t the o r i g i n ,

. Then =

i~~SoI ~ 2 rn-3 drd~

= _

1 2 ~ n-2 drd~ I~l~ J°n---~r

½S 2r-2dx + The e q u a l i t y Remark.

sign implies

It f o l l o w s

that

@2E(w.,~)

and represents

a continuous

We n o w d e f i n e

the q u a d r a t i c

(27

3~ , w h e n c e (2-n)~ = 2r~-~

that

is w e l l

quadratic

defined

f o r m on

~ = O ~ 6 ~1,2 (B ,JRn+l )

for

HI'2(B,~n+I)

forms

: S il vl - Lvi2}dx

- --T-Z a r

r

for

v 6 H1

,2

(B,~ n)

(2.8)

~ 2 n 6 H I' (B,~)

and

~2E(w*,q0) = H(V) + K ( D )

, respectively.

for

~o =

By

dx

(2.5), w e h a v e

(v,r])

o

Lemma and

2.3.

We

Proof.

have = 0

H(v) a.e.

We w r i t e a g a i n H(v)

where

v(e)

= v(re)

for a l m o s t all

> 0 on

for B

all

v 6 HI'2(B,~R n)

with

v

~ 0

.

x = re

, r = Ix I , I0Jl = I . T h e n

~vl 2dx + } r n - 3 Q ( V ) d r = f ~--~i B O defines

r 6 (0,1

] J Q(V) = I~I--II

an H 1 ' 2 - f u n c t i o n

, and

~vE2

{n-1)Iv[2}d~

on

S n-1

= {e : 10~i = I}

16

The

quadratic

form

at the

identity

ated

Q(~)

is the

map

id : S n-1

Q(v) ~ (S-n) cf.

[14].

Thus

second

variation

+ S n-1

~

I~I=I

, and

Ivl2de

of the

energy,

it is w e l l

evalu-

known

that

;

we o b t a i n ~ ~ l ~~v 12dx-

H(V)

(n-3) S B r - 2 1 v l 2 d x

,

B and if

v ~ O

, v 6 HI'2(B,~n)

n = 3 . If

n = 4 , we

H(v) and

for

infer

= ]B

n > 5 , we have

taking

Proof

once more

of T h e o r e m

4 (n-3)

62E(w.,<0)

Lemma

2.3 y i e l d s

(n-2)2/4

Then, K(q)

Therefore

Proof > 0

(2.6)

2.2.

= H(v) + K ( n )

such

that

> 0

> 0

2.1.

<

tiple

r° 6 of

(0,1) z , and

H(v)

> O

> O

and

,

therefore

>

o

account.

of

(2.8

for

~ =

if

v ~ O

(n-2)2 4

if

, we have (v,~)

, and

6 HI'2(B,]Rn+I)

a ~

n

implies

(n-1)/a 2

lql

2]

dx

if

> O

~ #0

~ # 0

Suppose

that

a <

• T h e n w e can

n

choose

an

the n u m b e r = 1{ ( n - 2 ) 2 4 ( n - I )0} a

Let

2

. Hence

again

I 2} ~vl dx

(n-2)

> O

2.2,

_> SB[]~ql 2

62E(w,,~)

of T h e o r e m

into

by L e m m a

(2.6)

4_(n-3) ] B~l~12dx ~v (n--~i~j

By v i r t u e

H(v)

from

{!$v 2_r-2 Zr

.(v) _> i by

~B ~~ V r 2 d x

implies

be a n u m b e r

such

2 +e

that

~

<

log r °

is an i n t e g e r

mul-

set q (x) =

C 4

t

r I - n / 2 s i n (-/L-glog r)

r for

O

_< r _< I o

r

_< r O

J

o

where

r =

]xl

Then

<0 =

(0,~)

6 H 1,2

N L ~(B,]Rn+I)

is a t a n g e n t i a l

,

17

vector

field

along

w,

, and

2 N = --e r

AN + a r

Therefore

~2E(w,,~)

Thus

w,

is u n s t a b l e .

the mapping

w,

the

class

Cro t

the

second

part

In o r d e r

of of

to p r o v e

= K(n)

Since

even

= -

~ =

yields

the

( X s i n ~ (r) ,a c o s ~ ( r ) )

Moreover,

(O,~)

part,

, r =

Ixl

we

mappings

chose

, in

of

E

of

w,

,

within

B ÷ Sna " T h i s

proves

an arbitrary

Cro t

mapping

w(x)

=

. Then

2~nrl • 2 -~-JO ~ ( r ) s i n 2 ~ ( r ) r n - l d r

+a

nrl"2 +--~Jo~

implies

~1 2,. . JoCOS ~tr)rn-3dr

variation

point

2.1.

first

wI3 B = w,i~B

< 0

is a " r a d i a l " critical

symmetric

~n I = -9-]'(n-1)sin2~(r)rn-3dr -O

E(w)

~x elxl-21~i2dx no-< I i-
an u n s t a b l e

rotationally Theorem

r ° _< r <_ I

for

cos

sin,~(1)

2, ~(r)rn-ldr

= 1 , cos~(1)

= O

. Therefore,

t 2 = JOn_--X~cos~ ( r ) s i n ~ ( r ) ~ ( r ) r n - 2 d r

< 2JO c ° s ~ ( r ) r n - 3 d r

+

f ~2(r)sin2~(r)rn-ldr

.

Hence I • 2 fO s l n ~ ( r ) r n - 3 d r

I n-2

=

~1 2 JoCOS ~ (r)rn-3dr

> I

4

- n-2 and

_

1-2(r)sin2~(r)rn-ldr

(n-2) 2~O~

thence ~n n-1 + -> ~ n - 2

E(w)

a

-

(n-2)

2 fO ~ 2 ( r ) s i n 2 ~ ( r ) r n - l d r

~°n r l . ~ 2 ( r ) c o s 2 ~ ( r ) r n - l d r +--~Jo

By 2.1

(2.4), is an

Proof

We

because, w

=

now

n-1 n-2

consequence

2.4.

This

first

note

for

every

(u,z/a)

a ~ I . Thus

Consider

= en 2

E(w,)

immediate

2.1.

I

a mapping if

have

of T h e o r e m

Theorem a =

we

we

of

result that

the

will

B

assume

thus

w =

into

the

first

claim

previous

estimate.

turn

to b e

it s u f f i c e s

mapping

of

an a r b i t r a r y

' and

out

to p r o v e

(u,z) : B ÷ S an

S n = sin

that

of T h e o r e m

a consequence the

assertion

' one

can

satisfies

E ( w +)

to s h o w

= E(w)

that

, we

can

for

define E(w)

_< E(w)

a = I mapping

w =

(u,z):

B + Sn

with

wlB B = w,l~B

+ We want

of

E(w,) assume

S E(w)

. Since

w.l.o.g,

that

w

=

z ~ O

(u, lzl) .

satisfies

•

18

Consider

the r e t r a c t i o n

R

of

the c l o s e d

upper

hemisphere

Sn

of

T

into

the o p e n

upper

R

for

T

hemisphere,

(vsin@,cos~@)

( v t , .... v n )

v =

,

defined

=

by

(vsinT~l~,cosT~)

Iv[ : I , 0 _< ~ _< ~/2

, 0 < ~ < I , and

set

w k = R1/k ~ w Then

tends

wk

to

w

in

H 1,2(B,IRn+I)

and,

in p a r t i c u l a r ,

lira E ( w k) : E(w) k÷~ Wk(B)

since

c S n = {(vsin'i,,cosg) : v 6 S n-1

from Theorems

1.1-1.4

that there

exist

O < ~ < T~/2~

uniquely

determined

B ÷ S n which are regular, harmonic and satisfy T as E ( h k) ~ E ( w k) Since there are rotationally on

B

with

W k iSB ' w e Obviously Thus

values

L2

A well

known

hi

E(h)

the o t h e r

a.e.

hand,

an < I

for

of T h e o r e m

It r e m a i n s

to p r o v e

of Theorem

2.4.

2.1,

are

~ lira E ( w k)

n h 7 , and

boundary

uniformly

weakly

then

hk :

i = Wki as w e l l k I~B ~B symmetric harmonic maps

rotationally

tO a m a p p i n g

h

in

values symmetric. bounded.

H 1,2

of c l a s s

and strongCro t

implies

= E(w)

therefore

< E(h)

whence

E(w.)

the f o l l o w i n g

result

Rotationally

hk

converges

theorem

S l i m inf E(h~')

On account

2.5.

on B

semicontinuity

E(w.)

Theorem

hk

are

maps

infer

h

and w i t h

hk

of t h e m a p p i n g s of the

and p o i n t w i s e lower

hemisphere

t h a t the m a p p i n g s

the H I ' 2 - n o r m s

a subsequence

ly in

On

in the u p p e r

conclude

, we

~ E(w)

for all

w

that has been used

symmetric boundary values on

~B

6 C in o u r p r o o f

that are

c o n t a i n e d in the open upper hemisphere can be e x t e n d e d to a rotationally symmetric harmonic map of

Sketch

of the proof

(see

B

into the open upper hemisphere.

[43]

for details).

map

w(x)

/

A rotationally

symmetric

\

= ( X s i n ~ (r),cos~ (r)) , r = /

B

into

Sn

is h a r m o n i c

if

Ixl

,

of

19

n-1 + ~

(2.9)

-

n-1

sin2~

: O

2r 2 holds

on

[0,1]

. This

(2.10)

if w e

a + (n-2)~-

introduce

conditions I .

be

a : (-~,O)

4(0)

Equation can

at

= O

(2.10)

is

linearized

=

, ~ =

B

the

two

B =

two

the

eigenvalues

[O,~] =

(2-n) 8 + ( n - 1 ) a

eigenvalues

n 2-

of

is

the

the

7

of

order

(-~,O]

: ~ ( e t)

equation

. The

l i m a(t) t÷-~ the

boundary : O

damped

pendulum

second

for

there

is

=

8

, ~ =

(0,0)

(2-n)8-

and

are

11

=

(n-l) (~-w/2)

(~,5)

=

('n/2,0)

, 12

:

I - n

whereas

n <- 6

. That

I

are

2-n 2

I ± 2

~

n -> 7

(0,0)

point

a =

system

one

for

point

spiral

infer

which

that

connects

trajectory

if

of

always

is a

l

, but

negative

saddle

point,

whereas

a stable

nodal

3 _< n -< 6

exactly

~(t)

+ ~/2

definition

for

is

a solution

of

Combining

these

the

of

Lemma

then

2.4.

of

with

one

the

If

as

t + --

line

and

invariant

(2.9)

W :B +

2.4

S na

= 0

a = ~/2 is

for

curve

. For

(z/2,0) point

in

, but

that

that

~(r)

for

we

arrive

is

complete.

if

at

the

is

~(r)

, the it

be

infinitely 7

the

maximal

[ O , r o]

, and

, is

stated

harmonic and of class

a

trajectory

n A

= }(ror)

assertion

(a,B)-

\

crosses

, r E

the

(a (t) ,~ (t) ) n ~ 7

(-~,t+)

note

is weakly

. Let

increasing

, where also

provided

results,

Theorem

t+

. We

(~/2,0)

a(t)

~(t)

t ÷

a(t)

and lim

6 . Moreover,

of

proof

(0,0)

(2.10) left

3 ~ n ~

cases

and

system

/

stays

=

.

then

plane

, a(O)

(n-l)sin2~

(a,B)

first

is p o s i t i v e

critical

a stable

n_> We

8n + 8

the

~(t)

to

and

points

of

_

and

on

correspond

(2-n) 8 +

DI/2

means,

by

w/2

first

= O

into

systems

equilibrium

The

÷

the

,

the

transformed

differential

into

B

is

n_~1 s i n 2 a

, ~(1)

the

transformed

a =

with

equation

often

in b o t h interval

, r E

[O,1]

,

a solution. above,

and

H 2, 2 (B,~n+1)

,

20

(2.11) where

~

denotes

gradient on

1 jB I ~

t ( w i~B ) =

The same statement

energy,

~ 6 C c ( B , ~ n)

e > O . Then

~0 = ~B~Bw

sn-1

, and

wl2de

holds for each rotationally

w : B + S na with finite smooth boundary data. Let

~w~ ~ J 2doj

= E(w I~B) - ~ J s B

the tangential

(2.12)

Proof.

I

(n-2)E(w)

symmetric, weakly

harmonic

or for each energy m i n i m i z i n g map with

, t 6~

with

6 ~ 1 , 2 ( B , S n)

Itl < e

, and,

for s u f f i c i e n t l y

for

w t(x)

small

= w<x+t~(x)>

,

we o b t a i n ~ t E ( w t ) It=O = ~B<~ w,~ since

~(x)

harmonic.

is a.e.

tangential

On the other

hand,

along

~0>dx = O w

, and b e c a u s e

an e l e m e n t a r y

SB{~I ~wl 2div~ - < ~ w , ~ S w > ~ a } d x Thus

the left hand

simple

side m u s t

approximation

~I~B = o If we c h o o s e

vanish

argument,

w

computation

is w e a k l y

yields

= ~ t E ( w t ) It=O oo n 6 C c ( B , ~ ) and, by a 6 Lip (B,]Rn) with

for every

also for every

i

J

~(1-~÷t)

~(x)

= I - t/~

= ~(Ixl)x for

tends

to zero,

we arrive

formula

w

or S c h o e n - U h l e n b e c k well

defined.

f "~

2de+

only

,

of

[77].

l~wl2de

(2.11),

on a c c o u n t

and has ~B

Thus

2

smooth

according

of

boundary

for

the b o u n d a r y

x = 0 .

values,

to a r e s u l t integrals

The same is true for r o t a t i o n a l l y

can be s i n g u l a r

O < r < I -~

~B

is e n e r g y m i n i m i z i n g

in a full n e i g h b o r h o o d

for

at

~wl 2 = I~wl 2+I~I If

= 1

~w~ I 2 d x - ~ I f r ~ l ~ w l 2 d x = Jr~l~B B

-~(n-2)~I~wl2dx B and this i m p l i e s

~(r)

O < t _< ~ , we o b t a i n

1(n-2)S~l~wl2dx B As

, where

w

is s m o o t h

of J o s t - M e i e r in

symmetric

(2.11)

[51]

are

solutions

which

21

Remarks.

I. If

w = w~

(2.13)

E(w,)

2. L e m m a

2.4 a l s o h o l d s

for h a r m o n i c We,

mappings

in p a r t i c u l a r ,

C

of Theorem

where

~ =

izl

n = 2 , and

w :B ÷ M

2.3.

Suppose

minimizer

, and

is a c o m p a c t

then

infer

w

that

K

w =

in

for all

of

B

(u,O)

continuous

mizer

of

harmonic therefore

real

We finally

(B,~ n+1) and

a n d of

on

.

which within

mapping

£

6 C ,

C . Then ~ n - 3

, we

and

infer

z 6 HI'2(B,~)

infer have

B

such

refe-

that

C . Moreover, of c l a s s see

If

Thus we

infer

that

that

, and

u

were

retraction

there

H 2 ' 2 ( B , ~ n+l)

immediately

, Stampacchia's

z = 0

lu I = I

is i m p o s s i b l e .

is n o

from

D C

there

(2.11-13)

is a m i n i exists

is c o n t i n u o u s

no

and

.

every minimizer

E

with

. We already

w.

implies

w =

of

map

~B

. Thus

a continuous

harmonic

on

within

(u,z)

(u,~)

2in

~ = O

C N H 2 ' 2 ( B , ~ n+1)

on

that

(2.11)

w = w~

we now

, we would

. Hence we

of class

coincide

c Sa n

E

w = w =

of

z > O

. Since

, whence

S n-1

of

C

q ~ 0

z ~ O

B

harmonic

show

the i d e n t i t y

E(w~)

w(B)

each weakly

must

M

n>dx ~ O

with

analytic

will

particular,

~,~

implies

within

mapping

solution

. Since

let

also

6, c o m m e n t s

~u12 + a - 2 1 ~ z 1 2

minimizer

E

is a w e a k

h(a,z)a-2~

sphere

that each weakly

is t r u e

manifolds

dimension

chapter

-~z

. From

the

E

~uI2 + a - 2 1 ~ z l

to b e c o n t i n u o u s into

of

Hausdorff

(see

, and

C . Then

-£u = h(a,z)

~ 6 HI'2(B,~)

w = w =

with K

: [I + (]-a 2) (z/a2) 2]

principle

relation

n = 2

= infcE

within a minimizer

B B-

(u,z)

fB<~

maximum

infer

Riemannian

for

E(w~)

E

is a g a i n

set

that

h(a,z)

where

known

an a n a l o g u e

into arbitrary

of

that w is r e a l a n a l y t i c in r e n c e s to t h e l i t e r a t u r e ) .

We

. Thus we

have

b e an a r b i t r a r y

there

~B

n1_--TE(w,! ~B)

=

for

on

1~wll2de = -..[~Bll~Wll2de ]~BI~ -~

(2.14)

Proof

~w = ~ 0

, then

(u,z)

u(x)

of c l a s s

know

S~BI~-~i2de = O

. Thus

within

that

C

and,

w =

, on account

is r e a l

in

C n H 2'2

analytic

(u,O)

,

of

E(w)

on

B-

K

=

22

and

satisfies

(2.17)

-Au

as w e l l

=

IZui2u

on

B-K

as

(2.18)

u(x)

Furthermore

= u,(x)

,

~--~ 3U(x)

= 0

on

~B

we have

(2.19)

-Au,

=

i~u,l ~

2U,

on

~B

for

m = 1,2,3,...

B - {0}

and ~ue Dr

(2.20)

n-1 A = ~---~0--a 4 -7

Note

that

Then

(2.17-19)

!mplies

(x) = o n I

~r +r-~'2Ae , w h e r e

Ae

is t h e L a p l a c i a n

sn-1

on

that 32u =

0

on

~B

Sr 2

and we also

8e ~ ~r u = 0 for and

and

~ = 0,1,2, .... set

'

get

-2 u = 0

on

If w e n o w d i f f e r e n t i a t e

r = 1 , we arrive

~B

(2.17)

with

respect

to

r

at

~3 u

Proceeding

this way we

~VU(x) ~r D Then we

apply

conclude

that

= O

of

analytic

continuation.

cOMMENTS

and REFERENCES

B

on

on

SB

the C a u c h y - K o w a l e s w s k a y a

in s o m e n e i g h b o r h o o d

The

~r 3 = 0

3B

, whence

~B

for

v = 1,2,...

theorem u(x)

and obtain

= u.(x)

for

u(x) x 6 B-

= u.(x) K

by

to the L I T E R A T U R E :

/ \ singular harmonic mapping w. (x) = ~ u . ( x ) , O ) , u, (x) = x/ixl , o f ] f f x £ ]Rn : Ixl < 17 into Sn a p p e a r e d f o r the f i r s t t i m e in H i l d e -

brandt-Kaul-Widman [34].

[

The harmonicity

], a n d proof

implicitely

already

was originally

in H i l d e b r a n d t - W i d m a n

carried

out

by e m p l o y i n g

a

23

stereographic duced

in

ping

projection.

[37],

pp.

w~ : B ÷ S n

proved

that

minimum

results

particular,

derived Wood

[92]

Kaul

[43] w e r e

derived

[17];

other

[43].

3 ~ n ~ 6 , but

2.1-2.4 for

2.4

see also

been

[37],

by S c h o e n

interesting

was

pp.

stated

and U h l e n b e c k properties

is an

mappings part

of the

and,

problem

case

571-573.

[43]

2.1-2.3)

have

in been

of a f o r m u l a

generalized

formula

an a b s o l u t e

of

harmonic

the D i r i c h l e t

intro-

authors

The m a i n

and L e m m a t a

is a s p e c i a l

recently

of W o o d ' s

2.5.

was

of the map-

These

tool

symmetric

in T h e o r e m

notes

furnishes

n A 7 . The main

Result

Lemma

idea

reproved

various

by J ~ g e r - K a u l

rotationally

that has

The b a s i c

cars-Seller-Burns

properties

(Theorems

[I ],[ 2].

by W o o d

stability

that

the N o n e x i s t e n c e

[53].

The b a s i c

is d e s c r i b e d

section

by B a l d e s

in t h e s e

of the

of w h i c h

of this

presented

for

provided

investigation

B ÷ S n , part

found

C~

approach

discovered

is u n s t a b l e

within

detailed

498-499.

were

w~

The

by K a r c h e r

and

in G a r b e r - R u i j s e n -

The

results

[78] w h o , i n

of h a r m o n i c

of J [ g e r -

addition,

maps

into

the

sphere. In the f o l l o w i n g Sn

we

(i) E v e r y

minimizing

into

energy

list

the N - d i m e n s i o n a l

N S+

is r e g u l a r except

~

:=

provided

(ii)

Every

Both

smooth N

into

S+

results

example

w~

terexample The next (iii)

set

Suppose Then

~

N+I

A 0

domain

~

of

into

~n

}

n ~ 7 , then dimension

u

is r e g u l a r

at m o s t

n-7

. For

is d i s c r e t e . every

constant

.

energy

if

these

and at the .

same

V

results

minimizing)

map

u

from

3 ~ n ~ 6 .

and by G l a q u l n t a - S o u c e k

that

results

are

are

due to S c h o e n

d(N)

a number

denotes

that

every

mappings

time,

[21].

optimal

been

proved

by

The J ~ g e r - K a u l

since

there

is a coun-

n ~ 7 .

three

[I]

. If

of Hausdorff

independently

d(2)

where

on h a r m o n i c

o f some

: IYl = I , y

3 ~ n ~ 6 ~

set

[78]

shows for

Define

~N+I

y 6

(in p a r t i c u l a r ,

have,

results

N u : ~ ÷ S+

map

is n e c e s s a r i l y

Schoen-Uhlenbeck

further

hemisphere

that

for a closed

n = 7 , the s i n g u l a r

~n

some

largest

M

is a R i e m a n n

energy

minimizin~

[78].

by

= 2 , d(3)

the

and Uhlenbeck

= 3 , d(N)

integer

manifold ma~

u

= [win(N/2 + 1,6}]

S I

for

of dimension

from

M

into

I 6 ~

N~4

I

.

n , and SN

for

that

is s m o o t h

n S d(N) . in

the

24

interior crete, n

-

d

and

(k)

-

If

(iv)

~n

of

. If

Z

is a c l o s e d

singular

and

d(3)=

from

~n

seems very

, every

n _< d ( N )

and

2

d(N)

energy

the

special

(iv)

For

eVery

o f the

or

examples

of

data

data

data he

for

every

also

U

N >- 4 , a n d

smooth

(x,O)

in B a l d e s '

proved

the

map

w

is not

Eells

from

stable

suppose

harmonic

map

nonexistence

following

result

existence

result

do

-&w

= w

of

of

contained

Lemaire

might

= 0

~

B

in

9B

into

into

S na

the e q u a t o r

Sn a ~ {z _> O} with w I~B = ~ ,

sphere

Sn N a

S an

and

w~

mapping

6 C °o (B,S n )

ellipsoid

of

w(x)

play

[ 15] the

confirm

same

this

impression.

exceptional

role

as

The the

boun-

for n+1 n-2

,

w

> 0

,

(n -> 3

, N =

I)

,

(n =

, N =

3)

,

for

for

harmonic

[70],

Wente

const)

on

2HWxl

mappings

[85], ~B

=

and

, then

^Wx2

,

w :B +

Lemaire Lemma

2.4

+ Sna " O n

N = 2

[48]

and

least

WI~Bl

on

two ~B

Jost

harmonic o

; compare If,

implies

w :B

, then

SN

[58].

each harmonicmap

at

map

2

Then

boundary

ko(~B)

AW

or

at m o s t

4) :

is a h a r m o n i c

{z = O}

dary

boundary

smooth

that

The

is dis-

is constant.

because

Satz

provided

boundary

u

dimension

minimizing

= min~[N/2],4~

2 < n < d(N) SN

that

[ 2 ],

there

and

into

(see

of

set o f H a u s d o r f f

~

N > 3

It

~

is c o n s t a n t .

SN

that

are

set

.

N >- 3

Let

n = d ( N ) + I , the

in g e n e r a l

I

into

(v)

M

the

that other

Brezis-Coron

mappings

for

w : B ÷

[5 Sn

the

2

results

example, w(x) hand, ] have with

~ 0 if

(or

same

Pochozaev

~ 0

(or

~ const)

wi~ B ~ O

proved

the

of

w(x)

that

and there

boundary

for n = exist

data

25

3.

Liouville

theorems.. R e m o v a b l e

ma! submanifolds

By a c l a s s i c a l on

~n

in

t h e o r e m of L i o u v i l l e ,

is n e c e s s a r i l y

analogous

Theorem

result

3.1.

manifold

X

U

into

be a h a r m o n i c

in some

Here

a Riemannian ~n

line e l e m e n t x = x(q)

manifold

X

manifold

which maps

X

or c o m p a c t

M , and

an

Riemannian

suppose

BR(p)

ball

two p o s i t i v e

topologically

d~ 2 = ¥~B(x) dx e dx B

of

M

that .

of

X

with

if

X

numbers onto

IRn

respect

is h o m e o I ,p

, and a

such t h a t the

to the c o o r d i n a t e s

, q 6 X , satisfies

x

and

~

in

~n

furnished with a metric erator

~

~

~ ul~I 2

. In o t h e r w o r d s ,

for w h i c h

X

is t o p o l o g i c a l l y

the a s s o c i a t e d

Laplace-

~n

Beltrami

op-

AX = y-1/2 ~ {y1/2ya6 ~6 }

is u n i f o r m l y

elliptic

~n

on

P r o o f of the theorem:

Let

. A simple manifold

clearly

is c o m p l e t e .

/ [ ul(x) . . . . .

\ uN(x) )

be the re-

u(x)

presentation

of the h a r m o n i c m a p

centered

p £ M . Then we

at

= U

with respect

to a n o r m a l

have

1 (u) ya8 ~ u i ~ u k AX ul and the r e a l v a l u e d I

A Xv

+ Yik

function

= u

i

A×u

i

a v = lul 2

+ y~B ~

B

= O

satisfies

ul ~B ul

= 7~8 [ ~ u I ~B ul - ul F~ k(u) ~ u i ~8 uk ]j By v i r t u e

function

manifolds:

of a s i m p l e

regular

harmonic

we shall prove

is said to be s i m p Z e

iI~I 2 ~ y ~ ( x )

for all

entire

section,

map,

, and if there e x i s t

X : X ~ ~n

A p p l i c a t i o n s to m i n i -

of R i e m a n n i a n

map

Riemannian

is a c o n s t a n t

to

mappings

a complete

U

morphic

a bounded

In this

is c o n t a i n e d

Then

chart

a constant.

for h a r m o n i c

Let

U(X)

the r a n g e

singularities.

~n+m.

of T h e o r e m

5.1,

formula

(5.16), w e i n f e r

that

chart

26

(3.1)

Axv

where [4

<: = m a x

and

Axv

X

by

_> O

is compact,

(3.1),

If,

{O , sup B R ( P ) K}

]u(x)] -
(3.2)

If

> 2 aK(lu[) gik(U) y~8 Z a u i

o n the o t h e r

hand,

immediately

d

tend

to i n f i n i t y .

An

inspection

lowing

Theorem

3.1'

no b o u n d e d

the c u t

w/2

where,

Proof.

BR(p)

On account

the

second stant

U

---const

the s t a t e m e n t

of T h e o r e m

(I .6) of T h e o r e m

the p r e v i o u s

whence,

is a c o n s t a n t

proof

yields

map.

3.1

1.5 by

Riemannian

functions

the

constants.

Suppose

a complete

Riemannian

manifold

into

is

contained

~

center

case whence

in s o m e p

letting

the

fol-

assumptions,

as b o u n d e d v(x) U(X)

in

the

is a c o n s t a n t

on

X

e(U) (x) ~ O

we

infer

does

that

/~R

~(p)

not

. Then of

once

again

from

0 ~v ~R 2 . Since functions,

v(x)

this

U

M , the

which

assume

boundary

~ R '2

, whereas

, and

that

the

map.

, and

subharmonic

~ R 2 , or

c SBR(P)

BR(P)

ball

that c a r r i e s

K = m a x { 0 , sup B~v(p)K }

by

contained U

manifold

. We m o r e o v e r

is d e f i n e d

of our

either

but

X

, or

we h a v e

proof

of T h e o r e m

that does

assume

in

we

a<(lul)

implies

see

with

that

carries

that

R' < R

> O

(3.1)

X

holds U

v(x)

. In the in the

is a con-

map.

Another 3.1

of

is,

v(x)

be a c o m p l e t e

is s u b h a r m o n i c

constants

case,

part

is c o m p l e t e l y

luJ 2

ball

therefore

implies

That

is simple,

o f its

as usual,

const . Then first

of

locus

ball

v =

only

map

U(X)

either geodesic

X

of which

meet

is a r e g u l a r

result.

Let

U(X)

range

, and

principle

the e s t i m a t e

first

subharmonic

is a h a r m o n i c

that

the

additional

X

from

BR(p)

.

e(U) (x) - const.

follows

of

X

the m a x i m u m

we o b t a i n

Since

a<([ul) > 0

on

~8 u k

that

not u s e X

X ~ ~n

T h e n we h a v e

3.1.

is simple,

, ca6

L e t us

the a p r i o r i and w e

= ¢'TYeB

consider

estimate

a second

(1.6).

For

proof this

of T h e o r e m

purpose

set

• fl(u ) = ae~ F~k(U) Sa ui ~B uk

we

27 f~ a~6 Sa u i 9 5 ~i dx

(3.3) for all

~ 6 ~I,2

We m o r e o v e r

N L=(~,IR N)

and for each b o u n d e d

open

set

~ c ~n

.

introduce

Q(u) Then

= f~ fl (u) $i dx

there e x i s t

=

numbers

aa5 ~a ui

~B ui

[ , ~ , and

a , with

O < I ~Z

and

a ~O

such

that (3.4)

and

~ lSul 2

that

-< Q(u)

f = (fl,f2

(3.5)

r

-

--

t

-< ~ 13ul 2

fN)

satisfies

f(u) I -< a Q ( u )

Moreover,

we infer

ul fl(u)

By v i r t u e

from T h e o r e m

~ a~5 ~

of

/~ lul

(3.6)

that

ui ~5 ui - a<(lul) gik(U) a ~5 9a ui ~B uk

-< / ~ R

u I fl(u)

5.1

< w/2

, there

is a number

a~ < I

such

that

~ a~Q(u)

holds. If we IRN

test

and

(3.3) w i t h

n 6 C~(~,IR)

~q Q(u) h d x = f~

This Lemma

implies 3.1.

% = R {u- ~ ], w h e r e

vector

in

I

+ ~ ~9 a~B Se lu- ~12 ~B ~ d x

(u I _ ~i)

. fl(u ) ~ dx

the f o l l o w i n g If

is a c o n s t a n t

, ~ ~ O , we get

~ 6~N

is

~

(a ~ + a i ~ l )

~Q(u)

~ dx

.

result. a constant

vector

with

l~I S

norm

(I - a ~ ) / a

then

(3.7)

f n a~ IR

2 ~

4u-

~B q dx

~ O

for with

The next

lemma

is a v e r s l o n

of the w e a k

Harnack

all

n 6Cc(IR

n k 0

n

,R)

.

inequality

for super-

,

28

solutions.

Lemma

3.2.

radii

v 6 H 1'2

Br

Let

r >0

D L~(B4r,~)

(3.8)

~

there

such

that

aag ~

v asqdx

a number

sup

B

Vr

v <

~B

=

centered

< O

for

60 > 0

which

(I - 6 0 ) sup

Br where

two c o n c e n t r i c

balls

at

0

in

~n

with

. Suppose

that

all

n 6Cc(IR,IR)

q->O

with

c+

exists

(3.9)

be

satisfies

IRn

Then

B4r

and

4r , r e s p e c t i v e l y ,

and

depends

only

v + 8 o{r

on

n

and

~/X

'

B4r

v(x) dx

stands

for

the a v e r a g e

of

v

on

the ball

r r

Without

loss

of generality,

R

we

:= sup

can

assume

that

lul Ik n

Moreover,

we

introduce

Ur

=

the

average

~B

u(x) dx r

of

u

Lemma

on

the

3.3.

as w e l l

ball

The

u

limit

:=

lim r~

as

(3.10)

!im r ~

In a d d i t i o n ,

sup

every

~B

Proof.

Choose

Lemmata

3.1

IRn

and

lu

~ 6JR n

-

~1

3.2

then

=

with

an arbitrary

i u - El Br

iu-~i

exists

2 dx

and

I~I

satisfies

= O

r

Vector

sup

(0)

~r

we have

(3.11)

for

B r = B r ( O ) c IRn.

IU

norm

vector

-

El

I~I

~ I -a a____~*

~ 6~N

with

a

imply

2

2 <-

I - a~

i~i-<-

(I - 8 o)

sup~ n

l u - El

+ 6o

~ B r l u - El

2

dx

• The

= R

.

29

_<

sup

lu - ~]

2

iRn As

r ~ ~

, we

obtain

(3.12)

lira r

In

~B

-~°°

particular,

tu- ~2

dx

=

supu
l u - ~r

r

for

(3.13)

that

~ = O

lira r ~

~S

, we

lul

2

get

dx

=

~u!2

sup

r

= R2

]Rn

Since ~B

] u - ~I 2 d x

=

~B

r we

conclude

from

(3.14)

lul 2 d x

- 2~ • U r +

I~I 2

r (3.12)

supN n

and

(3.13)

[ u - ~I 2 = R 2

that

+

lim r ~

I~I 2 -

~ " Ur

2 lira

exists

and

that

~" U r

r-~oo

Set

I - a* aR

~ -

exists

" Then

a vector

(1 + ~) R

arrive

u

. Now

T > 0

6N N

the

, and

which

vector

by

virtue

satisfies

~ = -T~

of

ILl

is

R = suPiRn = R

and

admissible

in

lul sup

(3.14),~

, there n[U+TU

! =

and

we

at lim r ~

Ur " u

lim r ~

Ur

= R2

,

whence

because

of

lUrl

,

luol ~IR

lu-~l

2 dx

=

. Furthermore

=

I~I

2 +

%

r as

r ~ ~

. Finally,

SUP

Thus

We

now

ploy On

the

can

the

account

with

proof

the

(3.14)

-

2 u r "u

~ 0

yields

l u - ~I 2 = R 2 +

n

of

Lemma

supply

a priori of

lul 2 d x r

the

3.3

coordinates

3.3, Lu

2~" L

=

[u

-

[I 2

complete.

announced

estimate

Lemma

is

I~I 2 -

proof

of

Theorem

3.1

that

does

not

em-

(1.6) there which

exists satisfy

some

point IL t

= R

q

in

BR(p)

as w e l l

as

(3.10)

and

30

(3.11)

Since d i s t ( U ( x ) ,q)

we

infer

from

(3.10)

Denote

by

Pt

dist(Pt,q) Now we

=

I

dist2(U(x),q)

dx

= O

with

Po

r , the

geodesic

= p

and

Pl

= q

" Then

(I - t) R the representation

to n o r m a l

I

and

coordinates

:= {t 6 [0,I]

:

is c l o s e d

~B

' t 6 [0,1]

introduce

respect

• lu(x) - u

that

lim r ~

(3.15)

< const

contains

find

a number

all

t 6 (to, t']

Then

we

: sup

{t 6 [O,1]

t'

: sup

t = O

with

ut(x)

which

IRn

I.~n

are

of

d(U(x),Pt)

lutl

at

if

such

Pt"

map

U

Clearly,

with the

set

_< ( I - t ) R }

_< ( I - t ) R

. Moreover,

t o < t' ~ I

the harmonic

centered

}

to 6 I

that

and

/ ~ suPlRn

to < I lutl

, we

< ~/2

can for

*) u

can

repeat

the previous

. In particular,

(3.16)

relation

=

lutl

s uPlRn

(3.13)

lim r ~

considerations

for

ut

instead

of

becomes

]B

lut 12 dx

:

lira r ~

r

~B

dist2(U(x),Pt

) dx

r

Since d i s t ( U ( x ) ,pt ) -< d i s t ( U ( x ) , q )

we

infer

fore

from

t 6 I

(3.15)

for

hence

U(x)

In

following

the

Theorem

3.1

we

space.

manifolds.

in

~)

the

first

shall

to d e r i v e

Grassmann us

(3.16)

that

t 6 (to, t']

sup

ipn

(I - t) R

lutl

Thus

we

,

_< ( I - t ) R obtain

and

I =

these-

[O,1 ] , a n d

--- q

Euclidean

Let

and

all

+

For

review

following

This

procedure

will

be

this

some

will

is

followed

apply

Bernstein

be

the Liouville theorems

purpose,

results

we

for

- type

for minimal

shall

study

Grassmann

result

stated

submanifolds

harmonic

manifolds

maps

G(n,m)

in of

into

that

used.

similar

to t h e

in sections

approach

6 and

7.

that,

in some

more

detail,

31

By

G(n,m)

of

all

This

we

n-

denote

manifold

to

on

Gin,m)

that,

moreover,

for

G(I,1)

the

following

On

every

locus an

R

BR(p)

both

G(1,n)

manifold

not

ray

for

the

meet

in

G(n,m)

The

manifold

BR(p)

is

G(n,m)

its

can

be

G(n,m)

tangent

space

inner

product

of

is

given

Y*

With

the

the

identify

simply

kind

a Riemannian

complete with

connected.

point

Po

locus

Sn

In

p

and

that

/~ R < ~

radius

satisfies as

of

space. . Except

addition,

, the

cut

" Let

K ~ 0

be

Po

of

G(n,m)

with

center

hence

also

. Thus /~R

quotient

/SO(n)

G(n,m)

of

K

G(n,m)

of

expressed

, Y>

=

invariant

transpose

respect

to

of

of

the

Y

tr(XY*)

metric

, and

normalized

G(n,m)

X , Y 6 TpoG(n,m)

, and p

and

its

every < z/2

denote

cut

ball

radius locus BR(p)

.

space:

xSO(m)

matrices. 6 TpoG(n,m

The )

,

of

G(n,m)

t r ( X Y*)

we

may

satisfies

choose

<

K = I

s ~ 2 , it

is

metric

corresponding

O < K ( X , Y)

If

sense

this

in

denotes

a

suitable

the

trace

way. of

Here

the

X Y *

K ( X , Y)

Thus

is,

of

by

normalize

is

metric

G(n,m) can be described by nxmPo 4 4 tangent vectors X = (x~) , y = (y~)

<X

matrix

, this metric

T

two

(3.17)

if w e

the

2

defined

IRn + m

known:

in

= SO(n+m)

=

consists

of

under

homogeneous,

curvature

, provided if

invariant

some

locus

that

planes)

invariant

n = m

conjugate

ball

is

for

is

at

n-

only

G(n,1)

are

sectional

regular

Every

and

first

conjugate

which

a uniquely

G(n,m)

geodesic

first

in

dimensions

(called

a symmetric,

starting the

the

Gin,p)

on

with

a closed the

is forms

results

bound

. Except

fact,

geodesic

. Then

does

, the

N = nm

metric

factor,

Gin,m)

can

coincides

upper

by

SO(n +m)

in

of

subspaces

a Riemannian

positive

. Thus

manifold We,

of

a constant

manifold

oriented

carries

transformations up

the

dimensional

2

two

K ( X , Y)

~ I

if

s = min

the

sectional

independent if

{ n,m }

n = I

~

curvature

tangent

2

or

vectors m

= I

, and

.

as

if

not

~

(3.17),

to

S = I

totally

,

trivial

< =

2

if

to d e s c r i b e

S --> 2

•

geodesic

balls

BR(p)

32

in a l g e b r a i c B~(p)

terms.

contained

purpose,

we

< p,q >

by

Yet

in

define

(3.18)

BR(p) for

where

e I,

...

of

p

and

q

f1'

"'"

'fn

it is e a s y

,e n

two

>

in terms

of an a l g e b r a i c p , q 6G(n,m)

symmetric

sets

condition. an i n n e r

For

this

product

= d e t (< ea,f 8 >)

and

p

maximal

n- planes

f1'

"'"

, respectively. of

to d e f i n e

and

q

denote

'fn

Then,

oriented

for g e n e r a l

, respectively,

orthonormal

bases

el,

...

bases

,e n

and

we h a v e

det (< ea,f8 >) (3.18')

>

= /det(<e~,eB>)

and

the f o l l o w i n g

Lemma

3.4.

(3.19)

B{(p)

B~(p)

Then

For

product

~p,q~

:

> cos s

~

}

p

and

=

m i n { l~qXl

s = min { n,m}

For

<>

(3.21)

q

of

~n+m

we

can d e f i n e

another

by

<>

3.5.

and

{q 6G(n,m)

n- planes

(3.20)

Lemma

=

{n,m}

c BR(P)

(unoriented)

inner

holds:

s = min

Let

/det()

=

: x6p

, ixT = I }

p,q £G(n,m)

and

<>

=

<>

-< i1

<

<>

,

we

have

and <>

(3.22)

Lemma that

3.6. is

Let

over,

suppose

e >0

such

be

that

~

for

a complete immersed

there

exists

TxX, p ~

A e

n - dimensional

into an

the

Euclidean

n- plane

Riemannian space

IRn + m

p 6 G(n,m)

and

manifold . Morea constant

that

(3.24)

holds

X

isometrically

s

all

x 6 X , where

TxX

denotes

the

(appropriately

oriented)

33

tangent

8pace

of

way

be i d e n t i f i e d

of

G(n,m)

such

the i m m e r s e d with

. If we

that

p

an

manifold

m-plane

coordinates

z n+1

by

x ; TxX

at

~n+m

in

choosec~r~esian

is d e s c r i b e d

X

= 0

, ...

can

in a n a t u r a l

, that is, w i t h an e l e m e n t I n+m ~n+m z .... ,z on , zn+m

= 0

and

can

thus

be i d e n t i f i e d w i t h ~n j then there exist fn+1 (x) , C ~ - functions f n + 2 ( x ) , . .. • f n + m ( x ) , fi : IRn ÷ ~ s u c h that X c a n be r e a l i z e d g r a p h o f the m a p p i n g f : R n ~ ~ m , f = (fn+1 , . . . . fn+m) :

X

(3.25)

Moreover,

=

{ (x,f(x))

these functions

x6R

n }

satisfy

I +

(3.26)

:

as

I 2 <- - 2

'l~f(x)

,

~ = I,

•

.~

sn

.

Since

(3.27)

are

T~(x)

the c o o r d i n a t e s

(3.28)

x

us

pings

a simple

now

specify

into

Theorem

3.2. X

U(X)

U

of

/~
The

U

some

Theorem

X , we

obtain

p(£) ~ I

constant

. Thus

X

is in

3.1

for

U : X ~ G(n,m) a Grassmannian

< = I

the

particular

case

of h a r m o n i c

map-

manifold.

Let

between of

The

that

is h a r m o n i c field.

on

~6 fl(x)

manifold.

is c o n t a i n e d

3.3. X

tensor

= 8¢~B + 8 a f l ( x )

be a h a r m o n i c manifold

in s o m e if

BR(P)

ball

s = I , and

map

G(n,m)

of a simple such

c G(n,m)

< = 2 if

that

Riemannian

the r a n g e

that satisfies

s ~ 2 , s = min

{n,m}

is c o n s t a n t .

link

Theorem

with

" 8S f(x)

~e ~8 S p(S) I~I 2

into

where

submanifolds

fold

~ yas(x)

a Grassmannian

manifold

Then

the m e t r i c

~ 61R n

and

particular

Let

of

i~12

f o r all

+ ~af(x)

= 8e6

minimal ~n+m

Gau~ map

parallel

G

is i s o m e t r i c a l l y

if and only

In p a r t i c u l a r ,

is h a r m o n i c .

submanifolds with

if

X

in

~n+m

mean

: X ~ G(n,m) immersed

more

general,

field)

the E u c l i d e a n

with parallel

of a minimal

space

mean

submanifold

of

is g i v e n

of an n - dimensional

into

is i m m e r s e d

the G a u ~ m a p

(or,

curvature

of

by

mani-

~n+m

curvature

~n+m

,

.

34

The

GauB map

G : X ~ G(n,m)

submanifold X

the

X

of

tangent

The mean

manifold.

The mean

standard

H(x)

X

It is

tangent on

Dv H 6 Txl~ +m

Theorem

Let

X

Suppose

that

field

immersed

to e a c h p o i n t

is s a i d

~

x

of

D

the n o r m a l

space

sub-

if

denotes

the

the o r t h o g o n a l NxX

pro.

type result.

n - dimensional

into

the s e c o n d

as a m i n i m a l

to b e p a r a l l e l where

indicates

onto

Bernstein-

trace of

is i m m e r s e d

v 6 TxX

, and

exist

X

H

vectors

immersed there

, is t h e

if

be a c o m p l e t e

is i s o m e t r i c a l l y

field.

zero

& ~n+m

the f o l l o w i n g

3.4.

, x 6 X

~n+m

Then we have

that

an i s o m e t r i c a l l y assigns

.

curvature

connection

with

the map which

field

f o r all

of

associated is

TxX

form on

= 0

jection

space

curvature

fundamental

(Dv H ) A

IRn + m

~n+m

a fixed

with

Riemannian

parallel p

n- plane

manifold

mean

curvature

and a number

a

0

with

(3.29)

> COS

, s : min

% such

{n,m}

< = i

~2-7~

if

'

i 2

s->2

planes

q = TxX

that

(3.30)

-> O

holds Then an

for

(appropriately

the a s s o c i a t e d n- dimensional

Proof. with by

all

Assumptions 0 < /~R

(3.19).

< ~/2

GauB map affine

On account

(3.21),

and

that

of Lemma

that, G

from Lemma

by virtue

into

and,

infer

G(n,m)

therefore,

theorem

3.6

of T h e o r e m . Theorem X

is p r o v e d .

(3.30)

imply

c B R (p)

3.4,

we have

with

and

3.3,

X

X

X is

is a n u m b e r

BR(p)

R

is d e f i n e d

< ~/2

yield

is a s i m p l e

is h a r m o n i c a l l y

linear

there

where

for all

3.2 n o w i m p l i e s

is a n a f f i n e

that

(3.30)

, and

of

~n+m

/~R

~ T x X, p ~ that

is c o n s t a n t

of

G(X)

(3.29),

0 < ao < Then we

subspace

c BR(P)

(3.22),

tangent

G : X ~ G(n,m)

~inear

(3.29) such

G(X)

Moreover,

oriented)

that

subspace

x 6 X

Riemannian

mapped 0 of

manifold

b y its G a u s s

is a c o n s t a n t IRn + m

. Thus

map the

map

35

In

addition,

fn+l

we

(x)

,

if

"--

infer

X

,

by

is

described

fn+m(x)

a short

(3.31)

in

x CN n

,

,

computation

< T x X, p >

the

over

form

the

(3.25)

n-

as

plane

graph

p

:

of

IRn

c

functions

N n+m

v

that

-

/% (x)

where

X(x)

(3.32)

The or

statement that

Thus

we

Theorem

of

the

fn+1(x), have

theorem

...

arrived

at

zi =

define

also

that

or,

that

there

more

z n+1

is a number

(3 . 34)

O

<

B°

= 0

B°

< c°s-S

vector,

functions. the

,n+m

previous

,

submanifold mean

theorem:

, z n+m

x 611{

X

IRn + m

that

field.

Suppose

too much from

. More

: 0

of

curvature

do not differ

, ...

a constant

functions

has a p a r a l l e l X

" $~ fi (x)

~a f i ( x )

is

of

i = n+1 , ...

of

+

linear

version

n - dimensional

planes

@aB

9f(x)

affine

the smooth

general,

plane"

are

,

=

that

following

fi (x)

the tangent

"horizontal that

the

a nonparametric

is minimal

XaB(x)

yields

, fn+m(x)

Suppose

3.4'

,

: det(xas(x))

precisely,

the suppose

with /~ ,2- ~-J ~

)

,

s = min

{n,m}

,

K =

{I

if 2

such

s =I s->2

that

(3.34')

where

If

m

=

I

and

condition furnishes

An entire

(3.35)

, we

on

IRn

have

solution

by

(3.32)

which

s = < =

(3.34') for

for all

-< Bo

is d e f i n e d

X(x)

linear f u n c t i o n s

3.3'

2

y(x)

I

simply M

=

, (3.34)

becomes

I the

~f

~I + ;~fL 2

an affine

yields 18f(x)

following

O

no I ~

....

restriction

const.

equation

on

IR n

are

fn+m(x)

n -plane

result:

surface

-

,

fn+1(x),

represent

of the minimal

div

Then

x £ n -{

Hence

in

for

IRn + m

B

Theorem

,

36

with

s u p ~n

This

is M o s e r ' s

stronger

3.2 of

(3.34)

by

is

be

Giorgi,

of

~ ~/2

to

. We

by

in

the

a harmonic

Simons

. This

the

which that

is

this

property

do

/JR

not

of

by

down

Theorem

the

assume

to

< ~/2

situation breaks

criterion

preceded

equivalent

condition

to h a n d l e

decisive

following

equator

it

n > is

8

not

other

replacing in

Theorem

the

methods

if

3.2.

the

equal-

This

can

example:

tinuous

clear

that

and

S2

,

simple K ~ I

contained

in

t 61k

~

, i.e.,

< = I

closed

the

theorem

,

manifold

the

Clearly,

Bernstein's

and are

~ ~3

cases

f : ~n

ya8

chosen that

~ ~m

as

=

O

n = 4

15,

minimal

,

into

the

u

is

not

Grass-

. Since

northern

map

does

and

3.3'

have

not

hold

,

..

(3.32),which m

=

3

are

found

surface

i = n+1

by

and

3.3 p.

the

defined

defined

f(x)

[55], of

~a fi }

are

is

Theorems

Osserman

~B { / ~ X ~ B

m

f : ~4

is of

O)

the

curvature

in which

Lawson

(3.36)

and

of

, it

pole

known

sint,

with S2

solution

y

(cost,

u(~)

hemisphere constant.

for

m

= I

On

the

.

hand,

where

of

is

=

u : ~ ~ S2

, p = norlth

Moreover if

map

S 2 = G(I,2)

the

B /2(p)

It

and

function.

geodesic

mannian is

590-591)

IRn

Louiville

the

u(t)

is

on

unable

the

pp.

Almgren,

replacing

are

fact,

admitted

([66],

lSf(x) I

, or

In

seen

theorem

De

~

report.

sign

easily The

of

by

~'~R

this

ity

Berstein

boundedness

in

is necessarily an affine-linear

< ~

results

uniform <

I~fl

con-

system ,

.

is

. Their

sharp.

a Lipschitz

n+m

not

,

linear

example

is

~n

on

if

the

n

map

by x JxJ ~ ( - ~ [ )

-

for

x % O

,

2 where (z 1 , z 2)

denotes

the

Barbosa

as w e l l

[16], ~3 f :

Hopf

map as

Theorem 5.4, ~p ~ of t h e

satisfies

sup

n

( Jz I J 2 _

~ : S3 ~ S2

results has

from

sketched

minimal

J~fJ

=

<

. On

Iz2 j2 , 2 z I ~ 2

the

geometric a proof

surface

system

other

hand,

measure that

each

(3.36)

)

using

theory, entire has

an

idea

of

Fischer-Colbrie solution

to b e

linear

if

it

37

Now we shall harmonic

derive

maps

a criterion

that, by virtue n+m

submanifolds

of

T o this

we must

end,

pact set

K

(3.37)

caP2,ro(K)

For

of

caP2

relations a n d its

If

r(K)

< ~

HS(K)

= 0

If

S

as

"caP2(K)

caP2(K)

the

for

Finally, if

if Z

X

is a n

= 0

manifold

of

n

we

without

IRn

, we define subset

N

(X,W)

that

Finally,

X

whereas let

set

K }

means

following K c IKn

are well

= 0

known:

implies

Z

of and

X

the relation

of

~ M

M

-caP2(S)=O"

S

with

"caP2Z

, respectively.

boundary,

K

manifold

~ = int X , t h e n

assume

and

boundary.

of

compact

chart

= O

. The

caP2(K)

Conversely,

on

.

n -dimensional

for e a c h

following,

of dimensions

without

subset

r ~ ro

Hn-2(K)

by

' ~ >I

of a compact

measure

= 0

of a com-

is d e f i n e d

n = 2 , caP2(K)

for all)

caP2(K)

s > n -2

for each

. If

of

to minimal

2 -capacity

caP2,ro(K)

: ~ 6 C c ( B r o (O)'~)

(K)

Hausdorff

of singularities

can be applied

relative

(and h e n c e

caP2(K)

all

is a s u b s e t

N W)

For the

= O"

the

IV~I2 d x

2 -capacity

, then

3.3,

, then

= cap 2

for s o m e

is a n a r b i t r a r y

caP2(Z

introduce

= inf { ~n

= O

between

of Theorem

K c Bro(O)

(n -2) - d i m e n s i o n a l

Hn-2(K)

that

first . If

n ~ 3 , we set

that

and

~n

for t h e r e m o v a b i l i t y

or without

= 0"

boundary,

means

. are

Riemannian

is supposed

is a n a r b i t r a r y

be a relatively

manifolds

to be a complete manifold

closed

subset

with

or

of

= int X. Then we will

Theorem which

3.5.

is

caP2(Z)

derive

Let

contained

the

U :~ -~

~ M

in s o m e

regular

= 0 . Then

Proof.

U

(3.38) we on

can

can

First we will

(3.38)

holds

for e a c h o p e n

such

be

a harmonic

result:

map, the

BR(p)

ball

of

to a h a r m o n i c

range

M , and map

on

U(~ -~)

of

suppose

that

all

~

of

.

that

e(U)

Q' c c

d vol

~' c c W

<

~

. It c l e a r l y

, where

local coordinates that

removability

extended

subdomain

for s u b d o m a i n s introduce

be

show

f

BR(p)

following

the harmonic

x map

(x,W) on U

~' on

suffices

is a chart on and normal W

to show ~

. Thus

coordinates

is d e s c r i b e d b y

u(x)

u =

38 (U 1(x) ,uN(x)) where N = dim ~I fl (u) = a a5 Filk(u) ? a u i ~ B u k . Then r

. . .

r

(3.39)

- $

As usual, and The

q = I

We c h o o s e on

following

Lemma there open

3.7. is

[ a aB ~8 u I ] = fl(u)

W and its subsets

respectively.

result

If

~o

S

some

will

AS before,

on

W

,

w e set

1 ~i ~N

be i d e n t i f i e d

function

with

n EC~(W,IR)

a aB

= / X y aB

K

x(W)

and

that s a t i s f i e s

x(S)

"

of

K c ~o

~k(X)

a compact

subset

~ 1

, and

a.e.

W

satisfy

on

W

caP2(K)

with

~k 6C~(W,IR)

functions ~ W

in

that

0 S ~k

~

I

all

well

f I~-oqk[2 dx

, and

= 0 , then

vanish

as

in

some

as

~ O

W

as

k ~ o~ .

We now take Lemma

3.7.

K = Z N supp(D) If we m u l t i p l y

integration,

we arrive

and choose

(3.39)

by

~ =

a sequence (n~k)2U

{~k }

a c c o r d i n g to a partial

and p e r f o r m

at

f Q(u) (q~k)2 dx

=

f u I fl(u) dx

W

W

- 2

f

aS5 ~ a ( ~ k )

~5 ui q ~ k ui

W

since third

(~k)2

6 C ~ ( W -Z,IR);

integral).

and t a k i n g

(3.4),

Employing (3.6),

(no s u m m a t i o n Schwarz's

f

we arrive

c

to

k

to the third

in the integral

On

W -

at

l~ul 2 (q~k)2 d x - <

c

W

where

respect

and the i n e q u a l i t y

lu(x) I -< R into account,

with

inequality

f

{~(q~k) 12 dx

W

is i n d e p e n d e n t f W

of

]~ul 2 (qDk)2 dx

k

. On a c c o u n t ~ const

,

of L e m m a

3.7, we infer

k = 1,2 ....

,

O ~ qSI

is well known:

is

,

.

~'

a sequence

set

•

that

39

As

~k(X)

~ I

a.e.

on

W

, Fatou's

lemma implies

that

l~ul 2 ~ 2 d x < W whence

(3.38')

f

for e v e r y mate

I~ul 2 d x <

subdomain

(3.38')

~' c c

yields

W , since

~(x)

on

m I

~'

. However,

esti-

(3.38)

Thus w e h a v e

u 6 L ~ ( W , IRN)

,

u 6 H Il,o c2( Q , ~ N)

Now we show that (3.40)

~

{ui ~

W

Let

u

(3.40')

D Z

in L e m m a

replaced

by ~

3.7.

~k

of

solution

~ 6C~(W,IR N)

,

i.e.,

{~k }

~ 6 C ~ ( W , ~ N) c

corresponding

(3.40)

and

(3.41)

to

hold with

~

ui {l~) ~k

W

~

~ S W

(u I" i ) 2 d x

S

of

k ~ ~ , since

(3.40')

is v e r i f i e d ,

"

converges

and

(3.41)

i~kl2dx}I/2

f W

to t h e follows

left h a n d s i d e by a n a n a l o g o u s

argument. In o t h e r words, into

U

c a n be c o n s i d e r e d

X , the r a n g e of w h i c h

By v i r t u e of T h e o r e m of)

U

1.4 w e

K

then y i e l d s

+

t e n d s to zero as

(3.40)

(3.39),

for all

a sequence

first e q u a t i o n

integral

of

S fl(u) ~ i d x W

~i + $ u l < ± } ~ k d X

integral

Thus

~u 6L~oc(W,jRnN)

for all

~

t h e first

(3.40).

= O

T h e n the e q u a t i o n s

" The

{ui ~

=

and c h o o s e

f U i l ~• e ~ k--d X W whereas

,

, that is,

is a w e a k

W

The s e c o n d

W

c

~ a~8 ~ u i ~ i W a ~

as s t a t e d

on

~

that

K = supp(~)

a.e.

~i + ~ u i ~ i } d x ~

and,furthermore, (3.41)

lu(x) i < R

as w e a k l y

is c o n t a i n e d infer that

is a r e g u l a r h a r m o n i c

map

from

harmonic

(a u n i q u e l y ~

map

in a r e g u l a r b a l l

into

X

defined ,

from BR(P) extension

q.e.d.

40

By

applying

fairly that

obvious

we

and

way

state

Theorem

strong

the

Let

3.5'

of its center

has

The

that

both

cases

Other

singular

vided

by

obtain

of

map

at

w~

x = O

harmonic

:B -{0}

3.5

in T h e o r e m

3.5'

maps

satisfies

=

a 3.5

caP2(Z)

= O

4~ R

~ Sn

is

net

in some

may

occur.

with

~ ~/2

(cos g(x) , s i n g ( x ) , O

, n ~3

, B =

sense

O)

,

K := , or

9BR(p)

{x 61R n : Ixl < I]

Moreover,

optimal,

non - removable

.....

,

in

locus

~ .

removable.

is

the cut

and

singularities

j log w(x)

in

Theorem

not meet

is c o n t a i n e d

map on all o f

that

Theorem

map with

does

U(Q -Z)

to a harmonic

Thus,

, we

version

BR(p)

R

Then either

singular

map.

lul 2

be a harmonic

the ball

and its radius

a singularity

harmonic

to

strengthened

~ M

where

K ~.

can be e x t e n d e d

Remark .

following

U :~ -Z

p

principle

proof.

c BR(P)

max { O,SUPBR(P)

U

the

without

U(~ -Z)

maximum

g(x)

w,

is

we

see

are

pro-

Ixl

n =

=

if Ix[ 2 - n

In

the

from

same

the

way

as w e

Liouville

non -parametric

Theorem

smooth

minimal

Let

3.6.

relatively

~

closed

functions

field.

system Suppose

In

extension

of

fi (x)

(3.36)

,

) or,

that

there

an

Theorems

connected with

3.5

domain

caP2(Z)

f 6C 3

i = n+1,

on

...

= O

and

in

3.4

and

theorem

3.4' for

3.5'

IRn , and

. Suppose

Z

be a

that the

Ci.e., more

~

,n+m

...

general,

planes

plane

= O ....

~o2 ~ -Z

such

for all is

,

,fn+m

satisfy

the minimal

has a parallel

the tangent

~o

x 6~ -Z

}

fn+1,

zn+1

,

submanifold

: x 6~-Z

is a number y(x)

fact,

from

n-dimensional

the horizontal

that

~

[ (x,f(x))

also

(3.34") ~)

theorems

derive

that is minimal

much from suppose

Bernstein

can

we

be an open

a nonparametric

iRn+m

surface

the

3.1,

surfaces

subset

X =

of

derived

n a

~)

zi =

describe

have

theorem

a

of

that

x 6 ~ -Z

sufficient.

X

mean

do not

,z n + m (3.34)

= O

and

curvature

differ

. More

too

precisely,

41

are s a t i s f i e d where fn+1 (x) . . . . .

clusion holds if

(3.42)

. Then

(3.32)

are real analytic in

the functions

~ . MoreoVer,

the same

con-

satisfies only

X(x)

/X(X)

is defined by

T(x)

f n + m (x)

~ Cos-S

( ----~---~ > 2 /~-~

and if the strict i n e q u a l i t y in

for all

x 6 ~ -Z

holds for at least one point

(3.42)

x 6 ~ -Z

Proof.

We once

again

infer

X ~ G(n,m)

is h a r m o n i c .

imply

G(X)

that

from Theorem

Furthermore,

c BR(P)

3.3 t h a t t h e G a u ~

assumptions

for s o m e b a l l

BR(P)

(3.34)

map

and

c G(n,m)

G :

(3.34")

with

/~ R ~/2

.

Let u(x)

b e the (x 1,

,x n) 6

at

p.

situation

3.5

defined

on

lined

~ -Z

Then

because

Theorem

of

(ul(x) . . . . .

representation

...

tered

=

X Z

on

~

U

. Yet,

6, w e s e e

only

3.5 a n d

3.6).

Thus we

to a H ~ i d e r

repeat

the

to s e e

x ~ v(x)

, x 6~

parametric

f , and the stronger) Theorem

has

zero,we

follows

U

z = f(x)

of

a continuous infer by some

. Now

[65], of our

of

well

pp.

Then

u

if

it is a n e a s y

with

the mapping

to t h e n o n -

which

corresponds

. The

continuity to all o f reasoning

f

finally

implies

regularity yield

is p r o v e d .

way

of the GauB

standard

known

theorem

from

(see L e m m a t a

extension

266 - 277)

in a s i m i l a r

is o u t -

u : ~ ~ G(n,m)

of

X

~

in

continuity

follows

~ -Z

~n

cen-

is n o t

1.4 t h a t

on

3.5.

on

over

continuity

~ >0

first assertion

assertion

agrees

BR(P)

, whereas

u(x)

map

of T h e o r e m

G(n,m)

that

for s o m e

harmonic

is t h e t r a n s f o r m a t i o n

implies

a n d M o r r e y (see

bounded

representation

x ~ ~f(x)

is of m e a s u r e

f E C I ' a ( ~ , ] R m)

the

weakly

therefore

to N i r e n b e r g

of Theorem

is e s s e n t i a l l y

the H~ider

Z

of the H~ider

Since

Then

on

, and only

however,

representation

f 6 C I (~,~m).

~"

on

proof

patch of

~f

over

u

x =

3.5 to t h e p r e s e n t

This,

the nonparametric

Z

Theorem

the proof

of t h e p r o o f

v

coordinates

for t h e

can extend

, where

coordinate

coordinates

is n e e d e d .

continuous

the map

,

are n o t d e f i n e d

review

~f(x)

arguments that

X

that

to t h e

apply

manifold

a ~B 6 L ~

N = n-m

to n o r m a l

directly

and

if w e

,

respect

and

both

map

matter

X

cannot

because

we

on

with

we

a ~B = / ~ ¥ ~

G

G

is a " s m o o t h

in section

u

of

uN(x))

the

~

u

.

that

that

results

due

analyticity

The second

by reviewing

to of

of

(and

the p r o o f o f

3.5'.

The Lipschitz and Osserman

continuous

but non-smooth

that was mentioned

before

example shows

that

f : |R4 ~ ~ 3 there exist

by Lawson minimal

42

submanifolds surface

of c o d i m e n s i o n

system

assump t i o n s Colbrie weak

[16],

Theorem

Theorem of

strengthened

theorem

U :X + M

Then

U

had b e f o r e

a priori Related maps

3.1'

of T h e o r e m

to T h e o r e m

that was

3.1

3.1

[35].

is due

that we gave maps,

by Choi

of the in our

has been

is the f o l l o w i n g

obtained

The

to M e i e r

consequence

for h a r m o n i c

Liouville

[9]:

the range of which is c o n t a i n e d in a

of a complete m a n i f o l d of

and

case where

been p r o v e d

theorems

M

by Cheng

can be found

[63] by Meier,

Systems

Reidel

continuous

- Jost - W i d m a n

as T h e o r e m

estimates

in

Publ.

M

Suppose,

moreover,

nonpositive Surveys [37];

curvature

this r e s u l t

of the literature

furthermore

as in the f o l l o w i n g

on Liou-

in the papers

paper by Giusti:

theory for nonlinear elliptic systems,

partial

differential

equations,

ed.

by J. Ball,

1983.

An interesting reGent l y

[8].

as w e l l

of n o n l i n e a r

has

[36],

Some aspects of the regularity in:

that every L i p s c h i t z

-

is a constant map.

the p a r t i c u l a r

[61]

Yet the

since F i s c h e r

is a complete R i e m a n n i a n m a n i f o l d with nonnegative Ricci cur-

vature.

ville

second proof

BR(p)

singularities.

to be an i m m e d i a t e

be a harmonic map,

regular ball X

stated

The

for h a r m o n i c

the minimal

analytic.

turns out

[63].

of

to the LITERATURE.

version

that avoids

found by M e i e r

For

is real

proof

principle.

and

that

isolated

not n e c e s s a r y

3.1 has b e e n p r o v e d by H i l d e b r a n d t

maximum

Let

nonremovable

5.3, has p r o v e d

(3.36)

our p r e s e n t

notes

(and even solutions

3.6 are c e r t a i n l y

and R E F E R E N C E S

slightly [63];

with

of T h e o r e m

solution

COMMENTS

(3.36))

n ~ 3

generalization

been derived

of the results

by J. K a m p m a n n

in his

of Cheng thesis

and Choi has

L i o u v i l l e - S £ t z e fur

harmonische A b b i l d u n g e n und F u n k t i o n e n auf R i e m a n n s c h e n M a n n i g f a l t i g keiten partiell

negativer Ricci-Kr~mmung

(Bochum

Kampmann

that,

the a s s u m p t i o n

proved

can be w e a k e n e d bounded

to the r e s u l t

to F i s c h e r - C o l b r i e

[35]. from

in Choi's

"Ric X A 0

subset of a c o m p l e t e

References due

to

Lemmata [35].

minimal

3.5 and

on

submanifolds

noncompact

for some predecessor).

where

"Ric X ~ O

is s u p p o s e d

in [35].

[54].

Theorem

3.3 that links h a r m o n i c

discovered

The B e r n s t e i n

Lemma

and d i r e c t proof was

by Kasue

by Rub and ~ilms theorem

on

to be a

M .

can be found

derived

Theorem

S

manifold

an e x p l i c i t e

3.6 were

was

X -S"

G(n,m)

[16];

The f u n d a m e n t a l

theorem, on

1983).

[72]

3.4 is

given

3.2 stems

maps

and

(see also

3.4' was p r o v e d

in

[71]

by Hilde-

X"

43

brandt

- Kaul

s i o n of

- Widman

a result

of

[35]

assumptions

(3.29)

superfluous

to a s s u m e

The T h e o r e m s due

3.5,

to S e r r i n ;

We moreover

partial tions

and

refer

due

3.4

imply

3.6 w e r e

Acta Math.

111,

equations

results

order.

found by Meier

linear

8.18

8.19

the S t r o n g m a x i m u m

We moreover the p a p e r these

note

[40]

authors

curvature

in

established

to the G a u s s

surface

to this orem

tends

the p a p e r

Theorem

t h a t any

Nitsche

[41]

note of

[35]

equa-

of L e m m a and

3.2

their for

was

derived

in

it w a s

applied

by

of constant

the f o l l o w i n g

in

mean

result was

IR3 , the Gauss

of

S 2 , must

by

solution

De G i o r g i

De Oiorgi and Stampacchia,

isometric

(see p.

Ishihara

isometric

that,

296, where

immersion

3.1 w a s for

the m i n i m a l

map of

either be a

immersions

Theorem

into

some

~

if

of c e l e b r a t e d

- Stampacchia;

sphere. [82].

Hn-l~)

in : O

investigations

Simon

R - E

ex-

. This

is

b y Bers,

On a theorem

Simon,

Zeitschrift

the-

m = 1 , Leon

equation

see L.

into

Related

the R u h - V i l m s

codimension

Mathematische

9).

found by Tachikawa

surface

i n all o f

in a s e q u e n c e

and

to m i n i m a l

of T h e o r e m

3.6 w e

solution

result

in

to m i n i m a l

to a n a n a l y t i c

the f i n a l Finn,

is

3.4'

obtained

generalization

Concerning

one

cylinder.

of T h e o r e m

is g e n e r a l i z e d

proved

where

[62]).

3.1'

and

is

Elliptic

1977,

t h a t is n e e d e d

surfaces

for i n s t a n c e

in some closed hemisphere

has b e e n

result

A slight

of c o m p l e t e

3.7

differential

out in

Schoen,

of constant mean curvature

or a right circular

sphere

10.

estimate

principle

and

it

[40]:

A generalization some

map

Lemma

Trudinger,

case of Theorem

Osserman,

II{3 . In this w a y ,

[62]. Lemma

of G i l b a r g - T r u d i n g e r ,

are c a r r i e d

that a particular

which is contained plane

8.26

(details

by H o f f m a n ,

in

A complete

3.5'

and

that

this m a d e

Springer

the H a r n a c k

from Theorems

of T h e o r e m

and

and nonlinear

follows

the p r o o f

(1964),

by G i l b a r g

instance,

ver-

.

247-302

Theorem

provides

X

observed

is s i m p l e ;

of second order,

on

For

X

of

strengthened

essentially

that

the s i m p l i c i t y and

is a s o m e w h a t

to K a s u e w h o

to the m o n o g r a p h

the b a s i c

of s e c o n d

Theorem

(3.33)

3.5',

see:

differential

can find

[35].

155,

199-204

(1977) . Finally other

we mention

results

mappings manifold.

which

that Eells

concerning

the

do n o t r e q u i r e

and

Polking

removability any

have

recently

found

of s i n g u l a r i t i e s

curvature

conditions

on

some

of h a r m o n i c the

target

of

44

4. E s t i m a t e s

for Jacobi

Throughout

this

Riemannian

manifold

ted by

<','>

point

p 6 M

A vector

fields

section of

. Then

we assume class

field

J

along

D

(4.1)

for all

a geodesic

Here

R(X,Y)Z

equation nothing

denotes

(4.1), but

integral equivalent second

the

at

to t h e

system

we

shall

of

the

for

c

with

c(O)

~ 0

is

and

both

form

tensor

of

M

. The

m

variation

coordinates,

linear

ordinary

of

linear c , is

the Dirichlet

the Jacobi

differential

equation

is

equations

order

the u n k n o w n

functions

a geodesic

we denote

+ M

the

second

l~r.s •.k + R klrs(c ~ c c

along

.

abbreviate

takes

curvature

c . In l o c a l of

is d e n o -

Jacobi equation o f t h e g e o d e s i c

equation

fdt

of w h i c h

t = 0 at some

= 0

the R i e m a n n

so-called

the E u l e r

t ~ 0

for

= O

T h e n (4.1)

+R(J,c)6

product

m-dimensional

if it s a t i s f i e s

is p o s s i b l e ,

(4.1'

scalar

c : [0, ~)

c

D

dot.

is a c o m p l e t e

, starting

times

d--{ d--{ J + R ( J , c ) $

If no m i s u n d e r s t a n d i n g D d-~ w i t h a s u p e r s c r i p t

of

c(t)

to b e a Jaoobi field a l o n g

said

M

C 4 , the

a geodesic

, is d e f i n e d

that

by

Jc

is a J a c o b i

c

~k(t)

, k = 1,...,m

span a 2m-dimensional

" In p a r t i c u l a r ,

field

= 0

of constant

_~D dt

the

tangent

length

= o

. Thus

linear

vector

lic(O) ll

R(c,c)c

the J a c o b i

space over c

along

~

fields which

of a geodesic c , since

= o

and

2 = 2
if

J

dr<% ~,J*>-<S,5 d--£1 We

therefore

and

J* 6 3

c

*> } : <J,s*>

t c>

0

, then

- <S,J*>

=-
obtain

<J,J*> - <J,J* > = const

for all

J,J*6

3

c

*>+

= 0

.

45

and

in p a r t i c u l a r ,

for

(4.2)

<J,c>

Suppose tion

now

that

llcll = I . Then,

can d e c o m p o s e

jT

, we arrive

= const

c : [0, ~)

jT

we

J* = c

for

+ M

at

all

J 6 J

is a g e o d e s i c

C

normalized

by the c o n d i -

by s e t t i n g

~$

each

a = <J,c>

Jacob±

field

and a normal component

,

J 6 J

c

j±

j

into

jT

a tangential

component

J± : j = jT + j ±

We claim

that both

(4.2)

implies

(~c)"

= ~c

The

jT

i = 0

= O

tangential

if w e part

and

, and take

jT

are Jacob±

c = O

=

fields.

In fact,

(jT).. + R ( j T , 6 ) ~ into

is of the

jT(t)

(4.3)

Ji

therefore

=

equation

(jT)-- =

(ac)'" =

account.

form

{at + b } c ( t )

where

(4.3')

a = <J(O),c(O)>

Thus

the g r o w t h

from

the

Hence

initial

we

can

the normal which, denoted

of

Jacob±

J(O)

span

jT(t)

can

easily

be d e t e r m i n e d

J(O)

of all J a c o b ±

These a

part

and

the g r o w t h

field~.

(4.3),

b = <J(O),c(O)>

tangential

values

control

by

by

the

,

are

fields

the e l e m e n t s

(2m-2)-dimensional

of

if w e can Jc

subspace

estimate

orthogonal of

J±

J

to

that

is

c

C

Unfortunately, yet on

they

can

there fairly

the s e c t i o n a l

tions

of

which

also

wherever

the

is no s i m p l e well

curvature

scalar

way

to c o m p u t e

be e s t i m a t e d of

M

differential

in t e r m s

. To this

end,

equation

f+~f

= 0

,

-(7

= < + ( )2

~ 6JR

satisfy

f

does

not vanish.

,

the n o r m a l

of u p p e r we

and

consider

Jacob± lower the

fields, bounds

solu-

46

In

particular,

value

we

introduce

the

solutions

s<(O)

We

and

<

c

K

of

the

initial

problems

i< +KS < = O

We

s

= O

and

, s<(O)

I

:

!c

+
/

(o)

: O

=

c<(0)

1

= o

have sK(t)

=

t

,

cK(t)

S < (t)

= -~=sin ¢ <

s<(t)

= ~sinh~'-~t

= I

,

c < (t) ,

= cosT~t

c<(t)

=

cosh~-/L~
if

< = 0

,

if

m

> O

,

if

K

< O

.

set t

that

is,

Lemma

tK

4.1.

that

some

is

~7~/¢-6 ~ (+~

=

the

Let

first

c : [0,~)

J E ]±

<

:>

o

~

_<

0

if

positive

zero

of

s<(t)

be a g e o d e s i c

÷ M

satisfies

[[Jil > 0

on

with

]l&ll = I , a n d . Finally

( O , t ~)

suppose

we a s s u m e

C

that, on

for

I't*

=

satisfying

some

number

{

c(t)

the

:0

K ,

~

t ~

t*

differential

the

}

sectional by

tke

curvature

inequality

K

of

K ~ K

. Then

inequality

d2

(4.4>

LIJI[+~IIJII > o

on

(O,t ~)

dt 2 Proof.

We

first

obtain d d~

(4.5)

whence

, I!~[CI<j,~> iJll = ii~ll

,

d2 'l,,j,,'~=~,"jiI-1+l;J,'1', ,~I'2-"JII-S<J,, ~>2 dt 2

~

,,

,-3],, li,JL/21[Ji

= ~,J;~q<J,b>+ilJIL and,

by

Schwarz's

inequality, d2

(~.~) The

Jacobi

dt 2 equation

we

arrive

,,

, ~1 ilJll>,L~i <J,~>

(4.1'),

<J,J>

on

the

other

= -

hand,

•

,12 - <j,5>21 f

at

implies

M

is b o u n d e d ],J]]

is

47

The

term

denotes

on

the

the

right

sectional

two-plane

~(t)

by

is n o t h i n g

but

of

c(t)

M

J (t)

at

and

c (t)

-KIIJ[~ with

. Thus

, where respect

we

K(t) to t h e

find

<J,J>'s '~I~ bl =-KI]J,I--> ~lisll

Finally,

we

side

curvature

spanned

(4.7)

Lemma

hand

(4.4)

4.2.

assume

follows

Let

the

that

from

assumptions

a(o)

= 0

and

of

and

d d--[

(4.8) Proof.

(4.6)

(4.7).

Lemma

t ~ <_ t

4.1 ,

K

~J~ll > O sK ] -

be

satisfied.

If,

moreover,

then

on

(O,t*)

Set

z = llJII's<-ilJlls< Then,

for

O < t < t ~ , we

= if w e

take

Jii"s<-i

(4.4

into

IJ 'i

(4.5)

Z(t)

A O

since

s< l'Jl' ÷0

~ < --

account.

Z(t)

Moreover,

obtain

Hence,

_> Z ( t o)

for

for

any

all

to E

t 6

(O,t ~)

, we

infer

that

(to,t~)

yields Jl

HJII" < ~15 and

therefore

jzL -
to ÷ +O

and

Z ~ 0

, we on

have

s~(t o ) ÷ O

(O,t ~)

. Then

and

the

Theorem be

a normal

over on

4.1.

Ft<

(4.9)

o : [© o~) ÷ M

Let Jacobi

suppose

that

= Lc(t)

~

field the

}

inequality

, whence (4.8)

Z ( t o) follows

÷ 0 from

z

s2 K be

a geodesic

c

which

sectional

: 0 -< t <

ilJ(O)lls<(t)

along

(O,t ~ )

llJ(to) II ÷ O

desired

]~EjE! dt

on

curvature

with

J(©)

satisfies K

of

~II-I ii

M

has

. Then

-< ilJ(t) II

for

all

t 6

[O,t<)

= O an

and let

S

. We m o r e -

upper

bound

K

48 Proof.

If

that

J(O)

= O , (4.9) o b v i o u s l y

IIi(O) II > O , w h e r e a s

such that

IIJli > 0

on

J(O)

(O,t*)

IIsJ[i(t0 ) _< ~sI

~2

As

to

tends

to

= 0 . Then there

We t h e r e f o r e

may a s s u m e

is a n u m b e r

t ~ £ (O, tK)

, and Lamina 4.2 implies

(t)

<

is correct.

for

0 < to

_< t

< t*

+O , the q u o t i e n t on the left hand side is an expresO ~ which, a c c o r d i n g to L ' H o s p i t a l ' s rule, is d e t e r m i n e d

sion of the kind by lim

J~2 2 =

to÷+O

lira ~tilJ]12

--d--T

s<

to÷+O

=

lim÷+o

~s<

~t~[IJ[12 d2

2

-

113(o)II

2

,

to

dt2S< since

d

2

,,,d 2

~s<(t

O) + O

,

2.

•

dt2S<(to)

d~tlS][2(to ) = 2 < S , J > ( t O)

÷ 2

÷ 0

,

ddt2211JI12(to ) = 2{ilJIt12+ < J , J > ~ ( t O) 2~[13112- < R ( J , ~ ) 6 , J > } ( t O) ÷ 21[J(O)112 and a s s e r t i o n J(t)

(4.9)

cannot vanish

is p r o v e d before

t

for

O ~ t ~ t*

, and thus

K

. We then c o n c l u d e

(4.9) m u s t hold

that

for all

t 6

[O,t) By the same reasoning, Theorem J

o

6 ]I

on

FT

4.1'.

{

=

c : [O,~)

Let

. Suppose

we can p r o v e

also

that

}

c(t) : O ~ t ~ ~<

re(t) ana

÷ M the

IIJIl"(0)

where

= 11s(o) I#

= llJli-l<J,3>(O)

(4.10)

~(t)

be

a geodesic

sectional

We

_< llJ(t)II

T

with

curvature is

K

the

llcll = K

first

positive

(t) +Ijs11"(o)s<(t) then

I , and

satisfies

let K

zero

< < of

,

o b t a i n *)

for

O < t < y<

and

(4.11)

iIJ (t)If_
where

0 < t*

*)for J(O)

<

for

all

t 6 [O,t*]

<

# 0 ; the case J(O)

= O is h a n d l e d

by a limit

consideration.

49

We n o w s h a l l

turn to a n o t h e r

c l a s s of J a c o b i

a l o w e r b o u n d on the s e c t i o n a l To this end,

let

XI,X2,...,X m point

c(t)

space

Tc(t)M

c : [O,~)

be

m

curvature

+ M

parallel

vector

U(t)

If we i d e n t i f y

~m

u •

by

[0,~ ) ~ ~ m

U

with

which,

at e v e r y

f r a m e of the t a n g e n t

<Xk'Xl > = 6kl

along

c

Tc(o)M

can be w r i t t e n

and introduce

= (U I (t) . . ... u m ( t ) ) T

we obtain a 1-l-correspondence and the v e c t o r

c

as

: uk(t)Xk(t)

u(t)

19m

along

and let

we have

and

field

fields

from

M .

y i e l d an o r t h o n o r m a l

. In o t h e r words,

Then every vector

derived

a g a i n be a u n i t s p e e d g e o d e s i c ,

of the g e o d e s i c ,

Xk = 0

of

field estimates

fields

between

U

To any re×m-matrix f u n c t i o n

along B(t)

the v e c t o r

= column

the v e c t o r

c

functions

g i v e n by p a r a l l e l

= (bl(t)~

function

which

u : [0,~)

÷

translation.

acts o n v e c t o r

func-

/

tions

u(t)

an o p e r a t o r ,

according

to

again called

(B(t)u(t)) 1 = --bl(t)uk(t) B \ /a c,t i n g on v e c t o r

rule (BU) (t) = (B(t) u(t) if

U +-+ U .

We,

in p a r t i c u l a r ,

vector

function

can a s s o c i a t e

I =

(4.12)

where

I +R

the m a t r i x

function

where = ckx k The w e l l k n o w n

=

which

satisfies

=

is d e f i n e d

and

s J k-l'r. = Rklr c c Ks

relation

=

Rc .

field

0

R(J,c)c

by the

Xl(t)

sr

of

U = ukXk

?

S : Rs ~i Rk klr

symmetry

the s y m m e t r y

I

Rc(t)

implies

c

fields

with every Jacobi

(j1,...,jm)

, w e can a s s o c i a t e

by

J = jkx k

a

50

Next

we

choose

a basis

J1,...,Jm

J 6 ] J(O) = O I of ] c Ic (4.3), t h e t a n g e n t vectors for

all

t 6

functions A(t)

(O,t k)

, defined

assume

to

the

and

(4.13)

where

I

matrix

function

denotes

satisfies

the

the

A A -I

:

Ricatti

also

end,

we

=

claim must

-R

,

vector

matrix

I + ...

that

A A -I

:

=

I

= _~A-1

and

S = - A A -I

(O,t<)

+ (AA-I) 2

- t -I • I + 0 ( I )

and

S(t)

prove

define

the

,

yields

, and

is

i(t)

:

as

from

(A-I) " = (4.13)

we

t +

+0

on

Tc(t)M

I + ....

a symmetric

operator

. To

this

that > =

if w e

vector

introduce

the

= vkXk(t)

with

saying

that

two

parallel

u = A -I ( t o ) U ° the

t

(O,t<)

o

, which In

pair

and

of

fields v

tanqent U(t)

vectors

U

= o

= ukXk(t)

= A -I (to)V °

, this

is

function

-

:

t =

can

+S 2

c

V(t)

for

the

the

I

t 6

and

on

:

for

But,

vanishes

A(O)

therefore

each

vanishes

be

" Then

. We

holds for every t £ (O,t K) and for k k o u X k ( t O) , V = V o X k ( t o) 6 TC(to)M.

to

Ik

by

. Moreover,

c

t"

R

<S(to)Uo,Vo

equivalent

now Jk

and

independent

,

= O 1 (6k)

_~(A-I).

S(t)

A(t)

4.1

linearly

:=

equation

of

~ = _~-I

(4.15)

since

. Let

vektors

= - i ( t ) i -I (t)

differentiation and

A(O)

matrix

S =

infer

We

,

unit

(4.14)

the

are

(I1,I 2 ..... I m)

= O

S(t)

since

Theorem

By

Xk(O)

Jc

satisfies

A+RcA

- A - I A A -I

K ~ <

Jacobi

subspace

by

invertible

which

m-dimensional

J 1 ( t ) ..... J m ( t )

A :

is

the

~k(0) =

with

if w e

corresponding

of

is

fact,

proved we

infer

by

showing

from

the

that

¢

definition

identically of

~

that

51

l i m }(t) t++O and,

on

the other

hand,

=

%

is c o n s t a n t

-

now

J

be

the

associated

an arbitrary vector

(4.16)

We

on

fix

over, by

because

+

normal

function.

~ = -SI

holds

,

of

= -

Let

= O

Jacobi

Then

or

= O

we

field infer

in

]c

from

' and

let

A = -SA

I

be

that

J = -SJ

(O,t<)

some

to

6

we define

setting

(O,tK)

and

set

U ° = U~Xk(to)

a parallel

vector

field

= U~Xk(t)

. Then

we

U(t)

k(t)

U

= ]IJ(to) ] F I j ( t o ) . M o r e -

along

claim

that

c

the

with

U ( t o)

= Uo

function

= <SU,U>(t)

satisfies

(4.16'

-k

provided t

that

-<-

on

s

~ _< K _< K

is a s s u m e d .

We

)

also note that

e _< ~

implies

-< t

K From

(J (4.16')

we

infer

that

~-~ (t)

<SJ, J>

lls[[2 (t)

holds

for

for all

t : to t £

(O,t)

which

is

the

to h o l d

other

Theorem

4.1

tO

, and,

together

on

hand, into

we

Theorem unit

arbitrary, with

iiJl12

this

(4.16),

inequality

we

arrive

is t r u e

at

- sq

(O,t K) by

=

was

Ii;j I12 =

repeating

account,

z Hence

_<

Since

'J',i

On

(O,t

we

flail" s<

-

the

proof

of Lemma

O

on

4.2

and

by

taking

obtain

lla "s<

>

(O,t<)

have

4.2.

speed

Let

geodesic

J

be

a normal

c : [0, ~)

÷ M

Jacobi . and

field

with

suppose

that

J(O)

= 0

~ ~ K ~

along ~

a

holds

52

on

the

~c(t) : t 6 ( O , t )

set

~ . Then

we

m~y

conclude

thqt

< J

(4.17)

~

It r e m a i n s hold

on

-<

to p r o v e [O, ~)

iijii2

< _ ~

(4.16').

, since

these

We

on first

relations

(o,t<) note

that

are

true

ii = I IIU,I for

t = t

and o

, and

= 0

U ,

are p a r a l l e l . Thus we

get ~

,

and <su,u> 2 ~ [Lsull2 = < s 2 ~ , ~ >

Furthermore,

(4.14)

yields

~t < S U , U >

= + <S2U,U> = + <S2U,U>

and

therefore

(4.18)

k ~ ~ +k 2

Consider

the

on

(O,t<)

function h = s k+s

which

then

satisfies

(4.19)

as we

h A hk

see

from i

if we

take

checks

the

,

(4.18)

=

s k +s

and

~

_ s " k + sek >

k + ~

+es

= 0

into

2 +

(s+es)

account.

By d i f f e r e n t i a t i n g ,

one

identity h ( t ) e x p i t- ] k ( s ) d s ) = h(E) + ]t (h-hk) (s)exp i s- ] k ( T ) d T ) ds "

O < ~ < t < t<

, and

S

~

thus

h(t)

by

-

(4.19):

t _> h ( ~ ) e x p ( f k ( s ) d s ) \E

AS and

E

tends k(e)

to

÷ -~

+O

, (4.15)

. Hence

we

h(t)

yields

k(s)

=-1+O(I)

, whence

infer

> O

for

t 6

(O,t<)

,

h(E)

÷ O

53

which

is e q u i v a l e n t

From

Theorem

4.2 w e

d-~ I s i.e.,

the

reasoning

to

infer

s2

function

llJII/s e

as

proof

in the

t 6

Thus

we

have

proved

Theorem

4.3.

Let

Then

thus Theorem

4.2

is p r o v e d .

.

.

.

.

.

is d e c r e a s i n g of Theorem

.

on

4.1

.

(O,t e)

yields

ff}
.

S

and

then

ii~(O) lisa(t)

the

same

a IIJ(t) Jl

(O,t<)

J

be a n o r m a l

speed geodesic

vature

and

that

j =

for

unit

(4.16'),

K

of

M

c : [0,~)

satisfies

i]Jl[

the f u n c t i o n

Jacobi

field

+ M , and

suppose

~ ~ K ~ ~

is d e c r e a s i n g

with

on

that

the s e t

in

(O,t)

s

(4.20)

K

llJ(t) II~ ii J(O)[Is

(t)

J(O)

for

all

= O

along

the s e c t i o n a l ~c(t) [

: t 6

a n d we

have

a cur-

(O,t)~ J

o

"

t 6

(O,t)

Supplements

I. W e was

first

only

If w e tes

note

used

to

instead

(4.9),

2. F r o m

that

the

insure

assume

(4.17),

that

and

e ~ K ~ <

completeness

the

and

of

existence c(t)

(4.20)

of

is d e f i n e d

will

hold

was

for

for

not

for

really

all

needed.

t 6

0 ~ t ~ R

It

(O,t<)

, the

estima-

0 < t < min{t<,R}

= K(t)IIJ±[l 2

~ILJ'II2 _< and

M c(t)

we

conclude

that

_< <jlJ±ll2

therefore

(4.21)

if w e

also

needed tion true 3.

<J,c> for

Let

pose

]lclI

assume

to derive = O

all

us

that , C(T)

that the was

Jacobi

once

by

/~ ctg/<-•

assume

" but

= c(T/r)

= I , whence,

nowhere fields

again

J 6 Jc

e _< 0 < <

statements

not

, J(T)

else J

. The

of

the used.

along

that

inequality Theorems Thus

c

necessarily -- J ( T / r )

these

with

~ _< K < <

and

<J,J> 5_(T ) < / c ~

ctgh

= 0

note

that

/~

T

,

all we

the

assumpremain

.

e < O _< ~

Itc i = I . T h e n

, and

was

and

statements

J(O)

(4.17),

_<

(4.21)

4.1-3,

we

J 6 Jc

, and define and

supr

tlC_]I

54

and

therefore r,~< ctg~,~r%_<

If w e

introduce

the

we

arrive

a

and

t

for

0 <- t < ~ / w <

,

for

O -< t < ~

,

(r)llJ(1) [I2 < < J ( 1 ) , J ( 1 ) >

~•

provided

that same

_< a

}

a<(r)-

(4.23)

the

ctgv<< t

= t/-0~ ctgh/-~o

at

(4.22)

By

< rC-~ ~ c t g h / - ~ } r t

functions

a K (t) = ty~ ae(t)

<J'J>(t)

I iiJ(1)~i 2 <- < J - J , J > ( 1 )

/k r < z

scaling

(r)l[J(1) ]I2

{a

_<

}

( r ) - I llJ(1) Ii2

.

argument,

we

derive

from

(4.9)

and

(4.20)

the

in-

equalities ,2 -2 2 i[J(O) il r sK(rt)

By

_< ~jiJ(t) ii2_
if

O < rt

< ~//K

.

setting b<(t)

we

arrive

us

that

Theorem

4.4. ÷ M

ture

of

K and

Whereas two

(section

these

Let

M

be

< n , the

previous

will

4.5.

Let

. Moreover,

Then

we

b<(r)

following

field

estimates

be used

with

suppose on

the

(4.22-24)

are

needed

for

the

J(O)

= O

that

the

arc

c

along

a geodesic

sectional

. Then,

if

6 and

7,

curva~ ~ 0

hold.

in sections

proof

of

the

uniqueness

the

follo-

theorem

8).

6 ~

from

22

-< [',J(O)[l

~ ~ K ~ ~

estimates

that ded

a Jacobi

satisfies

theorems

in t h e

r = lic(O)II , a n d

with

r/< the

J

results

Theorem J

sinh/io~t = ~--~t

b(~(t)

/~ r <

collect

c : [0,1]

wing

and

2 i!5(o)ii2 bK(r) -< IiJ(1)ll2

provided

K

~-~---

at

(4.24)

Let

sin/Kt -

c above

have

by

c : [O,P]

some

+ M

let number

be

the

a geodesic

sectional

< ~ 0

, and

[Icll = I , a n d

with

curvature assume

that

K

of 0

<

M

be

suppose boun-

p < ~/v~

.

55

-

(4.25)

Proof.

Let

s

ljJ(O) II a n d defined on

every in

a

s(p)

by

As

be

solution

of

= l]J(O)[I • T h e n

(~,,8)

of

Lemma

we

note

consider

in

infer

=

Z : [O,o]

+

where

{ _> O

J

on

does (G,B)

not

vanish.

whence

-< z (B-O)

+l;j(p)

iSl<(t)},

that -> O

[O,p]

on

-IiJ (o)11~ (p)41J{~)11 =

s

,

-il J (o)il41o (p)iI~< (p)

s(p)

(p)

=

s<

K

Now

we

are

going

trivially

then

J

that

are

has

Taylor's

to

prove

correct only

of

the J(t)

finitely

assumed

to

expansion,

of

the

theorem.

~ O

on

the

other

many

be

ordered

we

conclude

(p)

assertion . If,

zeros by

O

in

[O,p)

...

This

hand,

, say,

< tI < t2 <

assertion J(t)

: ±i[3(ti)

ll

,

i = 1,2 .....

Thus

we

arrive

Z(+O)

at

-< Z ( t 1 - O )

the

- Z(ti-O)

sequence

-< Z ( t 1 + O )

s

= 2s(ti)iiJ(ti)li

of

k

<

...

<

Z(tk+O)

Therefore

0 <

Z

(p-O) -

z

(+0)

<

(p/i J(p)'~ cH F . J. .(0). 1~ = llJ(p)ii ;iJli'(p-o) -

~

A 0

inequalities

z(t2-O)

s < (p)

llJ(p)il

k

. From

whence Z(ti+O)

~ O

tl,t2,...,t

< tk < p

that

iiJii'(ti±o)

IR

differentiable

that

s(o)

is

s(O)

function is

(O,p)

ilG(p)',I

satisfies

the Z

that

_ SK(P 1 ) {ilj(O),!]iSl<(p_t)

s (t) and

that

Iio(o)',l

have

s(t) and

we

= O

function

contained

4.2,

z (~+0)

we

we

:= siij'~i -s{{JiI. • T h i s

proof

Moreover,

s" + Ks

,i"

Z

intervall

the

- s
_< Z ( p - O )

, ,

56

lIJ(p).iF~t~(p)llJ (o)II fiJ(o) II s<(p)

-i!J(o)H I141"(+o)+ •

sK

{

}

'

= <s,J>(p) - <s,a>(o) -~-~K(p) [jJ(O)112 +IIJ(p)il2 + s-~(p)Is(o)ll lls(p)il, and the t h e o r e m Next,

is proved.

we i n t r o d u c e

the R i e m a n n m e t r i c

for

uj,vj

6 TyjM

, yj

:= +

6 M

T h e n w e c o n s i d e r two o p e n f o l l o w i n g condition:

(SC) Each pair exactly

This

(yl,Y2)

, j = 1,2 subsets

condition

implies

MI

and

M2

of

M

Yl 6 M I , Y2 6 M 2

of p o i n t s

one m i n i m i z i n g

.

geodesic

arc of

that the d i s t a n c e

M

that s a t i s f y

the

can be j o i n e d by

.

function

d i s t : M Z M + IR is of class Then,

C2

on

M I × M 2- {diagonal

for any f u n c t i o n

¢ 6 C3aR,~)

of

M × M}

with

6(0)

= O , the c o m p o s i t i o n

¢ ~ dist : M I × M 2 ÷ is a C 2 - f u n c t i o n

(this can e a s i l y

¢(t) = ¢(0) + t 2 ~ ( t ) we h a v e set

q~(t) and we o b t a i n

be i n f e r r e d

). Thus we m a y in p a r t i c u l a r

.t = JoSK(T)dT

, i.e.

, qK(t)

4.6. Let

(SC) , a n d suppose

MI

~ ~ 0

the H e s s i a n

¢ = q<

, where

~[1-ces/~-t]

' ~ > ~i 1

L t2/2

, K =

,

and

M2

be open

is an upper

D2Q<

, sets

QK: = q~ o d i s t in

M

that satisfy

condition

that

d i s t ( y ] , y 2) < ~//<where

=

choose

lemma:

a C2-map QK : M I x M2 + ~

Theorem

from Morse's

of

Q<

b o u n d for at

for all the sectional

(yl,Y2)

satisfies

(yl,Y2)

6 M1 × M2

curvature

of

the f o l l o w i n g

M . Then inequali-

57 ties:

(4.26)

D2Q<(v,v)

(4.27)

D 2 Q ~ (v,v)

f~r all y = T y ( M × M) MoreoVer,

=

(yl,Y2)

Ivl 2 if Yl = Y2 2 >< K(y) 2Q -
and for all

6 M I wM 2

if

Y2 ~ Y2

v = v I ~v 2 6 TyIM ÷Ty2M

~

we have

D20 (In (4.27),

v,v

> {I denotes

~]Q~(y)

y =

2

Proof.

For

i.e.,

Yl ~ Y2

' we define

(yl,y2)

unique

minimal

geodesic

if

the gradient of

6 M 1 xM 2 , let ej (y)

joining

at

as u n i t

a3nd

Y2

or

y .)

p = d i s t ( Y l , y 2)

6 Ty M Yl

QK

v = o÷u

. If

tanment

p > 0

vectors

at the end p o i n t s

,

of the v4

In

j

order

to m a k e

the

that

ej(y)

assume

the g e o d e s i c Let T v 3

choice

of the

points

into

two v e c t o r s the o u t e r

eI

and

direction,

e2 that

unique, is,

we

away

us d e c o m p o s e

each

and a normal

component

tangent v

vector ± : 3

v 4J 6 T y j M

T Vj = < v j , e j ( y ) > e j ( y )

,

into

a tangential

1 T vj = v j - v j

Then Vdist(Yl,y and

the

chain

VQ<(y)

rule

= O

if

a n d on a c c o u n t

D2Q<(v,v)

2)

= el(Y)+e2(Y)

if

p = O

,

VQ<(y)

= s K (p)e I (y) % e 2 ( Y )

of

= ,v>

a n d of

(p)Vdist

~vS<(P)

we obtain

p > 0

,

yields

=
DyeS

from

arc.

}

= s<(P)DvVdist+~vS<(P)Vdist

= s< ( P ) ~ v d i s t

= s<(p)

,

if

p > 0

part

58

D 2

Q<(v,v)

D2Q<(v,v)

(The

first

q<(k(y)) In c a s e in

that

terms

[22],

Let

: s

equation

: ~p2(y)

+ ....

Jacobi

we

c : [O,f)]

= ~

Z

and

apply

this

connects

Yl now

-< P, as the

~(p,~)

2

t3

+O(

) , that

p > 0

is,

verified.

the

this

,

~J :

Hessian

purpose,

i'(t,d)

of

we

be a

,

" = I [ICli

end

the f o l l o w i n g

= %-~(t,O) a~

result and the

sufficiently

two

= ~t is

express

For

L"(O)

to

Y2 two

~I (~)

the

(4.26)

that

vector

,

o/

the

L(c~)

the

distance

Synge's

use

function

Lemma

(cf.

(s~7ooth) v a r i a t i o n of a is,

c(t)

=

and

(uniquely

J1 (a)

c

:

< Z , c•> c .

)i

is g i v e n

dt

by

)d t + • Io p geodesic

and

. For

determined)

points

along

arc

2)

c : [O,p]

, we

have

÷ M

llc(t)ll

that ~ I

arcs

= eXpy I (~v~) lal

Z-

p : dist(Yl,y

geodesic

small

fie~ds

]oPi ~-~(t,a 9%,

=

minimizing

" Since

=

Z±

~{,~D..i,2 = jo\ti~-[" [I - < R ( Z , c ) $ , Z >

Consider

for

will

~ith

Zi

derivative

L" (O)

We

) Thus

1

q<(t)

if

(t,O)

~by

the

from

+ M

+ M

Z(t)

Then

,

(p) < V d i s t (y) , v > 2 + s < ( g ) D 2 d i s t ( v , v )

fields.

9 : [O,p] x [-S,~:]

(t)

~ : 0

122-123):

geodesic

Denote

if

follows

g > 0

of

pp.

!2 iv l

:

~2(~)

fixed

~

minimizing

and

o 2(a)

: eXpy 2 (ave)

, we

define

geodesic

~,(a,t)

arc

, i.e. , 9 (0,~)

, 0 _< t

that

connects

= o I (~)

and

= o2(a)

Then h

J(t)

defines

a Jacobi

field

J(O)

we

infer

from

is

a normal

jT(t)

Jaeobi

:

: ~(t,O)

along

= vI

e

,

d

Moreover,

~-77~.(a) da ~ J

0 <

t -< p

,

. Since

{at+b}c(t)

field. D

,

or%

: 0

J(p)

, that the

,

: v2

,

jT

0

; in o t h e r

equations

j = 1,2

,

words

J

59 imply that ~D B~ ~(t,a) Then, for

= O

for

t = 0

Z = J = J± , we infer from

and

t = p

Synge s lemma that

D2dist(v,v) = L"(O) = ]oD(i]J]i2-)

dt

Moreover, the Jacobi equation

J+R(a,$)c = 0 implies that

[IJ]] 2-

= IIDI[2+<~,j>

: ~d<J'J'>

whence, by Theorem 4.5,

• Ip

i

s<tp)

2

D2dist (v'v) = <J'J>lo -> s K (P) j=IL v ilv_il <231 - s<2(p)~!v%1~ [iv2, i,l Combining this estimate

with the previous expressions, we arrive at

D2Q<(v,v) _> sK(p)<e I (Y) 4~e2(Y),v I ~v2 >2 2

+ ~< ~p) Z111v~,I2- 211v~i '~v2,,I"2

>-~ > (I-<(0,/L~, iI2

(4.29)

lJ~

~2 i

sl<(P)t4A__l<ej (y),v>)

j=l J

+ 1 - sl<. (p))j!l I!v~'ll 2

(~-~<(0)] ~ '
However,

( J! < e j (y),v> )2 -< ( 21[iVjl)2 V ! T I ~ 2 2~ 'ivT 112 j~ jL--I 3

'

and therefore

~2

2

<ej (y) ,vj>/

On the other hand, VQ<(y) = s<(p)e I (y) ~e2(Y) whence 2 = s 2<(p)<j!1<ej_ (y),v>) 2 Thus it follows that

60

l+s D2Q<(v,v)

(p)

<

2 --22s 2 (p)

We m o r e o v e r infer from the d e f i n i t i o n

q<(t)

Q<(y)

, and of

= q<(p)

that I- s

(p) =
=
and t h a t

l + s (p) l + e (p K 2 = ~ ~ s<(p) 1-c (p

1

<~

=

l

~

-c< (p)

i.e.

l+s 1 -s

for

< > 0

(p

=
; but o b v i o u s l y

Thus we o b t a i n

1

--

Kq< (p)

<

Q<(y)

(p)

I Q< (Y)

s<(p)

the same r e l a t i o n s

are also t r u e

2 ~,2 ~ llvjli j=1

I D2Q<(v, v) -> 2 Q K ( y ) < V Q < ( Y ) , V > 2 - ~ Q < ( y )

for If

p > 0 , and

(4.27)

is p r o v e d .

v I = 0 , v 2 = u , we c o n c l u d e D2Q<(v,v)

for

p > 0

on a c c o u n t of

(4.29)

that

-> s K (p)<<e 2 (y),u>2 +[[u±l12> =

holds

for

the i n e q u a l i t y

s(

)l!ull 2 =

; b u t this e s t i m a t e

remains

true for

p = 0

s i n c e we

have D2Q~(v,v)

if

= llvli2 = lluli2

and

P = O . Thus a l s o the e s t i m a t e

of T h e o r e m

cOMMENTS

(4.28)

= O

is e s t a b l i s h e d ,

and R E F E R E N C E S

f i e l d s on g e n e r a l

to the L I T E R A T U R E

for J a c o b i

Riemannian

s p a c e s of c o n s t a n t

fields w e r e o b t a i n e d

manifolds

sectional

example,

[22], pp.

and D.G.

Ebin,

Co.,

178-182;

Comparison

curvature.

The m o r e d i r e c t

we a l s o r e f e r

theorems

Amsterdam/Oxford,

by c o m p a r i n g

Jacobi

w i t h t h o s e on s t a n d a r d m a n i f o l d s , The estimates

this w a y a r e u s u a l l y q u o t e d as Rauch comparison

Publ.

and the p r o o f

4.6 is complete.

The f i r s t e s t i m a t e s

say,

Q<(y)

theorems

to the m o n o g r a p h

in Riemannian

geometry,

derived

(see,

in

for

by J. C h e e q e r North-Holland

1975).

approach

to J a c o b i

field estimates

presented

in these

l e c t u r e s is due to H e r m a n n

Karcher.

We have learned

his ideas

from [6]

61

and

[52] as w e l l as f r o m p r i v a t e

lectures.

t i o n are b a s i c a l l y due to Karcher, Theorems

4.5 and

stated

4.6 w h i c h h a v e been d e r i v e d by J ~ g e r and K a u l

Karcher's

approach

than

old c o m p a r i s o n m e t h o d s .

the

Thus the r e s u l t s of this sec-

e x c e p t for the e s t i m a t e s

is q u i t e t r a n s p a r e n t and in m a n y w a y s m o r e

in

[42]. flexible

62

5. R i e m a n n

Let

normal

9(t,a)

coordinates

be a m a p p i n g > O

E [-~o,ao] , ~ ao Then J(t) = - ~ ( t , a ) a

9 : [O,R] × [-ao,a o]

, the c u r v e

c(t)

is a J a c o b i

field

÷ M

such

= ~(t,~) along

that,

for e v e r y

is a g e o d e s i c

c

. This

in

follows

M

from

. the

identities D 3~ 3t ~a

D ~

3~J : 0 ~t

and D D Z 3t 3cJ. where

Z

denotes

3t 3t 3a

D ~a

D Z = R(~ 9 ~ h Z 3t \~)t'~0'~/

an a r b i t r a r y

St 3~ 3t

vector

3~ 3t 3t

field

along

k,3t ~ / ~ t

~

O-

In fact,

we h a v e

\~a'~t/

or 3+R(s,$)$

This ideato

construct

First,

by m e a n s

normal

coordinates

and

TpM

its

identify

TpM

tion,

with

can be The

on

with

c(O)

= p

Let

q = eXppV

mapping

and

, and

then,

: v

vector

is the

in

"normal

chart"

with l

where

supposed

to be e x i s t i n g

some

by

point

space,

identification

Tv(TpM)

. Hence

Riemann of

via

transla-

each

Tv(TpM)

to the c e n t e r c

M

w e can

p

is

is the g e o d e s i c

with

"radial" , and

with

vector

U

pp.

136-137),

6 TqM

,

~ =

p 6 M

to

v

the

, ~

are d e f i n e d

linear

is an

by

(d e X p p ) v ( n ) is g i v e n

~ : U ÷ ~m

is a l i n e a r .

parallel

~,n

center

j : TpM ÷ ~m on

([22],

: <~,~>q

p 6 U , and by a m a p p i n g

j o exp~I_

introduce

satisfies

(d e X p p ) v ( ~ )

(~,U)

be

, where

lemma

÷ TqM

Tv(TpM)

=

p

following.

.

<~,n>p

n 6 Tv(TpM )

shall

T M . P with respect

v 6 TpM

by Gauss'

. Then,

(5.1')

U c M

for

in the

is a l i n e a r

spaces of

eXpp : T p M -> M

(d eXpp) v : Tv(TpM)

arbitrary

A

tangent

we

let

To(TpM)

= c(I)

c(O)

end, TpM

(5.1)

where

. To this

the m e t r i c

be used

mapping,

Since

with

eXppV

will

space.

the d o u b l e

exponential map by

M

tangent

all

fields

of the e x p o n e n t i a l

furnished

defined

Jacobi

= 0

by an o p e n

of the

isometry,

and

form eXpp I

set

~ = is

68

Let

el,...,e m

ponds

to

dean

space

TpM

the

, we

be

the

standard ~m

orthonormal base

. Since

may

TpM

, and

the

are

given

by

base

is

q = exppV

Let

c

and

let

and,

be

el,...,e m

vector

÷ M

fields

=

we

proved:

have

5.1.

v 6 TpM gacobi

For

, then,

: p

the

for

vector normal

eucli-

all

v

fields chart

6 on

(~,U)

(d e X p p ) e i

for

J

= p

. By our

every

}

and,

some c(t)

6 TpM

@(-,a)

J(t)

,

= eXpp(tV),

a geodesic

remarks,

v

:

= ~(t,O)

moreover,

eXpp)tvek

J(O)

for

. Then

defines

is a g e o d e s i c

with

= v

Tp M

in

= t~kXk(C(t))

with

~ = {ke k , J(t)

c

c(O)

previous

c

: t~k(d

+ M

along

and

vector

along

each

normal

(9,u)

chart

coordinates

with

c(O)

= p

= t~kXk(C(t))

= 0 , 3(0)

and

c(O)

defines

= ~kXk(p)

and,

=

a if

q

all

q 6 U

. Let

=

~(q)

XI,...,X m

be

ponding

to

the

chart

ponents

of

the

fundamental

Christoffel

brevity,

we

(~,U)

symbols

. Then tensor

of

the

using

We obviously

, we m a y

introduce

the b a s e

vector

fields

Riemann

on

gkl(q ) = <Xk(q),Xl(q)>q on

first

different

,

U

, and

and

second

Fikl(q) kind.

and For

U

corres-

a r e the c o m 1 Fik(q) d~note

the

sake

Fikl(X)

:= Fikl(<0 -1 (x))

of

,

etc.

= ei,

and

notation.

have (p)

Moreover,

p

set

gkl(X) := gkl(~p-l(x)) without

center

by x

the

the

corres-

, with

normal

for

of

j

of

Tv(TpM)

orthonormal

= eXppit(v+@~)

field

c : [O, ~)

field

=

c(O)

(d e X p p ) t v ( t ~ )

If

m

under

(0,...,0,I) with

XI,...,X m

an a r b i t r a r y

@(O,a)

is a J a c o b i

7~(t,O)

Lemma

c(I)

be

with

~ , p(t,~)

with

therefore

Thus

geodesic

each

as

which

.

~ = ~ke k

for

[O, ~)

the

TpM

identified

Xi(q) where

of

(I,0,..~,0),...,

TpM

consider

base

(d eXpp) °

is

the

=

0

.

identical

map,

whence

Xi(p)

,

64

therefore g k l (p) Let

c(t)

Then

= eXpptV

n(t)

=

or

~kl

k v = x ek

, where

:= @ ( c ( t ) )

gkl(O)

=

and

6kl j(v)

= x =

(x I

,x m)

satisfies ..i Fl -i -k h + ik(n)n n = 0

On

the

fore for

other c(t)

all

hand,

= -1

the definition of <~ i m p l i e s ~(t) = t x i i k and Fik(tx)x x = O in p a r t i c u l a r

(tx)

x 6 ~m

ik(O)

Let

F1 = F1 ik ki ~ = ek

v = c(O)

•

= rikl(O)

= O

1

or

rik

(p)

=

Fikl

(p)

:

0

" 1 n = x eI

, and

be

a radial

vector

that

coincides

X i(q)

=

= <ek'xlel >

(d e X p p ) v e i , w e

= xl6kl

v

infer

= xk

from

Gauss

lemma

(5.1), (5.1 ') t h a t

k x

Thus

= <6,T)> v = < ( d

we

e x p p ) v ~ , (d e x p p ) v n >

k

Moreover,

one

distance

d(p,q)

1 = x gkl(X)

also

is g i v e n

infers of

the

and

from two

we

Lemma

5.2.

on

the

=

have

set

If

Gauss'

x

k

=

len~.a p,q

x l g kl (x)

(see

6 U

[22],

with

~ 4.4)

p = c(O)

that

the

, q = c(I)

=

by

ix I : / @ k l X k x I

Hence

also

points

d(p,q) where

i = x gkl(q)

= < X k ( q ) , xlxIj (q)> q 1 = x gkl(X)

have x

eXppV

with

. Then <{'~>v

Since

there-

. Therefore,

F1

since

and

FI (O)xixk=o ' ik

denotes

= II~11 = IIvll = the

euclidean

IxJ length

of

the

vector

x 6 ~ m.

proved:

x

=

U c M

<0(q)

,

(5.4)

g i k (O) = @ i k

(5 • 5)

Xk =

(5.6)

d(p,q)

gkl

=

are

Riemann

normal

'

Fikl(O)

: O

coordinates

with

then

(X)X 1 Ixl

'

xk =

g kl (x)xl

1 Fik(O)

, '

= O

center

p

85

Moreover, tes

m

if

v = x em

the g e o d e s i c

6 TpM

expptV

, x =

with

(x 1 , . . . , x m)

c(O)

= p

6JR m , a n d

and

c(O)

if

c(t)

= v , then

deno-

~(c(t))

tx For

some

real-valued

function

fl (x) Then

the

Lemma

following

5.3.

If

~f

=

5.8)

(x)

~x 1

holds:

x = ~(q)

are

Riemann

5.9)

x i]l F i l k ( X )

5.10)

xlFlk(x ) = xlFilk(X )

5.11)

xixkFikl(X)

Proof.

normal

coordinates,

+ F i k I (x) } = 6ik - g i k ( X )

then

together

of

(5.5)

= xixkFkl(x)

the

formulas

and

(5.7).

with

(5.7)

yields

account,

(5.5)

implies

,

,

= xixlFikl(X)

By d i f f e r e n t i a t i n g

is a c o n s e q u e n c e

Let

, we write

k (x) (x) , x k ik @il x gik,l = 6ii gil gl (x) = _ gll(x) i k i 1 k 1 x x gik,l(X) = x x gik,l(X) = x x gik,l(X) = O , i k ik i 1 ik k 1 ik x x g I (x) = x x g l(X) : z x g ,l(X) : O ,

5.7)

into

f(x)

(5.9).

(5.5), The

and

we

obtain

identity

Finally,

(5.10),

= xixlFkil(X)

if w e

(5.11)

= O

(5.7),

and

(5.8)

rik I + Fil k = g k l , i take

follows

Fil k = g l j F ~ k from

(5.8).

and

us n o w

r e t u r n to the f o r m u l a s (5.2) of L e m m a 5.1. If c(t) = e X p p t V k v = x e , t h e n we i n f e r f r o m Lenmla 5.2 t h a t x = ~(q) with q =

c(I)

, and

field

J(t)

~k(1)

= t~kXk(C(t)

(5.12) Thus

we

: xk

J(1) = obtain

(5.13)

the

if

$(t)

)

= sk

t)Xk(C(t))

. Hence,

the

Jacobi

fulfills

~l+rik(x)~ixk

relations

llJ(O) Ii2 : 6 k l [ k [ 1

,

Hj(I ) [[2 = g k l ( X ) ~ k ~ l

and

<J(1)- O(1),J(1)> = £1k (x) ~ixkglj (x)~ J

(5.14)

= F i j k ( x ) ~ i ~ Jx k

We

also

note

that

BR(p)

Let

now

and

radius

R

r

:= d ( p , q )

=

Ixi

= lq 6 M : d(p,q) S R~ in the m a n i f o l d

M

= ilcll .

be the geodesic ball of center J w h i c h w e a s s u m e to be c o m p l e t e .

p

=

66

Suppose ter.

that

BR(p)

Then we can d e f i n e

the following Theorem the

does

also

Let

M

BR(p)

that

~ K

the

in

Riemann

the cut

normal

be

a complete

M

does

sectionaZ

~ ~ , where

< . Let

intersect

C(p)

locus

coordinates

on

of its cen-

U = BR(P)

, and

holds:

5.1.

ball

not

~

finally

not

Riemannian meet

cv~rvature and

~

K

are

vwo

R/<< < ~ . T h e n

we

manifold,

the

cut

of

M

real can

locus is

its

bounded

on

numbers

define

and

of

that

suppose

Assume

BR(p)

by

satisfy

a normal

that

center.

~ ~ 0

(~,BR(P))

chart

k

with

p

as

its

center

q 6 BR(p)

for mates

are

(5.15)

and

With

obtain

respect

Riemann

to

these

normal

coordinates,

the

~

x = ~(q)

coordinates

following

esti-

true:

fa

I

(Ixl) - 1 } g i k ( x ) ~ i ~ k

< Fikl(X)xi~k~l

}

< {a

(Ix[) - 1}gik(x)~i~ k

1 (X) x l ~ i ~ k ~ i k _ a0~(ix[)gik(X ) [i~k < ~ _ ri k

(5.16)

{ £ i k - a<(ixl)gik}~i~ k (5.17) Proof. tes

x The

(4.23)

(5.16) Theorem

inequalities and

(4.24)

5.2.

= ~d

Let

(p,q)

the

from

and

(5.17)

readily

4.4 and from

(5.15)

assumptions we

x

by v i r t u e of

Theorem

q

6 M

Let

from

identity

5.1

satisfied,

be

the estima-

(5.14),

and

(5.9). and

set

have

r = d(p,q)

with

c(t)

p , and set

= eXpptV

F(x)

_< R

, q = c(I)

= f(q)

(D2f)q (~'~) Since

and

of the

and

, and

for

~ 6 T M q

~ : ~kXk(q)

J(t) = t~kXk(C(t)) forms a Jacobi field J along and LtJ(1)il 2 = [i~il2 . C o n s i d e r normal c o o r d i n a t e s at

follow

(5.13)

a<(r)li~II 2 _< (D2f) q (~, ~) _< a0 (r)ll~ii~2

all

Proof.

(5.15)

<

of T h e o r e m

Then

(5.18)

for

<- gik (x)

can be d e r i v e d

I 2

f(q)

~

c

.

6 TqM with

x = <0(q)

. Then J(1)

=

with

center

Then

=
F (x) = 11 xl 2 , we get

(D2f)q (~'{) by v i r t u e

of

: 6ik~ (5.9).

i~k

-

FI (x)xl~i~k ik

We then

derive

from

= ~

ikl

(5.2)

(x)xl~i~k+gik(x)~i~k

that

67 <J(1),J(1) > =

and

thus

Theorem

(5.18) 5.3.

curvature

follows

Let

K

M

from

(4.22).

be a c o m p l e t e

of which

is b o u n d e d

K

on some

BR(p)

ball

Moreover,

let

be c o n n e c t e d contain

that

R/~K < 7/2

any p a i r s

BR(p)

arc

in

This

r e s u l t was p r o v e d

ql

that j o i n

Riemannian from

~

above

,

K

not m e e t

. Then

of conjugate

arcs

to

<-

does

by a g e o d e s i c

(D2f)q(~,~)

any

>-- 0

locus

ql " q2

BR(p)

in

o f its

center

BR(P)

of

This

arc

a n d it is s h o r t e s t

p

does

among

.

can not

all

q2

and

[44]. We d o n ' t r e p e a t

the p r o o f

but refer

[44].

In the t e r m i n o l o g y ball

BR(p)

of

of s e c t i o n

1, T h e o r e m

M , we can define

5.3 implies/

a normal

chart

that, %

on a r e g u l a r

(9,BR(P) ~ \

centered bining

at an a r b i t r a r i l y

the T h e o r e m s

Theorem

5.4•

M , i.e.,

that

~ <_ K

chosen With

some

qo

respect

to

f(q)

point

nates

BR(p)

BR(p)

x = ~(q)

COMMENTS

can

that are

, i.e.,

• Thus,

in

the

complete

R/~

< 7/2

introduc~ centered

~(qo ) = 0 =

by com-

result: Riemannian • Suppose

Riemann at some

mani-

also

normal arbitrarily

(0,0 .... ,0)

these c o o r d i n a t e s ~ the e s t i m a t e s ( 5 . 1 5 - 1 7 ) hold. If, 1 2 (q,qo) , w h e r e d(q,qo) d e n o t e s the d i s t a n c e o f from

as i n f i m u m

the c e n t e r

of

length

qo

point

o f all

curves

of the joining

normal q

coordi-

qo

and

), then

. Moreover,

[22]

= / x i x i < ~I/<

estimates

(5.18)

are

satisfied

for

all

.

to the L I T E R A T U R E

differential

notes

the

~ E TqM

and R E F E R E N C E S

the l e c t u r e

ball

that is

/

BR(p)

= ~d

a n d all

For the basic

of

, < _> O , a n d

d ( q , g o) = ixl

q 6 BR(P)

qo

at the f o l l o w i n g

. Then we

BR(p)

on

BR(p)

of

we arrive

~ ~ 0

q 6 BR(P)

(defined

within

some

point

be a r e g u l a r on

x = ~(q)

point

moreover,

K <_ ~

for

coordinates

chosen

5.1-5.3,

BR(P)

Let

fold

for

sectional

,

two p o i n t s

points,

the

by

the cut

contained

by J o s t

manifold,

geometric

notions

and results,

by G r o m o l l / K l i n g e n b e r g / M e y e r .

we refer

to

68

The e s t i m a t e s coordinates

of C h r i s t o f f e l

s y m b o l s w i t h r e s p e c t to R i e m a n n normal

that we have p r e s e n t e d

Hildebrandt-Kaul

[29].

Rauch c o m p a r i s o n t h e o r e m s

as s t a t e d in

Our b o u n d s on the C h r i s t o f f e l to e s t a b l i s h a p r i o r i

estimates

found in p a p e r s of J o s t

[22].

for h a r m o n i c m a p p i n g s Further applications

(see [48,49]

in

f o u n d e d on

s y m b o l s w e r e u s e d in [29] and in

larity of w e a k l y h a r m o n i c maps.

[62,63].

in this s e c t i o n w e r e d e r i v e d

The d e r i v a t i o n g i v e n in [29] was

[32-35]

and to p r o v e reguc a n , f o r instance, be

for f u r t h e r r e f e r e n c e s )

and M e i e r

69

6. E x i s t e n c e

proof,

harmonic

In this

section,

first

if

} : 8X + M

we

theorem

}

U

the e x i s t e n c e

boundary

1.1

We b e g i n

by

solving

that n

boundary

U(~)

fl C2(Q,~A)

of c l a s s

harmonic

map

and w i t h

and

it r e m a i n s

with

problem

product

with

chart

CI

6 ~I,2(~,~4)

~(~)

c BR(p)

The

on

Sobolev

1.4.

space

the p r o p e r t y with

structure

the a s s o c i a t e d

the

, whence

1.3 and

X

of the

1.4, ~

with

of

space

that

1.3.

U 6 H I'2 N

Theorems

(x,W)

Namely,

such

of T h e o r e m

(1.2).

In fact,

two.

U-¢

u : X + ~N

Hilbert

1.4.

Theorem

is h a r m o n i c

to p r o v e

set of m a p p i n g s

and

latter

, we a p p l y

HI,2(~,M)

the v a r i a t i o n a l

by a s c a l a r

1.3,

} : ~X + ~! . By v i r t u e

Thus

as

BR(p)

c BR(p)

u : X -1 6 H ] ' 2 ( W , ~ N) for e v e r y n {±°Ca n } , ~+ = x 6 ~ :x A O. . T h e

is d e f i n e d

for w e a k l y

of the

mapping

ball

of a weakly

C°(X,A~)

is d e f i n e d

1.1,

consequence

of class

values

is proved.

H I ' 2 ( 9 , ~ N)

W c~+

X , with

is of class

Theorem

an d r e g u l a r i t y

the T h e o r e m s

in a r e g u l a r

is an e x t e n s i o n

prescribed map

prove

is a p r e s c r i b e d

(Q,6~) , ~ = int

Here

shall

is an i m m e d i a t e

is c o n t a i n e d

Then we obtain L

principle,

maps

the

}(~X)

maximum

on

range HI,2

N (~,~ ')

norm

[[Ul[2 = ] ~ l u [ 2 d v o l + ~ e N ( U ) d v o l

where

the

invariant

eN

is d e f i n e d

in local

coordinates

by

1 ~B~ (u I• X -I)D (ul' X --I ) e N (u) = ~Y °a ° B ° The

subspace

to c h e c k

~1,2(~,~N)

that

is the H 1 ' 2 - c l o s u r e

all p r o p e r t i e s

of S o b o l e v

retained.

For

example,

element

of c l a s s

L2

such

difference Since

function,

within

that

belongs

to

the c o m p o s i t i o n

element

we U

some

coordinates

ball

around

p

nition

HI'2(~,M)

for its e l e m e n t s To s i m p l i f y

BR(p)

that and

the n o t a t i o n a l

with

nature

has a t r a c e

same

trace, then

a C1-map

that

are on

~X

their

we

apparatus

that

is a n o t h e r require have

its r e p r e s e n t a t i o n . The

shall

not r e q u i r e

is m o r e

if w e

U :X ÷ M

H I ' 2 ( ~ , ~ N)

. Later

does

that

the

unambiguously

and

to

H 1'2 N L~(~,M)

U

have

be a m a p p i n g

belong

of a local

It is easy

.

HI'2(~,M)

regular

C c ( ~ , ~ N)

H I ' 2 ( ~ , ~ N)

of an H 1 ' 2 - f u n c t i o n

HI'2(~,M)

and

spaces

of

if two e l e m e n t s ~I,2(~,~N)

can d e f i n e of

HI'2(~,M) of

every

of

same

its

the c o n d i t i o n

another U(~)

satisfactory. we introduce

the

that

an

image

in n o r m a l

applies

indicate

H 1'2-

following

to defic BR(p)

70

CONVENTIONS.

Any e q u a l i t y

holds almost

everywhere,

preted

as

ass suplul

s y s t e m in

X

sign between whereas

t

-

and

U o X I

notation will In o r d e r

if

and

suplul

S

at hand,

its c o o r d i n a t e of

U 6 HI'2(~,M)

s h o u l d be i n t e r -

X

and

S c X

the lack of g l o b a l

linear

w e s h a l l n o t dis-

representation

X(S)

and

coordinate

c ] R n]

x =

, or b e t w e e n

-1 (W) . T h i s a b u s e of

not c a u s e any c o n f u s i o n .

to c i r c u m v e n t

we i n t r o d u c e

that equality

if the c h o i c e of a local

for the p u r p o s e

t i n g u i s h b e t w e e n a p o i n t on X (x I ..... x n) , b e t w e e n a s u b s e t U S

the a b b r e v i a t i o n

. Moreover,

is i m m a t e r i a l

two H I "2-maps m e a n s

for any

p 6 M

and any

R < i(p)

structure

in

H 1,2(9,M)

(= i n j e c t i v i t y

r a d i u s of

p ) the set BR(P)

By m e a n s BR(p)

of a n o r m a l

with

= {U 6 H I ' 2 ( ~ , ~ ) :

chart

the convex,

,,(~'BR(p ) ]

weakly

sup~dist(U(x),p>

centered

s e q u e n/t i a l l y

at

closed

p

_< R}

we can i d e n t i f y

s u b s e t of

H 1,2(~,]RN)

d e f i n e d by {u 6 H 1'2 N L~(~,lqN): sup~,ul

T h e n the e n e r g y fxe(U)dvol

functional

e(U)

on

BR(p)

by

E(U)

=

1

= 2gik(u)YaB~au198uk

lower s e m i c o n t i n u i t y

e a s i l y be c a r r i e d following

can be d e f i n e d

, where

(6.1) Well known

E(U)

< R}

over

results

(cf.

[65],

to the p r e s e n t

situation.

U k 6 BR(p)

converges

Theorem

1.8.2)

T h u s we o b t a i n

can

the

result:

If a sequence

of mappings

on

U 6 BR(P)

~ , then

weakly

to

U

in

H 1"2

, and

lim inf E ( U k) ~ E(U) k÷~ By the s t a n d a r d m e t h o d of the c a l c u l u s I) w e can n o w d e r i v e

Lemma

6.1. S u p p o s e

6 BR, (p) , there

the f o l l o w i n g

that

~R' (p)

is a s o l u t i o n

EIul

of v a r i a t i o n s

(see [65],

chapter

statement:

is a r e g u l a r o f the

in

ball

variational

Ipl o {u

in

~ . Then

to e v e r y

problem

c

121, MI}

,

71

Let

~

be a t a n g e n t i a l

f a m i l y of v a r i a t i o n s

Ut(x)

t i o n shows

that the

U £ ~(p)

in d i r e c t i o n

mild regularity

vector

= exPu(x)t~(x)

first variation of

~

of

if

U , ~

U . We c o n s t r u c t

. A straight-forward

of the D i r i c h l e t

exists,

provided

integral

that

~

the

computaE

at

satisfies

some

condition:

6E(U,~) Moreover,

a l o n g the m a p p i n g

BR,(P)

= ~tE(Ut)It=O

is a r e g u l a r

ball,

and if

with r e s p e c t to some s y s t e m of n o r m a l

u , ~

are r e p r e s e n t a t i o n s

coordinates

on

BR,(p)

,

then

(6.2)

6E(U,T)

where 6e(U,~)

= gik

= ]6e(U,Y)dvol X

(u)y~B~ u i ~ k

1

,

(u)7~@~ui~ uk~l

+2gik,l

~

B

and gik, l = ~igik The i d e n t i t y

gik,l

1 (U)y~@~ 2gik,l

= rlik + r l k i ui~

implies

that

I y~B$ ui~ u k I B uk = 2 F l i k ( U ) ~ B +2~iki(U)Y~ = Flik(u)y~SZ

ui~ u k 6

ui~Buk

Thus

•

(6.3)

6e(U,T)

If a s o l u t i o n tained

in

= gik(U)7@@8

U

of

8R(p)

with

the m i n i m u m R < R'

(6.4)

for all t a n g e n t i a l and v a n i s h on fies

vector

~

6E(U,~)

: O

fields

~

along

(P]

uZ8 8

of L e ~ m a

6.1

is con-

that are m i l d l y

regular

that

U

representation

). It, h o w e v e r ,

ukgl

@

of

T

satis-

c o u l d be t h a t the m i n i m u m

U

of the set 8R, (P) in L~([~,M) so t h a t we w e r e U in all p o s s i b l e d i r e c t i o n s . In this c a s e w e c o u l d

i n f e r that

(6.5)

holds

problem

, we o b t a i n

(that is, any n o r m a l

@ 6 ~I,2 N L ~ ( ~ , ~ N)

lies on the b o u n d a r y n o t a l l o w e d to vary only

.

u1~@~, k + r l i k ( U ) y ~ 8 ~

6E(U,'~:) -> O

for all v e c t o r

fields

~

along

U

such t h a t

U t = exPut~

stays

72

within

BR(p)

We w i l l

actually

show

values

%

solution

, each

therefore,

satisfies

MAXIMUM

PRINCIPLE.

tained

in some ball

of the m i n i m u m

Proof.

Let

respect

under

of

(6.4).

Suppose BR(p)

problem

u(x)

suitable

(P) For

that

D 20

in the

purpose,

R < R'

of Lemma

coordinates

with

assumptions

the boundary

with

(P]

lies

this

be a r e p r e s e n t a t i o n

to n o r m a l

£ Cc(~,~)

that,

on

on the

interior we

prove

value

. Then

every

of a s o l u t i o n centered

BR(p)

in

U

~

p

is c o n -

solution

U

BR(p)

of

at

and,

following

function

also

6. 1 is c o n t a i n e d

BR, (p)

of the

boundary

(P) w i t h

. Then,

for

, the m a p p i n g

u t = u - tnu has

the

R'

same

for

H1'2-boundary

values

O N t < sup~

an a d m i s s i b l e

variation

. This of

U

E ( U t) - E ( U )

If w e m u l t i p l y

this

as

u

means

for

by

the v e c t o r

coordinates

field

is g i v e n

-6e(U,~)

~

by

alone @ = -nu

lutl

=

(]-t~)lu I

is the r e p r e s e n t a t i o n problem

O < t <<

t , and

~E(U,~)

where

ut

for the m i n i m u m

Z O

inequality

and s a t i s f i e s

that

let

(P)

of

, whence

I

t + +O

, thenweobtain

_> O

the m a p p i n g

U

with

respect

to local

. Then

= gik(u)uky~B~

ui~Bn

+ n¢~ [~ik(u)~u% Buk + ul~lik (u)~ui~u k] and, we

by v i r t u e

infer

of L e m m a

-6e(U,Y)

=> ~I y a ~ a l u ]

where

~ =

max]~ O,suo~BR,

~

~/2

, and

R'<

5.2

(see

(5.5))

and T h e o r e m

5.1

(cf.

(5.15)),

that 2~B n + m s < ( l u [ ) ~ s ~ - i k ( u ) ~ a u m ~ s u k

(p) K 1 f . Since

therefore

a

f~y~

%$(lul)

BR, (p) ~ O

lu]2~Bn

on

dvol

is a r e g u l a r ~

~ O

. Thus

ball,

we obtain

we h a v e

73

for

all

n £ C ~ ( ~ , I~) c

with

(6.6)

for

SRa~

all

~ 6 HI'2(~,Iq) a sS

The

D A O

function

v

is

of

~ h O

y~

,

essential

known

and

supremum

easy

to

of

prove

its

of

class

~-

= 9 N

HI'2(f~,]R)

v =

get

we

have

set

]ul 2 = d i s t 2 ( U , p )

on

~

. Denote

by

= sup~v

trace

on

vJ

:= m a x { v ( x )

for

( v < k)

finally

9~

. Then

it

is w e l l

that

~ (x)

is

we

~ O

, where

H 1 '2

class

k°

the

approximation,

v~Bndx

with

= /~

. By

,

all

~+

= ~

-k,O}

k

_> k

. Consider

N

{v > k}

o

,

the

measurable

sets

9 ° : D N { v : k}

Then

By

n(x)

= O

~(x)

: v(x)

a well

a.e.

known

-k

on

~-

= O

result

(x)

,

on

due

= v(x)

-k

a.e.

~+

on

Ro

to M o r r e y

we

infer

that Q+

-

n(x)

= O

~ D(x)

a.e.

= 8 v(x)

on : O

~

,

~ n(x)

a.e.

on

n°

= ,

~ v(x) ~

=

a.e.

on

1,2,...,n

Thus faeB(x)Z =

on

f a~[3(x)~ ~+

account

~i ~

=

...

of =

Poincar6's . By

n~B~dx

= ~+a~B(x)~

v~B~dx

(6.6)

Sn N = O

assumption,

= laeS(x)~

since

on

that k

o

< _

v~sndx

~ A O

a.e.

inequality

n~Bndx

~

and . Since

n(x)

R2

= O

, and

is,

U

is

contained

in

° 2(9 6 H I' ,~) n 6 HI'2(~,~)

a.e.

on

therefore

su%lu(x) that

n

~ O

BR(p)

i -< R

,

Q

, or

. Hence we v(x)

we

infer ~ k°

obtain from a.e.

on

74

Now we values

% :

regular

ball.

over,

3X + M o f c l a s s

there

By Lemma

that,

t h e proof of Theore~

can complete

on

These

is a n u m b e r

6.1, w e

can

E(U)

÷ rain

account

CI

with

c a n be e x t e n d e d R'

find

> R

t(~X)

c BR(P),

to a m a p p i n c

such

a solution

in

1.3. W e c h o o s e a r b i t r a r y

that U

the b a l l

BR(p) BR(p)

ER, (p)

is a

. More-

is r e g u l a r .

of

ZR' (p) N {U - ~

o f the m a x i m u m

where

of class

boundary

principle,

6 ~1,2(~,~4)}

is c o n t a i n e d

in

~R(p)

. Hence

E (U t) __> E (U)

for

= eXPu(x)tP(x)

Ut(x)

vector

~1,2

field

alona

n L ~ ( d , m N)

U

, Itl ~ t O <<

, the

]x~e(U,~')dvol

4e(u,~)

Now we

turn

regularity. We

start

is g i v e n

by

~

is an a r b i t r a r y

of which

is o f c l a s s

restrict

H~ider

ourselves

that, o n

continuity

U 6 H I'2 N L ~ ( Q , ~ N )

= O

,

(6.3).

to t h e proof of Theorem We

by observlng

to p r o v e map

representation

~

. Thus

(6.4')

where

I , where

1.4. F i r s t w e s h a l l v e r i f y interior

to

the c a s e

account

o f the

n a 3 .

of Schauder

theory,

first derivatives

the r a n g e

of which

it is s u f f i c i e n t

of a weakly

is c o n t a i n e d

harmonic

in a regular

ball U(~)

First we show arbitrary x x°

that

point

element Green

coordinates of

X

on

function

is c o n t i n u o u s

G

(O,...,O) [2

. Let

known

~

. To p r o v e

continuity

, and set

a~

da 2 = Xa_(x)dxadx ~ = /~ X ~

. In

~

be

we

o f the o p e r a t o r L = -~

It is w e l l

on

at an

x of Q , w e for o n c e i n t r o d u c e n o r m a l c o o r d i n a t e s o small neighborhood Q* with smooth boundary, such that

in a s u i t a b l y has

U

c BR(P)

(see[23])

0 -< G(x,y)

{aaB3

that,

_< K I I X - y l

for

2-n

}

n _> 3 ,

,

x,y

G

satisfies

6 ~

the

line

consider

the

75 (6.7)

K21x-yi 2-n ~ G(x,y) ~xG(x,y)

if

[xl ,]Yl ~ P

S K3[x-y!1-n

and

B2p(O)

c 9*

x,y 6 ~

for suitable constants KI,K2,K 3 > 0 . For y ~ ~ and we define the mollified Green function G°(x,y) by

0 < o << I ,

C (x,y) = #B@(y)G(x,z)dz where ~sV(Z)dx . ~ s V ( Z ) d z , and Obviously, GO(-,y) 6 ~1,2 D L ~ (Q~,]R) G° (x,y) S K4 Ix-yl 2-n

(6.8)

B (y):=(x

6 I~n : Ix-yl

(we may assume that

-< d} .

K I : K~ )

and (6.9)

2~

a~6~sG°(''y)dx

= {B (y)~dx

for

~ 6 HI'2(~*,IR)

For the sake of brevity, we introduce the expressions (6.10)

fl(v) = aeSNlk(v)~

for any representation

v

vi~svk

,

q(v) = a~@~ v ~ B v - v l f l ( v )

of the given mapping

U

with respect to nor-

mal coordinates on BR(p) . By virtue of Theorem 5.1, we have (6.11)

2~

where m

< on

and

-@

BR(p)

aK(Ivl)e(u)

< q(v) _< 2/~ a@(l}vJ)e(u)

are the smallest nonneoative

, since

Iv! S 2R < ~ / ~

numbers such that

.

We fix a normal coordinate system around the center ball

BR(p)

and reserve the letter

u

e ~ K

p

of the regular

for the representation of

U

with respect to these coordinates. By virtue of (6.3) and ~ , gik(v)ae@~

(6.4), we have

vi~@ @k + Fik I (v) ae5 3evl~ 8vk@l] dx = 0

for all 9 6 ~1,2 n L~(N*,~ N) and for each normal representation v of U If ~ 6 ~1,2 N L ~ ( ~ , ~ N ) , we can take ~k = akJ(v)~ j , and, because of ~59 k : gkJ(v) 98~J +gkJ,m(V)$@vm~J

,

we get [... ] = a~6~ v i $ 5 ¢i +a~5
76 Moreover,

gik gkj = @Ji

implies

o~ik k j,m = -gik,m gkj

, and

r gik,m =Fmki + ~ m i k = gkrFmi + F m i k hence

'

• ~j gik,m gk3 = mi +Fmik gk3

If we combine these identities,

we arrive at

['''] = a~53~ vies %~ - fl(v)%l That is, each normal representation

~*fa~B3

(6.12) for all

v

U

satisfies

v±~B~idx = ~.fl(v){idx

~ 6 ~1,2 N L ~(9*,RN)

Consider now

(6.12) for the particular

centered at

p , and let

~ = uGq(.,y)

q(u)GO(.,y)dx Since

of

lu(x) I < R < ~zv~

= _ 1~ ae8~

for

representation for

u

of

U

that is

y 6 ~* . Then we obtain

[u[2~8GO(.,y)dx

x £ ~* , we get

a (lul) -> a (R) > O K K and, by virtue of (6.11), (6.13) Let

4a<(R)4.e(U)GO(',y)/~dx~.

w 6 HI'2(~*,R) aeB~ w ~ % d x

-f~.ae6$e~lul2~GO(',y)dx

be the solution of

= O

for all

~ 6 C~(~*,l~)"

,

w - ' 'luL2 6 H1'2 (~*,m)

C

Then 4a (R) f e ( U ) G C ( . , y ) ~ d x ~ and, on account of

(6.14')

S a~85 {w-[uJ2)~sG~(',y)dx

(6.9), we arrive at

~*Se(U) G° ( ' , Y ) ~ dx<_ #Bq(y)IW-lUI2)dx

Invoking Fatou's lemma and a theorem by Lebesgue, we find

77

(6.14)

4a<(R)~

On account with

e(U)G(',y)/Tdx_<

of the m a x i m u m

a lower

w(y) -[u(y)I 2

principle,

semicontinuity

we have

argument,

w(y)

we arrive

R2 ~ e(U)G(.,y),~dx< ~2~ - 4a< (R)

(6.15)

for a.a. _< R 2

y 6 ~*

on

~ * . Together

at

for all

y 6 ~*

In particular,

(6.16)

lira ]Br(O)e(U)G(-,O)/-y r+O

Now we choose

r

o

> 0

such

that

B4r

dx : O (O) c ~*

. Set

o T2r = B2r(O)

- B r(O)

,

O < r < rO

~r = { T 2 r U ( X ) d x where

u(x)

Consider starting troduce UI~

is the s t a n d a r d

new normal

Then w e apply

coordinates

coordinates (6.12)

!yi < r / 2 , o

l~nl S c/r (6.17)

with

c

to

centered

will

v = vt, r

< q < I-r, independent

S a~6D~(Ivl2q)~

first

integral

to

U

centered

at

Pt,r by

,aa~

a~SG ~(',y)$ ~B

(6.18)

n ~ I

on

~ G ~ ( •, Y ) Ivl 2dx vl~sqvldx

(y) Ivi2dx

if

= O

0 < J <-dist2(pt,Pl)

]

we get )

_< 4 R l d i s t < 4RIdist

of

~ = vG~(-,y)q

o In the s e c o n d

p .

v = vt, r

q : Dr 6 C ~ k B 2 r ( O ) , ~ ) , of r . We o b t a i n dx-~

at

; the r e p r e s e n t a t i o n

and to t h e ~ / \ t e s tvector

Gq(-,y)

is equal

of

, O ~ t ~ I , on a geodesic normal coordinates tu r . We in-

be denoted

+ 2S q(v) G ° (-,y)qdx + 2 f The

representation

the points Pt = Pt,r 6 BR(p) at p that have the standard

in these

where

normal

(

U(x),p t U(x),PI>

)

- d i s t ( P t , P I)

,

Br(O),

78 Moreover,

formula (5.17) of Lemma 5.1 implies that dist U(x) 'Pl

<- be(R) iu(x) -u: I

Thus there is a constant

such that

i < U(x),p t > - dist2(pt,Pl ) ,i <- c~u(x)i -u}i! idist2

(6.19) (in fact,

c > O

I

c = 4Rb (R) ).

In view of (6.9), we obtain if a~BS~SBGC(',Y)[v!2dx-dist2(p t pq) I < cr-lfT2rl~xGO(x,y) I ]u(x )

--I

for some constant e (here and in the following, c will usually denote a generic constant that is not specified). Moreover, the fourth integral in (6.17) can be estimated from above by

~

11/2;.

ao-

2

l 1/2

C{/ G ~ (',y)e(U)~ T x i lJ (.,y)r- dxf ~2r " T2r on account of Schwarz's inequality and of (5.17).

By combining the last two estimates with (6.17), we obtain for that Iv(y) 2 _< dist2(p1,Pt ) -2{ q(v)G~(.,y)dx [

+ cr -If

~ -~ O

!

i8xa(X,y) ilu(x) uldx

T2ri

~~

f

+ c {~T2rG(.,y )e (U)~fdxll/2< f G(.,y)r -2~dxf~1/2 " T2r Poincar6's inequality implies that r2 U(X) -u i dx _< cr2f T2r T2r

f

18u'j2dx

whence, by (6.7) and by Schwarz's inequality, _

!

I~

r IfT2r!~xG(X,y)] iu(x) -uldx < j _ C{T~ Here we have used the assumption IY[ < r/2 also implies ]" G(-,y)r-2dx _< c T2r and

__

I I/2

e(U)G(',O)/y dy'j 2r that, on account of (6.7),

fT2rG(',y)e(U)~y-ydx ~< C~B2r(o)e(U)G(-,O)/~y dx

79

Thus w e a r r i v e

at

~v(~) ~2

(6.20)

dist2(pl,pt)

-2/ q(v) G ( - , y ) d x B2r(O)

]1/2 dx~

f L B 2r (0) Let

y 6 Bp(O)

have

]xl ~

and

0 < p < r/2

Ix -Yi + IYl ~

Consequently,

by

Then,

for

x 6 B2r(O) - B 2 p ( O )

, we

Ix -y' + p ~ 2!x -Yl

(6.7),

e(U)G(.,y)~/~dx

-
B2r-B2p

f e(U)G(-,O)q/~ B2r-B2p

-
e (U)G(. ,O)n¢~

dx

dx

B2r w h e r e we h a v e

set

f q(v)G(.,y)dx B2r

B s = B s (O)

for a r b i t r a r y

s > O . Moreover,

= ~ ... + f ... _> 2 a K (, s u\p l v B2p B2r-B2p B2p - cf

by

(6.11),

]f e ( U ) G ( . , y ) q / y /B2p

e(U)G(.,O)~-

dx

dx

B2r Set

(6.21)

£(r)

= S e(U)G(-,O)/~ B2r (O)

On a c c o u n t of

(6.16), w e h a v e

lim £(r) r÷+O the i n e q u a l i t y

We infer

from

(6.20)

(6.22)

Iv(y) I2 <- d i s t 2 ( p l " P t ) - 2aK

dx

= O .

e(U)G(.,y)/~

dx

2p + c{E(r)+

for a l m o s t that,

all

for s m a l l

y 6B

(O) , w h e r e O < p < r/2 . H e r e c denotes a number P r , is i n d e p e n d e n t of t,y , and p , where O ~ t ~ I o

Next w e c h o o s e

an a r b i t r a r y

W e c l a i m that,

for all

r/2

s/~}

number

t 6 [O,1]

R'

with

R' > R

and

and all s m a l l e n o u g h

a<(R') r

and

> O p <

, ,2 = , iv(y) I iVt,r(y ) ]2 ~

To this e n d we i n t r o d u c e

the L i p s c h i t z

(R,)2

for a.a.

continuous

y 6 Bp(O)

function

Q

.

80

h(t) Obviously some

h(O)

t 6

lim s u P B 'Vt,r(y) ] p÷O p

~ R < R'

[O,1]

Choosinq
:= lim sup r+O

. Hence,

, t h e r e w o u l d be a

6 > O

such that

a

h(t)

r

> O

we

and all

p÷O

[vl 2 -< d i s t 2 0

is s m a l l

enough,

thus

leads

< R '2

since

lim s(r) = 0 r÷O

contradiction:

lim suPB IVt,rl2 p÷O p

R '2 = h 2 ( t o ) = l i m sup r+O

This

h ( t o) = R'

sup~ i i oZO Vto,r , for some Po(r)

p

the d e s i r e d

Having

for

find t h a t

O < Po(r)

_< R 2 + c { s ( r ) + ~ / ~ ( r ) }

reach

s u c h that

R'

(6.22) y i e l d s

l i m SUPB

provided

t 6 [0,1]

were greater than

t o 6 [O,1]

(R'+~)

<

for all s m a l l e n o u g h

>O . T h e n

if

,

proved

that

h(1)

~ R'

we r e t u r n to

. Hence we

< R,2

(6.22)

and take

t = I

to lira s u P B :V1,r 12 ~ c ~ ( r ) + ~ ( r ) } p +O p <

whence

lim sup dist(U(x),U(y)) p+O x,yEBp

~ lim]suo dist/U(x),pl)+sup dist
Thus Note

U

is c o n t i n u o u s

that w e h a v e

at

x

o

not e s t i m a t e d

the m o d u l u s

u s e d the p r o p e r t y but,

lim e(r) = 0 w h e r e r+O so far, w e h a v e no w a y to c o n t r o l

s e c t i o n we s h a l l

acquire

= 0

sufficient

of c o n t i n u i t y

S(Xo,r)

also d e p e n d s

this d e p e n d e n c e .

knowledge

s i n c e we h a v e

on

e

In the

in o r d e r

on

x o following

to d e r i v e

a priori estimates. The n e x t s t e p is to prove H ~ i d e r

continuity

L e m m a 6.2.

representation

U :~ +

BR(P)

M

,

, with

Suppose

u = u(x)

Let

that

the

U(~)

range

respect

u(x)

(6 .23)

be

the

of which

to n o r m a l is

defined

OSC B

is

coordinates for

of

U . This

of

contained

a weakly in s o m e

BR(p)

on

x 6 B 2 p ( X O)

(Xo)u -< R/bc0(R)

, -e >-- O , and

e _< K _< K

on

regular at

map

ball p

.

, x O 6 ~ , a n d s a t i s f i e s ~)

*) K >- 0

from

harmonic

centered

20 As usual,

follows

BR(p).

81

Let.

noreo~)er,

with

local

p

coordinates

(6.24)

on

be small on

enough

B2p (Xo)

there

and

exist

n.N.R.k.~.~

~ 6 ~n

positive . and

< . such

e(U) IX-Xol2-ndx •[ Br(X O)

(6.25)

e(U) iX_Xo[2-ndx

(6.26)

that

x

can be

identified

that

<_% ~I[~ 2

, where

numbers

so

a ~ . and s u p p o s e

h!~I 2 _< a a ~ ( x ) ~ a ~

for x 6 B2p(Xo) O < I_< ~ . Then

> 0

the

l

constants

o.c.c*

. and

and

that

c**

V

satisfy

depend

only

that

< c(p) 2u[ e(U) fi x - x B P (xO)

'2-ndx

-

_<

c.(r~)2~

for all

0 ]

r 6 (O,p]

Br(X o ) and

(6.27)

[u]o,Bp(Xo)

(Here,

c**p -~

[u]o,Bp(Xo ) := sup{lU(X)-U(Y ) I : x,y 6 B (xO)

!x_yl~

~

x ~ y}

.)

'

Proof. Fix some number r with O < r ~ p , and let G(-,y) be Green's function for the operator L = -~B [ a a ~ a ] on ~* = B2r (Xo) . Then the positive constants K 1 and K 2 in (6.7) can be chosen independently of

r ~ I ; that is O ~ G(x,y)

~ K1[x-y] 2-n

O ~ GS(x,y)

(6.28)

~ K11x-yl 2-n

K21x-y[ 2-n ~ G(x,y) Moreover, (6.29)

where

for

x,y 6 B2r(X o)

,

for

x,y 6 B2r(X o ) and

for

x,y 6 Br(X O)

e < 2 r - lY-Xol

we have fT2rl~xGe(x,y) 12 L i ~ K4 r2-n for

iY-Xol ~ r/2

i

T2r := B2r(X O) - B r ( X o) , and fB2r(X O)a a~9 ~96G C(x,y)dx = ~B¢ (y) ~ (x)dx

(once again,

GE(x,y)

= {B (y)G(x,z)dz

is the mollified

Green function).

Choose q £ C c B2r(Xo),m with O _< q _ 7r/4

. T h e n , ~n(x) T*2r

:=

{x

in the exterior

~ O

:

~

.Ix - x

<

°

[

of the domain

-~7r~

<

Set Ur and let

p

(centered respect

that has

c M

centered

)

U(x),p

at

S boj(R)

inequality

u

as its u-coordinates

r

the r e p r e s e n t a t i o n

v : v(x)

coordinates

from Poincar6's

u(x)dx

BR(P)

p ). Denote by

to normal

(6.30) with

~T2r

be the point of at

i v ( x ) ~, = d i s t we infer

=

of

U

with

p . Since

i

u(x)

-Ur!

that

~T2rlV(X) ]2dx A c o n s t - r 2 • ~ l~u(x) l2 T2r

a constant

independent

of

r . Now we test

(6.12) with

% ~ G a ( - , X o ) , O < e < r , thus o b t a i n i n g

aC~,B~ { ~,V , 12} q(v) G£ (.,Xo)~dx + / i 9 BG s (-,Xo)dX = I + TT B2r (xo ) B2r(X O )

2f

where I = f

a~6$~BGs

= -S 2a~Svi~ T2r Assumption Then,

by

(6.31)

For

and

~iv(x) ] _< R

~ (x) --- I

on

O < s < r , the m o l l i f i e d

Green

~ g~[3~ = O

B2p(X O)

(cf. T h e o r e m

function

for all

-< I +II

m = G s (-,Xo)

~2 ~T~r,lV ] 2 ~GS(.,Xo) I dx 2 G8

(',x O)

12

dx+/

GS(',x O) T2r

satisfies

o ~ 6 HI'2(T2r,fR)

r~ = mlv12< _ ,~2 , w h e r e ~ 6 C ~(T2R,I9) c ' 6(x) ~ I , and by applying the usual C a c c i o p p o l i reasoning,

-< const{r-21 Iv] T2 r

5.1).

,

dx+ # ivlzdx B c (x o )

c~6 fT2r a

19~I _< c/r

on

B E (xO)

2a<(P0] ~e(U)GS(.,Xo)/y B2r (xo )

By i n s e r t i n g

,

v i ~ s D G £ ( - , x o)dx

(6.23) implies (6.11)

the e q u a t i o n

that

(',x o) Iv I2dx

2]

Svl 2dx

}

on T*2r ' we o b t a i n

83 On account of

(6.28) and

(6.30), we then get 2

r 2 (2-n)

~2 ilvI2iDGS(',Xo){~ I dx <_ const r

~

~~

2

~u'

+ i~v2}dx

T2r

On the other hand, IIl-< c ° n s t [ r - 2 ~ - I f T 2 r [ V ] 2 d x + ~ r l V I 2 1 ~ G s! ( '2 ' dx °x) ] iJ for each

6 > O . By choosino

6 = r

n-2

, we infer that

E [i ~2 + [Dv} 2} dx iIl ~ const r2-n(! ]i~ui T 2r if we take once aoain (6.30) into account. Moreover,

we derive

from (6.28) and

(6.30) that

lZ [ ~ constf [~vi,G/GC(',Xo)r-][v],J~(',Xo)dX T2r 2r T2r const r 2-n ~ {l~ul 2 + i~v!2~dx T2r Hence it follows

from

(6.24) and

e(U)Ge(.,Xo)dX

~ const r2-n~ {l~u[ 2 T2r

Br(X o ) As

(6.3]) that

+L~vl2}ax

s ÷ O , we infer that e(U)G(-,Xo)dX

_< const r2-nf ~ ~ul 2 + !~v!2}dx T2r "

B r (x o) Since

lu(x) I <_ R

and

Iv(x) :~ < R

f e(U) IX-XoI2-ndx Br(X o)

A C •

for

, we get

f e(U) IX-Xo!2-ndx B2r(Xo)-Br(Xo )

if we take Theorem 5.] and the estimates Here

x 6 B2r(X o)

on Green's

,

function into account.

C

is independent of Xo,r , and U , but depends only on n,N,l, Let us add C'fBr(xo)e(U) [X-XoI2-ndx to both sides of this inequality, and divide the result by I + C . Then we obtain ~(r) ~ @~(2r)

where

@ = C/(I+C)

6 (O,1)

, and

~/(R) :=B/(Xo I x - x O)e(U) l2-ndxR

"

84

Iterating

this inequality,

we ~et

~(2-mp) For

r 6 (O,p]

Choose

< @~(p)

, there is an integer

~ > 0

such that

@ = 2 -20

m A 0

such that

~ Since

~

2 -~-I

< r/p ~ 2 -~°

is nondecreasing,

we see

that ~(r)

and

(6.25)

Let now

< ~{(2

G(-,y)

estimate

Finally,

be the Green function of

Let now already

and

L

U

follows (6.26)

from

(6.25)

if we take

together with Morrey's

be a solution of the m i n i m u m problem

fied for each

U

is continuous

x° 6 ~

provided

on

that

. In view of

(6.15)

Dirichlet

U

into account. growth theo-

(6.23)

is satis-

is small enough.

Hence

on

of the first derivatives

loss of generality we can assume that

tions of Lemma 6.2.

. Since we have

is H~ider continuous

We finally will prove the H61der continuity U . Without

(P)

~ , condition

p(x o) > 0

we can apply Lemma 6.2 and obtain that

U

~ . of

fulfils the assump-

Then we have

f aeB(Xo)~ BQ (xO)

ui$B~idx = f [ a ~ B ( x ) - a e B ( X o ) ]~ ui$8%idx Bp (xO) + /

(

for all

B2p(X O)

(6.27), and the proof of Lemma 6.2 is complete.

shown that

(6.32)

on

e(U) IX-Xoi12-ndx ~ f e(U)G(X,Xo)dX B2p(X o)

(6.26)

(6.25)

rem imply

< 220~(Q)(r)2o

is proved.

KIBp(Xo) the

0) < @ ~ ( p )

~ 6 H I ' 2 n L ~ B P(X O) ,19N

)

fl (u) ~idx

Bp (xo) . Secondly, there is a number

cI

such that

(6.33) Let

w

(6.34)

Ila~(x) -aaB(Xo) ! -< clO be the solution of the Dirichlet

-~[a~B(Xo)~

w] = O

in

for all

x 6 Bo(X O)

problem

Bp (xo)

,

w = u

on

~Bp(X o)

From the linear theory of elliptic operators with constant coefficients, the following results

are known:

85 (6.35)

W £ L~
and

supB (Xo)iwl

-< s u p B

P

P

(6.36) (6.37)

wcc°'

(

(Xo)S

)and

wl ,B

(Xo) lUl

-< c2[u]o,Bp (xo)

t

[~wl2dx~ -< e3(r/p)n f ~l~wl2dx for all r 6 (O,p) Bp (xo) f2 2 ~w - (~w I dx iSw -(~W) r dx ~ c4(r/pln+2~ I P Br(Xo) I Bp(X o) f

and

B r (xO ) (6.38)

where we have set (~w) r = (~w)

= ~ ~w(x)dx Br(Xo)

Xo'r

From (6.37) we infer that (6.39)

+ ~u-~wl2dx f iZul2dx <- c5(r/p)n~ !~ul 2dx c5~ (Xo) B r (xO) B@ (xO ) P for p 6 (O,r) (6.32) and (6.34) imply that, for

Equations (6.40)

~ 6 HI'2 A L~(Bp(X o) ,I~N)

[ aaB (Xo) ~ (ui-wi) $ 8 ~idx Bp (xO) = f

fl(u)~Idx

[a~S (xo) -a ~B (x)]~ ui$~cidx +

B 0 (xo)

Bp (xO )

From the estimates of Lemma 6.2, we obtain (6.41)

suPB (Xo) [u-w I ~ c6P p

Then, by inserting lity, we arrive at

~ = u -w

f laul2dx < n-2+2~ (Xo) - c7P Bp

,

in (6.39) and by applying Schwarz's inequa-

n-2+3~ f ''13u_~wi2dx ~ CsP Bo(X O )

(6.42)

,

if we take (6.33) and (6.4]) into account. Together with follows that (6.43)

f ~ul2dx ~ c9{(r/p)n s !$ul2dx+p n-2+3~} Br(X o) Bp(Xo)

Without loss of generality, we may assume that derive from the following lemma that (6.44)

f Br(Xo)

uE2dx

for all r 6 (O,p)

n -2 +30 < n . Then we

elorn-2+3of. +p -(n-2+3o) f •

(6.39), it

Bp(X o)

i~u[2 }

for all r 6 (O,p)

86

Lemma

6.3.

Let

~(r)

be

a nonnegative

and

nondecreasing

function

that

satisfiea

<0(r)

with >0

_< A [ ( r / p ) a

A,B and

~ O

+slop(p)

and

c(a,~,A)

+Bp B

for

0 < B < a 2 0

such

. Then

some

p 6

there

exist

for

Proof.

all

For

r

with

O < p ~

0

Po

~T Y

some

X

• If we

with

set

< r s

and

O

B < X

so : T~

Iterating

this

p ~

Po

< T <

k

c'

= O

For

k+l T

=

< e

, then

TY~(p)

inequality,

<0(~kp)

where

~

kB

<

c

(TB-TY) -I

r 6

=

numbers

a °

that

s < so

" provided

I

we

, and we

, we

(O,p]

ao(a,5,A)

~(p)+

have

+Bp ~ then

get

T E

for

O

such

(O,1)

,

provided

obtain

for

each

that

integer

that

2A~ ~

that

< s < ao

+Bp 8

_< T k Y q 0 ( p ) + B p S T

_<

all

~}

M(Tp) -< ATa[1+87-a]M(p) Choose

and

that

~p(r) _< c { ( r / p ) B ~ ( p ) + B r holds

(O,Po]

O

k

 O

Po

that

k-1 ~ TJ(~-8) j=O

(k-1)B

<5--~<X

+B0 >

I . Evidently,

this

estimate

also

holds

for

.

any

r £

p < r

(O,p]

~ Tkp

, there

. Then

T

exists

(k+l)B

a nonne~ative <

(r/p

)B

integer

k

such

that

, and

(kp) <

+Br B} -6

Thus Let ning

the us

return

that

(6.45)

asserted

led

to

inequality inequality

(6.44),

f B r (x o )

we

[~u]2dx

follows (6.44).

derive

the

with

c = c'T

Repeating

suitably

often

the

reaso-

(O,p)

and

6 [

estimate

) n-2+28 < c11 (S r

for

all

r

[

(O,1).

87 Then,

if we once

again

fB for all

P

insert

(x o)

[ = u -w

I~u-Sw[2dx

in

(6.32),

s c12(~)p n-2+3~

hand, 2

f

f

dx -<

i~u - (~u) r

Br(Xo)l

~2 ($W)r!l dx

DU-

B r (x O )

~2 l~u -~wl 2 dx + 2f B r (xo )

_< 2f i~w-(~W)ri B r (x O ) (6.38)

yields 2

[

fBr

that

~ 6 [O,1)

On the ether

and

it follows

(Xoli

i 8w

-

(~w)

dx <- c 4 ( r / p ) n + 2 f

r

i~w-(~w)

BO (Xo) ,

2 I

Pi

Finally, 2 f I S w - (~W)p B@ (x O)

_<

dx -< ~2

12

2f Bp (x o)

<_

~w-~u-

~u- (~U)p! dx +2f

Bp(X o) i

[$w-3u[2dx

Pl

B p ( X o)

~(r)

:= f $uBr(Xo )

Thus,

is a n o n d e c r e a s i n g ~(r)

Choose

dx

Bp (x o )

!~u-(~u) !2d x + 2 f

2f

($w-$u)@

function

_< A(r/p)n+2~(p)

some

B 6 (2/3,1)

on

[O,p]

+ B p n-2+3~

. Then

(~u)

i2 } dx ri

that

satisfies

for

0 < r -< p

n < n - 2 + 35 < n + 2 , and Lemma

6.3

yields <0(r) < cr n-2+3B that

B +p - (n-2+38)

is, ,2 f i ~ U - (~U) I dx -< const Br (Xo) Xo,r i

for all H~ider

r 6

(O,p]

continuous

continuous

on

Q

and

functions

a priori

estimates

due to C a m p a n a t [o

6.2, for

n-2+35{

> O . A well

with exponent

As in the proof of L e m m a into

p(x o)

r

known

! } 1+ [ ~ul~2dx ~p (Xo) characterization

implies

that q

y = (I/2)[(n-2+38)-hi all of the last e s t i m a t e s

VU

. Note,

however,

that

VU

= 3B/2

of

is H~ider ~

I

>

O

can be c o n v e r t e d in e x p l o i t i n g

88

(6.40),

we have

used

a hound

of the

form

)f(u) I ~ constl Su] 2

that

requires

an e s t i m a t e

of

the

form

1 (u) i < c o n s t [Fik i If we w i s h cannot nates 131, The

that

the

use n o r m a l that was

Lemma

last

introduced

harmonic

mappings

that

we have

nowhere

map, The

the same

U

; it w a s

range

are

on

to e m p l o y [50];

R,N,m, another

cf.

solutions

and type

also

<

, we

of c o o r d i -

[20],

proof

that

is c o n t a i n e d

boundary

U 6 C°(X,M) to y i e l d

regularity

of the m i n i m u m

to k n o w

of w h i c h

be e x t e n d e d

the i n t e r i o r

in the r e g u l a r i t y

for p r o v i n g

to s h o w i n g

can easily

complete

sufficient

U(~)

is true

ourselves

depends

in J o s t - K a r c h e r

considerations

Actually of

only

but h a v e

pp.

130-

2.

previous

perty

constant

coordinates

in a r e g u l a r

although

the

of

U

of

pro-

harmonic

ball

BR(p)

. We r e s t r i c t

previous

continuity

for

(P)

the m i n i m u m

is a w e a k l y

regularity

H~ider

problem

used

U

proof

U

techniques and

VU

at the

boundary. Let

xO

choose p

of

be

an a r b i t r a r y

a geodesic

arc

the r e g u l a r

ball

point

Pt

on

~X

, and

set

' 0 S t ~ I , that

BR(p)

Po = p

'

which

q = U(x o)

connects

contains

U(~)

q

. We t h e n with

the

center

, and w e a s s u m e

that

Pl = q

Thus dist(p,pt)

For

each

t 6

[O,1]

of t h e b o u n d a r y

and

value

Lw = 0

where

v = Vxo,t

normal

coordinates

Well that

known w

Repeating

in

= vt

r

small

Sr(Xo)

is the

centered

results

the

for

enough,

we consider

the s o l u t i o n

problem

from

is c o n t i n u o u s

= d i s t 2 ( U ( X o )'pt)

= t dist(p,q)

at

,

representation Pt

the t h e o r y at

~X ~

w - Ivl 2 6 H 1 ' 2 1 >S r ( X,o ) ,]9

' and

of l i n e a r

S r ( X O)

of

S r ( X o)

w i t h r e s p e c t to r := ix E X : d i s t ( x , x o)

U

elliptic

p.d.e.'s

; in p a r t i c u l a r ,

"

argument

,

tell

lim w(y)

us

= w ( x O)

Y+x° that

led to

(6.14'),

and u s i n g

(6.11),

w e see t h a t

89

(6.46)

f
where

G~

is the m o l l i f i e d

~-2a

Green

¢sup [Vl)S~ e(U)G~(-,y)/~ < ~Sr (Xo) r (Xo)

function

of

Sr(X o)

Let

h(t) This

function

number

R'

for all > O have

is c l e a r l y

Lipschitz

such that

t £ [O,1]

such

:= lira sup!vt(x) I x-~x o

that

R < R'

. In fact,

h(t O)

= R'

a {sup iVto])
continuous,

and

a (R')

if this w e r e

. Hence

> 0 , and

for

(6.46)

and

not s o , t h e r e

r

small

If

that

(y) { O I

< W t (y) O

h 2 ( t O) -< Jim sup w(y) Y+X O a clear

h(t)

w o u l d be a

enough,

some

<- R' to

r > O , we

_< 0

for all

v 6 int S (x O) : r

= d i s t 2 -< R 2

contradiction.

Thus we have

h(1)

N R'

and hence,

a {sup Iv II) > O . Hence we K k S r ( X O)

for

find

Wl,r(y)

÷ dist
d i s t O

for s m a l l

Iv1(Y) [2 S Wl,r(y) Since

Choose

12

Ivt implies

.

(y)

o ÷ O , we o b t a i n

which

< R

yields

{!V(X)!2 -w(x)}dx B

h(O)

> O . We c l a i m that

for all : 0

as

s m a l l enough, r

that

y 6 int S r ( X o)

y ÷ x°

and

Iv I (Y) I =

at

lira dist( (y),U(Xo) ) : O Y+X O that is, COMMENTS

U

and R E F E R E N C E S

The r e s u l t s Widman

is c o n t i n u o u s

[34].

of this

section

The M A X I M U M

[29] by H i l d e b r a n d t

mates

of sections

o

6 ~X

have first b e e n p r o v e d by H i l d e b r a n d t - K a u l -

PRINCIPLE and Kaul.

has e s s e n t i a l l y Besides

been derived

the basic

4 and 5, the m a i n t e c h n i q u e

Jacobi

in the

field esti-

for its proof has b e e n

by S t a m p a c c h i a .

The r e g u l a r i t y tained

x

to the L I T E R A T U R E

paper

developed

in

theorem

in a r e g u l a r

for h a r m o n i c

ball,

has

mappings, the range of w h i c h

first b e e n

proved

in

[34].

In this

is consection,

90

we have d e s c r i b e d to employ derived Widman

the techniques

estimates

for the Green

[24].

pointed

general

The b a s i c

ditions

strategy

K ~ ~

sumptions,

of weak

a weakly

is proven,

lows ~) d i r e c t l y

f :

VU

Academic

from

is H ~ I d e r

from

iteration

pp.

procedure

the b o u n d a r y

of Green's

to the o r i g i n a l

approach

function.

Once

conditions

on

the c o n t i n u i t y

systems

fol-

developed

8.

given

various

as-

Linear and Quasilinear in

[30];

the present

possibilities

of this s e c t i o n

[20]

to show

is b a s e d

also elaborates

estimates to Tomi

on

Still

another

method

and Uralzeva,

inequalities

another

in the i n t e r i o r

[84] as w e l l

and at

as e s t i m a t e s

that is closer

has been

given

for vector-valued func-

und Angew.

Math.

309,

in S c h o e n

[75],

section

Analyticity

Math. Phys. 7!,

these

they a c t u a l l y systems

not furof the

under suitable

elliptic

Chapter

of L a d y s h e n s k a y a

to the s t a t e m e n t s

Comm.

~)although

con-

192-220

(1979),

in par-

204-211.

Borchers/Garber,

model,

to h a r m o n i c

and of G i a q u i n t a - G i u s t i - M o d i c a , and is

gradient

Reine

~)contrary

the

one-sided

and its d e r i v a t i v e s

6.2 were

The paper

tions, J o u r n a l

cular

1968,

the v e r s i o n

Variational

pp.

as

how

in g e n e r a l

that,

in their m o n o g r a p h

in H i l d e b r a n d t - W i d m a n , f~r die

U

of n o n l i n e a r

an old idea due

on the g r a d i e n t

of

There e x i s t

to derive

However,

obvious

one-sided

is continuous.

of C a m p a n a t o

that uses

will

is to show

U

Press

160-163.

proof has been w o r k e d

in general,

just implies

regularity

continuous;

technique

[20],

[20].

operators and Gr~ter-

to be a p p l i c a b l e

...,fN)

is

large).

map

and U r a l z e v a ~ )

[86],

ideas

Lu = f ; yet an a s s u m p t i o n

proof

from the theory

taken

the powerful

ticular,

(f]

of

(in the

harmonic

in order

ideas of the proof of Lemma

v e r s i o n was

taken

ones

the h i g h e r

Elliptic Equations, The m a i n

side

elliptic

[30],[87],[88],[31].

show that,

solutions

in the r e g u l a r i t y

by L a d y s h e n s k a y a

that

of linear

[60], W i d m a n

I, it is by no means

on the target m a n i f o l d

The key step

U

5 of section

on the right hand

One of the basic

for the r e g u l a r i t y

and W i e a n e r

counterexamples

and not t w o - s i d e d

of

function

h a s to be m o d i f i e d

In fact,

regularity

type f

in remark

technique

mappings.

nish

in [34].

hy L i t t m a n - S t a m p a c c h i a - W e i n b e r g e r

out by H i l d e b r a n d t - W i d m a n was

used

authors

adaptation

(1980),

p.

formulate

carry out their

in d i v e r g e n c e

case.

only

of solutions of the

proofs

their e s t i m a t e s for weak

U

: ~n ÷ SN

of

Rn

into

for C2-solutions,

solutions

harmonic

and Garber

of the L a d y s h e n s k a y a - U r a l z e v a

monic m a p p i n g s

23, and in

nonlinear o-

300

form that comprise

The m e t h o d of Borchers

3, p. O(N)

of d i a g o n a l

mappings

is b a s i c a l l y

technique

as a partian elegant

to the case of har-

the N - s p h e r e

SN .

91

The

"algebraic"

Campanato systems

Lem~ma 6.3 is d u e

technique

can he

and

found

to M o r r e y

of applications

and Campanato.

to g e n e r a l

in the p e n e t r a t i n g

lecture

Details

nonlinear

notes

o f the

elliptic

by Giaquinta,

Multiple integrals in the calculus of variations and nonlinear elliptic systems~Princeton U n i v . The on

regularity the one hand

other

hand.

ly e x p l o i t

of

H61der

[76,77]

[18]

been

reproved

to

property

as

first

of

U

and

later was

fairly

general

and J o s t - M e i e r

on Green's

that

then

result

maps

by Schoen-Uhlenbeck

[18]

avoid estimates

inequality

derive

harmonic

applies

1.4 has

authors

the m i n i m u m

stationary

1983.

a n d by G i a q u i n t a - G i u s t i

These

to a r e v e r s e whereas

theorem

Press

in case of

a monotonicity

proved by Price systems,

the

very much

tailored

to h a r m o n i c

mappings,

but

strongest

results.

In

[76]

with

the r a n a e

o f the m a p p i n o

it h a s

been

shown

mizing

map

U : X ÷ M

n = dimX

. If

that

fact,

in

can be

has

n = 3 , then

at most E

and,

covered

the s i n g u l a r

function

and

for

VU

set

for t h e s e

of

a single

E

of each

[51] l e a d s , p > 2 ,

(that,

for

the m e t h o d

[76,77]

it p r o v i d e s

further

by

the H a u s d o r f f

consists

formula

[69]). W h i l e

technique

the

and strong-

[18]

to L P - e s t i m a t e s

[76,77]

[51] o n the

is the

assumption

chart,

also

bounded

dimension

at m o s t o f i s o l a t e d

that

in

[]9],

energy

mini-

n -7

, if

points.

92

7.

A prio__ri e s t i m a t e s

The

first

aim of this

for h a r m o n i c

section

maps

is to p r o v e

Lemma

6.2,

it s u f f i c e s

monic

map

U : ~ ~ M , 9 = int X , that m a p s

of

M • We,

tinuity point

of

in fact, U

xo 6 ~

restrict

in a s u i t a b l y

coordinates

a point

x

domain

can

of

of

~

~

will

be

B2d

=

the m o d u l u s

our

small

. Then we can once

of l o c a l

The

to e s t i m a t e

Theorem

its

local

chosen

into

again

~

ball

BR(p)

of con-

of an a r b i t r a r y

that

shall

of

of a har-

a regular

to the m o d u l u s

neighborhood assume

By v i r t u e

of c o n t i n u i t y

attention

again

X , and w e

and

~

1.5.

~

carries

a chart

not distinguish

between

coordinates.

as b a l l

{ x : Ixl ~ 2d )

or as b a l l s

B 2 r ( X o) contained

in

We

all

retain

(6.12)

which

B2d

:

{ x : ix- Xol < 2r }

t h a t in the

the n o t a t i o n s states

that

following

of s e c t i o n

each

normal

is

thought

to be fixed.

6. M o r e o v e r ,

we

representation

v

recall of

U

equation on

BR(P)

satisfies

(7.1)

f

a ~$ Z

vi ~B
f

=

B2d We, in a d d i t i o n , i n f e r of

U

centered

(7.2)

at

from p

fl(v ) ~i dx B2d

(6.14)

that

the p a r t i c u l a r

representation

u

satisfies

4aN(R) f

e(U) G(-,y)

/~ydx

_< w(y)

-

lu(y) I

B 2r(Xo ) for all

y 6 B 2 r ( X o)

(7.3)

. Here

f

w

a ~B 3

is the

solution

w ~8 ~ = O

of

for

all

~ 6 C c~(B 2r(Xo)19)

B 2 r ( X O) w-

and on Set

G(.,y) B2r(X O )

denotes

the G r e e n

lui2

function

£ Hl'2(B2r(Xo),~)

for

the o p e r a t o r

,

L = -38{a~B~

}

83 2 M(r) The maximum

= sUPBr(Xo

principle

w(y)

Choose

Yo 6 ~ B r ( X O)

(R)

for

such

IXol

that

all

y 6 B 2 r (x o)

fu(y o) I2 = M(r)

e ( U ) G ( - , y o) ~ d x

. Then,

~ M(2r)

by

(7.2)

,

- M(r)

B2r(Xo )

of

(6.28),

K2ix-

and,

t _< 2 d -

implies

f

K

account

then

for

K M(2r)

4a

On

) lul

we

yl 2-n

have

~ G(x,y)

for

x , y 6 B r ( X o)

,

therefore,

(7.4)

r

2-n

e(U) / ~ d x

f

{M(2r)-M(r)

<

Suppose

that

B 2 r ( X o)

]

,

(n ->3)

4 a K (R) K 2

B r (x O ) c B2d

, and

M i = M ( 2 - 1 r o)

set

,

i = O,1,2,...

Then Po E i=I Thus For

we may each

io(P0)

(Mi_ I - M i )

M ° - Mp

< M

-< R 2

O

conclude: , there

Po = 1,2,...

SPo

=

such

exists

an

index

i O = io(Po)

with

I <

that

Mi

-1

- Mi

O

<

R2/po

O

-i o Then,

r = 2

for

ro

r

2-n

, we

infer

S

from

e(U) / ~ d x B r ( X O)

Hence

we

can

Lemma

7.1.

there

exist an integer

(7.4),

_<

that

I

R

2

Po4a~(R)

K2

state

For every

>0 iO :

and

every

{I ,2,

x° , r°

with

B 2 r o (xo) c B 2 d

" " " 'Po }" P o = [I/e] + 1 , such

,

that

94

(7.5)

r

2-n

e(U) } ~ d x

f

R

_< ¢

2

4 a<(R) K 2

B r (xo ) -i o holds

r = 2

for

Remark.

ro .

If w e k n o w

that the f u n c t i o n

%(r)

: r 2-n f

e(U) y ~ d x B r (x o )

is m o n o t o n i c ,

i.e.

(7.6)

,

~(r)

then we i n f e r f r o m

-< ¢(r')

(7.5)

for

e(U) / ~ d x

c o n s tI

~

Br(X O ) Estimate

(7.6)

and by P r i c e curvature.

has b e e n d e r i v e d

(7.6')

}(r)

The estimate tonicity

We n o t e

in

X

formula

that

-< e cr' ~(r')

(7.6')

if

follows

slightly

(7.6')

is f a i r l y

the m o n o t o n i c i t y

formula

(7.6').

Lemma

u(x)

7.2.

Let

coordinates

be at

the p

L = - ~B {aa~ ~c~ ]

tion f o r

(7.8)

centered

B2r (xo)

in

f

O < r ~ r o -
for s u b d o m a i n s

constant Price

that

with

We,

1

-<

U by h

with

+

y 6 B 2 r ( X O)

will not use

respect

G(x,y) be

f

to

Green's

the s o l u t i o n

f(u) G(.,y) dx

local func+~

of

. Then

min {R2,osc

4 a K (R)

: h(y)

mono-

considerations;

h - u [ ~1,2(B2r(Xo),ikN)

B2r(X O ) all

(although

.

therefore,

of

denote

B 2 r ( X O) , a n d

on

and

for

Ikn ,

(7.5).

representation

. As before,

satisfies

o

the f o l l o w i n g

involved.

e(U) G(.,y) / ~ d x

u(y)

of

sectional

[69] has s h o w n

0
simplify

B2r (x o)

(7.9)

~

nonpositive

a l s o from the g e n e r a l i z e d

~n c o n j u n c t i o n

(7.7) w o u l d of

if

that

(7.7) o b v i o u s l y

y e t the p r o o f

Lh = 0

[36,37]

with

In c a s e of a g e n e r a l m a n i f o l d , stated)

,

log

[69] for s p a c e s

not e x p l i c i t e l y

_
o

that

r 2-n ~

(7.7)

0 < r _
lul 2} B2r (x o)

95 Proof.

According

to

(6.15),

R2/(4 a (R)) . The second conjunction

with

the left h a n d -

estimate

stated

side of

in

(7.8)

(7.8)

follows

is b o u n d e d

by

from

in

(7.2)

the i n e q u a l i t y w(y)

s

lUl

sup B2r(X O )

In order

to prove

identity

(7.9),

we insert

aC~B ~a ~ 3B GP(" 'y) dx

I

=

in

~Bp(y)~dx

B2r (x O ) for all the

test function u dx

@ = u- h , whence

=

~

Bk(y)

h dx

+

~

Bp(y) =

~

=

h dx

+

tends

holds,

if we also

~B GO(''Y) dx

dx

B2r(X o)

}

h dx

to zero,

- ~h}

a aI3 Zau 3~ G 0 ( . , y )

~

+

~

Bp (y) p

a ~5 { g u B2r(X o)

Bp(y)

As

<0 6 ~ I , 2 (B2r (Xo) ,IRN)

dx

f(u) G 0 ( . , y )

B2r (Xo)

we infer

from

take L e b e s g u e ' s

the results

theorem

of section

on d o m i n a t e d

6 that

(7.9)

convergence

into

account. Len~na 7.3. notation points

in

In a d d i t i o n

T2r = B2r(Xo) BR(p)

respectively,

to the n o t a t i o n s

- Br(X o)

c M , the

, and

of Lemma

p

Pt

,

u - coordinates

7.2,

we

introduce

the

, denote

the

, O < t ~I

of which

are

~r

and

tUr

'

where

Ur = f

U (X) dx T2r

Let nates

Vt(X) on

dependent

be

the r e p r e s e n t a t i o n

BR(p) of

centered

U , t, r ,

4 f

aK(IvtJ)

at

and

Pt

U

with

" Then

o~"

there

respect exists

to n o r m a l a number

x ° , such that

e(U) G(.,y)

¢~]dx +

Ivt(Y) i

Bp (x ° )

_<

(I - t) 2 R 2 + C* { [r 2-n

f

e(U) ~yy dx ] I/2 B2r (xo)

+ p

2-n

J" B2r (x O )

e(U) ¢'ydx 1 f

J

coordic* , in-

96

for

all

y 6 B p / 2 ( x o)

Proof. 6.2

Let

and,

q

for

be

the

= v n Ge(-,Y) (7.11

and

the

sake

0 < p ~ r .

same

cut-

off

of b r e v i t y ,

, y 6 B p / 2 ( x O)

4 ~

a B2r(Xo)

function

set

v(x)

, 0 < 8
as

in the

= vt(x).

, in

(Ivl) q e(U) G e ( . , y )

proof

Then

(7.1)

and

we

insert

obtain

dx + ~

<

of L e m m a

Ivl 2 dx

~ I + II

,

Be(y )

where aa~ ~a ~ ~B G s (" 'y) Ivl 2 dx

I = ~ T2 r and II = - S

2 a ~5 D

vi ~

TI v i G e ( . , y )

dx

for

O -< t-O

c lu(x) - J r I

usual,

c

will

in

from

(6 .I 8) a n d

,

x 6 B r ( X O)

,

the

following

be

constant. (7.10)

quite

. As

+

obtain

which

proceed

as

is n e a r l y

in s e c t i o n

the

same

6 because

as r e l a t i o n we

there

(6.22).

have

used

Yet the

bound

(7.13)

I$ x G ( x , y ) l

stated

in

weaker

a priori

but

also

Thus

(6.7).

on

the

estimate

some

we will

employ

If w e w o u l d

H61der

slightly

inequality

S

here

since norm

alter

of our

(6.29)

~ K 3 I x - yl 1 - n

apply

this

,

estimate, we would

K3

does

the

coefficients

approach.

not

depend a

Instead

only a5

of

on

(7.13),

get

a

1

and

we

shall

,

I~ x G S(x,y) I 2 dx ~ K 4 r 2-n

for

l y - Xol

~ r/2

,

T2 r which

is

estimates

nothing

but

following

In a d d i t i o n ,

we

shall

a Cacciopprali (6.31) apply

which

are

inequality obtained

for by

the

GP(.,y). same

(cf.

idea).

the

Z

,

97

0 < G¢(x,y)

as w e l l as P o i n c a r & ' s

_< K I I x - y l 2-n

~ c r2

of

T2r

(6.9) and

I ~

I~ul 2 dx

f

T2r By v i r t u e

x,y 6 B2r (Xo)

inequality

l U - Ur 12 dx

f

for

(7.12), w e h a v e

(I - t) 2 R 2

+ c r

-1

f

2 lu _ ~-R]

]DGS(.,y)

2

dx

T2r II ~ c r -1

IDvI 2 dx

G ~ (.,y)

f T2r

On a c c o u n t of S c h w a r z ' s w e then a r r i v e

inequality

a n d the p r e v i o u s l y

stated estimates,

at

I + II < ( I - t) 2 R 2

_< (I

f

+ C

r2-n

- t) 2 R 2 + c

(13ul 2 + 13vl 2) dx } 1 / 2

f

L

T2 r

r 2-n

~1/2

e{U) / ~ d x

fB2r (x o)

For

e ~ 0 , w e then infer f r o m

4 f (7.14)

a B2r (Xo)

(7.11)

that

(ivl) h e ( U ) G(.,y) / ~ d x

2 Iv(y) I

+

K

( I - t) 2 R 2

+ c

r 2-n f

e(U) / ~ d x

ll/2 f

B 2 r ( X O) holds

for all

have proved

y E B p / 2 ( x o)

in s e c t i o n

The i n t e g r a l

, because

U

is a c o n t i n u o u s

mapping,

as we

6.

on the left h a n d -

side of

(7.14)

is n o w d e c o m p o s e d

in the

f o l l o w i n g way:

f

...

=

f

B2r By H a r n a c k ' s

inequality,

the p a r a m e t e r s Theorem

1.5,

+f

.

.

.

B2r - Bp

t h e r e is a c o n s t a n t

n , N , ~ , 9 , ~ , and

~

which

K5

which

appear

only depends

on

in the s t a t e m e n t

of

such t h a t

G(x,y) Therefore,

... Bp

~ K 5 G ( x , x o)

w e find

for

x 6 B 2 r ( X O) - Bp (x O)

,

y 6 B p / 2 ( x O)

98

4 [

a

B o (x o)

(Ivl) e(U) G ( . , y )

~dx

+

Iv(y) I

< ~/2

_< (I - t)2 R 2

+ c ~ r 2-n

~

e(U) /~dx[[

%

Finally,

e(U) G ( . , x o) dx

C I

B 2 r ( X o ) - B D (x o)

we h a v e

O ~ G ( x , x o)

0 _

, which

Finally, L e t us

+

9

B2r

we

~ KI ix _ X o l 2 - n

then

turn

introduce

implies

~ KI p2-n

for

x 6 B 2 r (x O) - B p ( x O)

,

(7.10).

to the p r o o f

of T h e o r e m

I .5:

the n u m b e r T

I > O

,

2R /< and

se t

the

smallest

t I = < , t 2 : 2< integer

such

, ...

, tm_ ! =

that

m7 ~ I . D e f i n e

[ buo~K (,~j R ) ~/ 6 : m i n { .I-(I-T) 2.] R 2, ( 2R ~----rw~,

(m-1)7

~] _ '

, tm = I , w h e r e 6 >0

, ~ >0

m

is

by

£ : 2 2 - n rain { I, (

m(n-2)6 ]'2 ~ c~2

where

c~

is the c o n s t a n t

that

appears

in L e m m a

7.3.

Set

R2 k 4 a K ( R ) K2 By v i r t u e

of

Lerrm~a 7 . 1 ,

2o 6 {1,2,...,po}

'

Po

for =

each

[k/c]

ball

+ 1

,

B2r

such

(Xo)

that

c

for

B2d r

, _ itoh e r e

= 2

r°

equality (2r) 2-n

S

e(U) U ~ d x

< s

B2r(X O ) holds.

Then,

(7.15)

{

by d e f i n i t i o n

r 2-n [

of

s , we arrive

e(U) / ~ d x

<_ m i n

B 2 r (x O ) Introduce

now

the b a l l s

Bo,

BI,

...

, Bm -i

I -
{

I,

2m ( n - ~ ) c~

Bi = B p i ( X o ) For

at

'

Pi = 2

, we have 2-n (m-l) (n-2) 2-n Pi-1 < 2 r

by

S

is

an

the

index in-

99

Then L e m m a

7.3 implies

For

y 6 B i , I S i ~m

every

4 S

a

the f o l l o w i n g

estimate:

, we have

(ivtil) e ( U ) G ( . , y )

/~dx

Bi-1

I/2

_< (1-ti)2 R 2 + e*

-e(U) / x d x ]

[r 2-n f

2-n + 0i_ I f

B2r (x o) <

(1-ti)2 R 2

+ e * 2m(n-2)

e(U) / T d x } B2r (x O)

{ r 2-n S

e(U) / ~ d x 1 B2r (x O )

_< (~-t i ) Then

2R2

+ 6

it follows

from

4 S Bm_ I

the d e f i n i t i o n

a K (Iv t

of

6

that

I) e(U) G(. y) vJ~dx + m

Iv t (y) I m

(7.16) R

b

(R),2

K

~

for all

• bTr n )

y 6 Bm

and that 2

4f (7.1 7)

a < (Ivt. I) e(U) G(.,y) ¢ ~ d x

lvt. (y) I

1

Bi

< R

2

1

for We now c l a i m

+

y 6B i

and

i = 1,2,

...

...

,

,m .

that

(7.18)

a<(Ivt

L)

on

_> 0

Bi

,

i =

1,2,

,m

1

holds. First we verify

Vtl (x)

(7.18)

for

i = I

= dist(U(x),Ptl) -< R + r R

In fact,

< dist(U(x),p)

=

for all

+ d i s t ( p , p t I) x 6 B2r

2/~

(x o) o

and therefore

a<(IVtl i) -> O

Suppose

no~ that

(7.18)

holds

on

for

B2r O(x o)

, hence

i = j , I _< j _<m-1

also on

. Then

BO

(7.17j)

ira-

100

plies

s u P B ' Ivt. I 2 _< R 2 , w h e n c e 3 3

that

IVt9+1 (x) I =

d i s t (U (x) ,Ptj+1 )

_< d i s t (U(x)'Pt3") Ivt.(x) I + TR

-< for We

x 6Bj

Thus w e h a v e

therefore

obtain

verified

aK( Pv t

+

d i s k ( P t j , P t j + I)

-< R + ~ R

(7.18)

I) -> O

, on

also

_<

for

i = j + I

Bm_ I , w h i c h

in t u r n

yields

m

R b<(R) Ivt m

-< 2

for

(y) l on a c c o u n t

of

O n the o t h e r

y 6 Bm

,

(R)

(7.16). hand,

OSC

b

B

inequality

(5.17)

u <- b~ 1 (R) O S C B m U

of T h e o r e m

S

5.1

implies

2 b ~ I (R) s U P B m d i S t

that

(U,p)

m

=

2 b~1 (R) s u P B

Ivt m

R i _< - m b (R)

or

R

OSCBgm(Xo where

I _<

i

Thus we have For

1

such

that

[k/s]

+

for

Pm = 2-m r

= 2- i ° - m r o

I

proved:

m +

=

S

O

) u ~ -be(R )

[k/s]

+ I

,

there

exists

a number

p

with

2 -1 r O

_ < 2 p _
R

O S C B 2 0(xo) u -~ This

is

a uniform

estimate

fact,

we can now apply

There

exist

numbers

of

Lemma

c->O

holds

for

in

B2d

and

O

[u]o,Bp(Xo)

b

~,

B2r an~

(x o) 1

is

an

depend

arbitrary only

on

ball

contained

n ,N ,R , X , ~ , ~ ,

101

From

this

result,

w e can

readily

derive

Theorem

1.5 by

a standard

rea-

soning. By r e l a t e d boundary.

arguments, We,

for

U :X ~ M

Let

harmonic is

of

instance,

in

U

some

. Then

for

the

establish

is

smooth

ball

X

, we

depends

only

on

at the

theorem:

manifold

. Suppose

M , and

~ 6 (0,1)

every

estimates

Riemannian

on

of

a priori

the f o l l o w i n g

c(X, N , R , e , < , 0,

c

constant

establish

of a compact

and

regular

[u]0,X where

also can

be a m a p p i n g ~ = int X

on

contained

values

w e can

let

~

have

an

also be

X

that

that

U(X)

the

is

boundary

estimate

[~]o,~X)

X ....

~o

, and

on

the

H~Ider

[~]o,~X

seminorm

Here we have

set dist (U(x),U(y)) [U]~, X

:= sup

: x,y 6 X

x % y

dlstO(x,y) and

[¢]o,~X

We will [20].

omit

U

this

Proposition

COMMENTS

derive

tained

(cf.

by t h e s e

established

remarks

authors

quotient

it is a d v i s a b l e

and

M

[20]

Sperner

[79],

attractive more

and

we refer

subtle

the r e a d e r

estimates

of

estimates

first

and

this k i n d

to u s e

in

(see,

known

to

for

in-

Karcher

many

approach

to h i g h e r

itself,

Yet

they

are d e r i v e d

of g e o m e t r i c a l

satisfactory

[50].

beautiful

were

later

The p a p e r

geometric

[50]

ideas

ob-

by a ap-

assumptions bounds

on

of and

the b o u n d s

and p a r t i c u l a r l y

bounds

one

derivatives

to L a d y s h e n k a y a

6).

a number

more [33],

Other

contains

because

U

order

as few d i f f e r e n t i a b i l y

[32],

Hildebrandt.

and by J o s t -

were

for

for

higher

at the e n d of s e c t i o n

Geometrically

established

by G i a q u i n t a -

the

technique, w h e r e a s

as p o s s i b l e . time

more

are n o t o p t i m a l

difference

the f i r s t

Instead

a priori

for

plications X

contains

estimates

our

result.

to the L I T E R A T U R E .

estimates

. In p r i n c i p l e ,

Uralzeva

'

7).

as o n e has

also

S

defined.

of this

paper

and R E F E R E N C E S

soon

can

the p r o o f

In fact,

stance,

As

is a n a l o g o u s l y

'

were

on for

in the p a p e r found

by

is p a r t i c u l a r l y which

should

be

exploited.

A very

elegant

order

estimates

for h a r m o n i c

mappings

102

has

re c e n t l y

Bochner

been f o u n d by Schoen

formula

the sectional explains

estimates

regularity The b a s i c

leads

of

~4

idea is due

by T o l k s d o r f

[83]

and M e i e r

[63].

estimate

maps

now a second m e t h o d

[7].

technique maps.

the

case,

if he

leads

if one ap-

[69].

in the case of w e a k l y

to C a f f a r e l l i

to h a r m o n i c

gradient

In the general

harmonic

of Price

estimates

clear how C a f f a r e l l i ' s

interior

out how

r 2-n fB r (Xo) e(U) / ~ d x

for

that there exists

and a priori

to be a p p l i c a b l e

to some

(smooth)

result

He first points

is nonpositive.

condition

for

the m o n o t o n i c i t y

We finally m e n t i o n

quite

e(U)

how a smallness

to a priori plies

for

curvature

[76].

for proving h a r m o n i c maps.

In the b e g i n n i n g

should

be m o d i f i e d

This d i f f i c u l t y

it was not in order

has b e e n o v e r c o m e

103

8. P r o o f

Let

X

3M

of

the u n i q u e n e s s

and

M

is e m p t y

interior

once

again

whereas

of

X

theorem

X

be c o m p l e t e

is s u p p o s e d

Riemannian

manifolds.

to be b o u n d e d .

Denote

Assume by

~

that

the

, = int X

The

aim of this

Theorem

8.1.

and map

X

section

Let

is to p r o v e

a regular

where

< ~ 0

is an u p p e r

BR(P)

Then

the f u n c t i o n

ball

BR(p)

bound

following

be m a p p i n g s

U I , U 2 6 C°(X,M)

into

the

that are

of s e c t i o n a l

that

q~(dist(U1(x)'U2

in

v'< R < ~/2 ,

curvature

@ = @ ( U I , U 2) : X ÷ ~

........

harmonic

. In p a r t i c u l a r ,

K

of

M

is d e f i n e d

on by

(x))

(8.1)

satisfies

the i n e q u a l i t y SuPX@

AS

in s e c t i o n

the

1 2

q<(t)

that

5,

= ~t

function

for

S suPgX@ q<(t)

< = 0

,

= ~(1-cos/~< t)

for

< > O

is,

Corollary.

Ff

UI

the u n i q u e n e s s

of T h e o r e m

8.1.

Lemma

Let

8.1.

[O,1/<) tion

by

I

q<(t)

t = ~oS<(T)dT

q~(t)

Thus

is d e f i n e d

~1

and

U2

agree

result

" 42

of T h e o r e m

be f u n c t i o n s

, j = 1,2 , < ~ 0 , a n d

of class

differential

C2

that

operator

on

~X , then

1.2'

~2 ÷ ~

•

is an i m m e d i a t e

of class

suppose

satisfies L : C 2 (~,~)

UI = U2

that

C2(~,~)

that

with

e : [O,I/~)

~ ~ . Finally,

consequence

+ ~

denote

is d e f i n e d

tj(~)

c

is a f u n c by

L

the

by

2 Lw = div(e-~Vw)

Then, no

for

:= i p

(8.2)

every

function

6 ~ : w(p)=

,

where

W 6 C2(~,]R)

~ :=

with

O! , and

L ( e w)

= ~w

~. e ~ ~ J j=1

on o

w ~ 0 , we

have

VW = 0

on

104

as

well

as

L(e}w)

(8.3) Proof.

_> & w - 2 ~

i[Vw[~ + w

2 [ (~ °gj)A~j j=1

on

~-

o

Since L(e~w)

= div(e-~V(e~w)) = Aw+wA~+

= div(Vw+wV~)

we first obtain L(eCw) Moreover,

= Aw

on

for C2-functions

o

h :JR ÷ ]R

and

f : ~ ÷ IR , we have the iden-

tities V(ho f ) = Thus L(e~w) On

~-~

o

= Aw+w

(h' o f)Vf

,

A(h o f) = (h .... f)[IVfil2 + (h' o f)Af

2 2 2 ~ ~(~j)IiV~j]i2 + w Z ~(~j)A4j + [ ~(~j) j=1 j=l j=1

, we have by Schwarz's

inequality

~(4j) >_ _wi~(4j) [21[V~jlI2 - ~ 1 and hence,

by virtue of

~] > ~2

,

j=1" ~ <w~(4j)l'Vgj'12+~(4j)) This inequality

yields

->- 2--wIllVwIl2

the second estimate.

We now turn to the proof of Theorem

8.1.

For the rest of this section we shall assume Theorem

8.1 are satisfied.

w :~ + ~ (8.4) for

x 6 ~

= q~
n° = w n ° ={x

We, moreover, estimates

Then we specify

that the assumptions

the functions

~I

' ~2

of '

by

~j(x)

tions made

i[vw[12

-I

(O)

, j = 1,2 , w(x) is given by

6 ~ : UI(X)

note that the functions

in Lemma stated

in

8.1.

= q~(dist(U I (x),U2(x)))

= U2(x) } 41

In addition,these

, 42

, w

functions

satisfy satisfy

the assumpthe following

105 The functions (8.4) satisfy

Lemma

8.2.

2

(8.5)

nw=

~I " 9 2 , w 6 C2(9,]R]

IIdujll 2

[ j=l

Aw > ~

(8.6)

on 2

~o

that are defined by

'

IIvwll2- <w E lldUj[I2

o~

o

j=1 &~j _> (1-<~j)ildujll 2

(8.7)

(We o b s e r v e to local Proof.

that

Ildfli2 = T ~ B < $ a f , $ ~ f >

coordinates

We i n t r o d u c e

on

for

f :~ ÷ X

holds with respect

~ .)

the h a r m o n i c

mappings

F,F I,F 2 : ~ ÷ M × M by F(x)

= (UI (x),U2(x))

,

Fj(x)

= (p,Uj(x))

,

x

6

Then we have W = Q< 0 F

~j = QK ° Fj

where Q< This

implies

:= q< ° dist

the f o l l o w i n g

identities:

Elvwlk2 =IIv(o
= [

ct IIV*jll2 : ]IV(Q< = Fjl]I 2 : /T
< Fj,~ <

£w = A ( Q K ° F)

We infer

with respect

from T h e o r e m

to

j !) .

4.6 that

&w = II dl~ll 2

on

Q

,

o

Aw > ~ilvwll 2-<wlJdFjl~ A~j >- (I-K,~j)ildFj~2 II Then

the s t a t e m e n t

F.> 2 a J

= [D2QK(~(xF,~cF ) cz = ~ D 2 Q < ( ~ a F j , ~ F j)

£~j = A(Q< ~ Fj) (no s u m m a t i o n

2

of the lemma

,

, j

follows

=

1,2

on a c c o u n t

of the i d e n t i t i e s

~

106

lldF112 = ~IdF 1 '~I2 +~/~idF21l 2 We

finally

Proof

turn

to the

of T h e o r e m

-log(1-Kt)

lidF j112 = iidfj [[2

8.1.

, which

We c h o o s e

satisfies

the

function

e,, = e,2

, and we w i l l

2 ~ 0J ~ ~j = - i o g ( I - ~ i j=1

=

e : [O,I/<)

apply

the

function

0

of T h e o r e m

8.1

as

e(t)

Lemma

=

8. I to

) (I-K~2)

= -iog[c°s
÷ ]R

takes

the

U2))]

form

0 = e w and

Lemma

8. I i m p l i e s L@ = £ w

on

~

, O

and

2

L®> W- wllVwll2+wj=1 We,

therefore,

infer

from

1-<~j £}j

the e s t i m a t e s

L0 2 0

In o t h e r

words,

the e l l i p t i c that

the

function

operator

L

~

The

and R E F E R E N C E S

results

of this

These

authors

along

a harmonic

have

of L e m m a

8.2

that

on

N C2(~,~)

, and E.

Hopf's

is a s u b s o l u t i o n

maximum

principle

of

implies

~ sup~x@

to the L I T E R A T U R E

section also

~ - ~o

(8.5)-(8.7)

0 6 C°(X,~)

on

SUPx~ COMMENTS

on

have

derived

mapping

been

proved

a maximum

U :9 ÷ M

by J ~ a e r

principle

, ~ = int X

and Kaul

for J a c o b i

, that

is,

for

[42]. fields solutions

of £J + R U J

= O

,

where RUJ

and, tions

moreover, of

Theorems Sperner

they have

the h e a t

:= y ~ B R ( J , ~

also

equation

U)~BU

established

associated

with

,

a maximum harmonic

principle mappings

for

solu-

(cf.

[42],

A and B). [79]

has

in the same

spirit

as J ~ a e r

and K a u l

used

Karcher

J

107

estimates harmonic

for Jacobi mappings.

fields

of g e o d e s i c s

to d e r i v e

a priori

estimates

of

108

9. M i s c e l l a n e o u s

One might

results

conjecture

two-dimensional smooth.

that e v e r y w e a k l y

manifold

In o t h e r

~

words,

into

an

harmonic

map

N- dimensional

U :~ ~ M

from a

manifold

M

is

supp Y c c

~

should

the r e l a t i o n s

E (U)

< oo

and 6E(U , Y) = O

for all imply

tangential

that

proved

U

vector

is a r e g u l a r

although

several

First

we m e n t i o n

every

isolated

the r e s u l t

singularity ~

domain

result

to all c o n f o r m a l l y

(9.1)

must

7(u)

satisfy

has

by

=

(9.2)

the

F(u,p)

U

with

This

conjecture

do s u p p o r t

Sacks - U h l e n b e c k

invariant

fF(u,Du)

[73]

harmonic

In

be r e m o v a b l e .

that

be of

map.

results

of a weakZy

F(.,p) 6 C I (IRN)

proved

necessarily

along

harmonic

[24],

map

is s t i l l

un-

states

that

it. which

from

Grater

two- dimensional

a two - d i m e n -

has

extended

this

integrals

dvol

, F(u,.) 6 C 2 ( ~ 2N)

m I [pF 2 < F(u,p)

GrHter

~

remarkable

sional

which

fields

< m2TP] 2

the i n t e g r a n d

on

F(u,p)

,

and

IRN x IR2N

of

such

an i n t e g r a l

must

form

= Gik(U) pi . pk

+ Bik(U) d e t (pi,pk)

where i

P Secondly, If

we state

U :~ ~ M

into M

'

" det(p1'pk)

the f o l l o w i n g

above

away

on

from

satisfies

theorem

harmonic

Riemannian

from

is b o u n d e d

U = U ( x l , x 2)

i k = Pa Pa

is a w e a k l y

a complete

is b o u n d e d

U(~)

k

"P

zero,

due

mapping

manifold

u(~)

i k i k = Pl P2 - P2 Pl

such

, and then

from

that that

U

the c o n f o r m a l i t y

to G r a t e r

[25]:

an o p e n

set

~ =

the s e c t i o n a l

curvature

the i n j e c t i v i t y

is a s m o o t h relations

map,

2

radius

provided

of

on that

109

I/alu

(9.3)

II 2 =

I ] s 2 u [I 2

,

<SlU

, s2u>

= o

~.e. on

Moreover, Euler

Gr~ter's

equations

for weak

result

to

can be carried

(9.1,9.2).

H- surfaces

In

U :~ ~ M

over

to w e a k

[25],

GrHter

in a

3- dimensional

solutions

has w o r k e d

out

of

the

the d e t a i l s

Riemannian

mani-

fold. There

remains

weakened. [75].

the q u e s t i o n

This,

He w a s

i n fact,

able

as

to w h e t h e r

is the c a s e

*)

(9.3)

that condition

to p r o v e

can be r e p l a c e d by the weaker a s s u m p t i o n of the D i r i c h l e t integral The meaning Schoen

of

U :~ ~ M

as f o l l o w s .

dz 2

is a h o l o m o r p h i c iable

"stationary

,

~

U

theorem

is a stationary point

First

point"

he proves

will

be

that,

defined

shortly.

for a s t a t i o n a r y

point

Hopf differential

, the a s s o c i a t e d

~(z)

or a t l e a s t

found by Schoen

in Gr~ter's

(9.3)

that

omitted

has been

E .

the n o t i o n

proceeds

can be

as r e c e n t l y

Lt~lu112

=

quadratic

11~2ull 2

-

differential

z = x I + ix 2 . It s u f f i c e s

with

-

2i < ~IU

respect

for our purpose

, ~2 U >

,

to the c o m p l e x

to a s s u m e

that

var-

n =

{z : Izl < I] If

0

vanishes

Grfiter's the

result.

zeros

of

one proves by virtue harmonic The

n

regularity

~'

from

that

on

U

n

n , then

~

has

and

call

is a r e g u l a r

at most

U

is s m o o t h isolated

the r e s u l t i n g

harmonic

Uhlenbeck

theorem,

map U

on account

zeros

open

set

no

into

of must

in

n

~o M

of

. Remove

" Then whence,

also be smooth

and

. proof

for

U

on

no

is r e d u c e d

to G r ~ t e r ' s

theorem

by

artifice:

be an arbitrary

harmonic

on

Otherwise

#

o f the S a c k s -

the f o l l o w i n g Let

identically

function

simply

v ( x l , x 2)

v ( x l , x 2)

= 1 ~Re

S

connected

on z

n'

f(~) d~

domain

in

n° , and denote

the

by

,

z = x

1

+ix

2

o where

f(z)

is h o l o m o r p h i c

f2(z) *)

The

reference

[St]

in

and

satisfies

= - 0(z) [75]

has

in

~'

to b e r e p l a c e d

by

our

reference

[25].

110

9v ~z

Then td x ~

I 2 f

--

that

and

n

W := (U,v)

satisfies

9(z)

It is e a s i l y Gr~ter's

theorem

and hence Grater which

on

[26]

has

by

( u , v l , v 2)

the n e c e s s i t y

and

and It

=

~(z)

~z ~

call

~z

that

and

must

have

the S a c k s - U h l e n b e c k

and

z = x

which

v 2 ( x l , x 2)

1

+ ix

which

on

~'

are

proof

theorem.

is d e f i n e d the r e a l

and

2

,

z 6

~,

satisfies

,

connected

I 44

_

W

: ~'

~

subset

of

~

.

,

M xR2

therefore

can

by

a

= O

compact

~tE(Ut)

the

(9.2);

to d e f i n e

~ E(U,Y) with

(ii)

is s m o o t h

of S c h o e n ' s

W : ~' ~ M x ~ 2

be

satisfies

smooth.

same

the

Thus

reasoning

of the v a r i a t i o n a l

the k i n d

U 6 HI'2(~,M)

(i)

U

of

U

assumptions is

smooth

of on

o one

points

is of

We

Thus

modification

-~(z)

simply

~z

that

of s t a t i o n a r y

:

~v ~--g

on

out

finally

the a s s u m p t i o n s

yields

follows

therefore

We

into

[~'

smooth.

function

~v ~

theorem

of w h i c h

,

is a h o l o m o r p h i c

it

turns

~'

function

= ~ + re(z)

computation

thus

from

because

satisfies

to a p p l y

v 1 ( x l , x 2)

zs a n a r b i t r a r y

Gr~ter's

be

the mapping

, where

~'(z)

An easy

on

the f o l l o w i n g

p a r t s of the s m o o t h

<0(z)

[2'

=- 0

therefore

considers

V(Z)

and

}2

conditions

W : R' ~ M xIR

discovered

GrNter

where

that

and must

even avoids

imaginary

{$v ~-~ (Z)

map

~o

Namely, W =

a harmonic

the c o n f o r m a l i t y

+ 4

checked

defines

see

points

stationary point of for a l l

tangent

support

in

for

the

(9.1),

smoothness

the i n t e g r a n d

[26].

the s t a t i o n a r y

I = O t=O

establish

integral

of

E

fields

~

are of c l a s s

U t = U o ( i d + t~) 6 C c (,~2)

.

if

vector

~ , that

E

and

along C~

for a n y

U

~'

111

As S c h o e n

has p o i n t e d

p o i n t of harmonic

E ; but,

regular

on the o t h e r hand,

by p r o v i n g

maps

harmonic

mapping

it is by no means

point.

of the Hopf d i f f e r e n t i a l

harmonic

(9.4)

each

map also is a s t a t i o n a r y

The h o l o m o r p h y smooth

out,

U . For s t a t i o n a r y

It

could

clear

very w e l l

9(z) dz 2 points

is a s t a t i o n a r y

of

is w e l l

if a w e a k l y

be not the case. known

E , Schoen

for

proceeds

that

O

~,

( l181u II2 _

II $2U il2) ~i n

+ 2 < 91U,~2U > ~2~ } dxl dx 2

O :

f , { ( II~IU ,12 - I, ~2U If2) ~2~

- 2 < ~IU,~2U > ~ I ~ } dxl dx2

and (9.4') for all follows

n E Cc(~%',|R) from

(O,~ (x I ,x 2) weak lemma

, where

~, c c ~

(ii) by c h o o s i n g , respectively.

Cauchy-Riemann implies

that

disc and

Yet the e q u a t i o n s

equations @(z)

is an a r b i t r a r y

¢(xl,x 2) = (n(x1,x2),O) of the

(9.4)

L I -function

is a h o l o m o r p h i c

function

and

in

~ . This

¢(xl,x 2) =

(9.4')

are

~ , and W e y l ' s on

~'

the

112

10. B i h l i 0 g ~ a p h y A., Stability

[ I ] Baldes,

and uniqueness

from a ball into an ellipsoid, A., Die l'quatorabbildung

[ 2 ] Baldes,

dungen singuldrer

properties

Math.

ins Ellipsoid

Riemannscher

of the equator map

Z. 185,

505-516

(1984).

und harmonische

Mannigfaltigkeiten,

Thesis,

AbbilBonn

1984. [ 3 ] Benci,

The Dirichlet problem for harmonic maps

V., and J.M. Coron,

from the disk into the Euclidean [ 4 ] Br&zis,

H., and J.M.

Rellich's [ 5 ] Brezis,

[ 7 ] Caffarelli,

systems,

vol.

Canad.

[12] Eells,

[14] Eells,

Sympos.

[15] Eells,

Soc.

(1980),

147-151.

theorem for harmonic maps, Proc. Amer.

(1982).

and conjugate

I__44, 320-328

points

in symmetric

spaces,

(1962).

di estremali

di tipo ellitico,

discontinue

Boll.

Un. math.

per un problema Ital.

IV. Ser.

(1968).

J., and J.H.

Soc.

I(3, 1-68

Board Math.

J., and L. Lemaire, Math.

[16] Fischer-Colbrie,

of the sphere,

8_~6, 109-160

of Riemannian mani-

(1964).

A report on harmonic maps,

Bull.

London

(1978).

J., and L. Lemaire,

hemispheres,

Harmonic mappings

Sampson,

J., and L. Lemaire,

Conference

Math.

of some

theorem for harmonic maps, Proc.

E., Un esempio

135-138

Math.

Ast6-

(1982).

folds, American J. Math. [13] Eells,

for harmonic maps in

theorems for weak solutions

36, Amer. Math.

R., Minimum

variazionale ~,

(1984).

3~5, 833-838

8_~5, 91-94

J. Math.

[11] De Giorgi,

and

to appear.

almost flat manifolds,

Pure Appl.

On the Liouville

Soc.

[10] Crittenden,

Comm.

Liouville

Pure Math.,

Math.

Regularity

L.A.,

S.-Y.,

H.J.t

3/7, 149-187

(1981).

nonlinear

[ 9 ] Choi,

of H-systems

Math.

Large solutions

Gromov's

Preprint.

solutions

Pure Appl.

Comlr~. Pure Appl. Math.,

P., and H. Karcher,

risque 81

[ 8 ] Cheng,

Comm.

H., and J.M. Coron,

two dimensions, [ 6 ] Buser,

Multiple

Coron,

conjecture,

n-sphere,

Selected Sci.,

Examples

Z. 185,

145,

Confer.

Set.

50

(1983).

of harmonic maps from disks

517-519

D., Some rigidity Acta Math.

topics in harmonic maps, N.S.F.

CBMS Regional

to

(1984).

theorems for minimal 29-46 (19RO).

submanifolds

113

[17] Garber,

W., S. Ruijsenaars,

E.Seiler,

and D. Burns,

solutions of the nonlinear o-model, Ann. Phys.

On finite action

119,

305-325

(1979). [18] Giaquinta,

On the regularity of the minima of

M., and E. Giusti,

variational integrals, Acta Math. [19] Giaquinta,

[20] Giaquinta,

31-40

(1982).

The singular set of the minima of

M., and E. Giusti,

quadratic functionals, Ann. 11, 45-55

148,

Scuola Norm.

Sup. Pisa Ser.

(1984).

M., and S. Hildebrandt,

A priori estimates for harmonic

mappings, Journal f~r die Reine u. Angew. Math. v M., and J. Soucek,

[21] Giaquinta,

IV, vol.

336,

124-164

(1982).

Harmonic maps into hemisphere, Pre-

print. [22] Gromoll,

D., W. Klingenberg,

and W. Meyer,

Riemannsche Geometrie im

GroBen, Springer Lecture Notes Nr. 55 (1968). [23] Grater,

M., and K.-O. Widman,The Green function for uniformly ellip-

tic equations, m a n u s c r i p t a math. 37, 303-342 [24] Grater,

M.,

(1982).

Conformally invariant variational integrals and the

removability of isolated singularities, man. math.

4/7, 85-104

(1984). [25] GrHter, 239, [26] Grater,

M., Regularity of weak H-surfaces, Journ. 1-15

Math:.

M., Regularit~t station~rer Punkte von konform invarianten

Variationsintegralen, Preprint

19~4.

R., Harmonic maps of manifolds with boundary, Springer

[27] Hamilton,

Lecture Notes Nr. 471, [28] Hellwig,

Reine Angew.

(1981).

1975.

G., Partielle Differentialgleichungen, Teubner,

Stuttgart

1960. [29] Hildebrandt,

S., and H. Kaul,

Two-dimensional variational problems

with obstructions, and Plateau's problem for H-Surfaces in a Riemannian manifold, Comm. Pure Appl. Math. 25, [30] Hildebrandt,

S., and K.-O. Widman,

187-223

(1972).

Some regularity results for quasi-

linear elliptic systems of second order, Math.

Z. 142,

67-86

(1975). [31] Hildebrandt,

S., and K.-O. Widman,

On the H~lder continuity of weak

solutions of quasilinear elliptic systems of second order, Ann. Scuola Norm.

Sup. Pisa

(IV), 4,

145-178

(1977).

114

[32]

Hildebrandt,

S., H. Kaul,

K.-O. Widman,

Harmonic mappings

Riemannian man~folds with non-positive Scand. 37,

257-263

[33] Hildebrandt,

S., H. Kaul,

[34] Hildebrandt,

curvature,

Math.

(1975). K.-O. Widman,

problem for harmonic mappings 147, 225-236

sectional

into

Dirichlet's boundary

of Riemannian manifolds,

value

Math.

Z.

(1976). Widman, An existence

S., H. Kaul, and K.-O.

harmonic mappings of Riemannian manifolds,

theory ]'or

Acta Math.

138,

1-16

(1977). [35] Hildebrandt,

minimal surfaces,

Inventiones math. 62,

S., Liouville

[36] Hildebrandt,

approach

and K.-O. Widman, Harmonic mappings and

S., J. Jost,

to Bernstein

P r i n c e t o n Univ. [37] Hildebrandt,

269-298

theorems for harmonic mappings and an

theorems, Annals of Math.

Press,

(1980).

Studies

Seminar on Diff. Geometry,

S., Eoniinear

elliptic systems and harmonic mappings,

Proc. of the 1980 Beijing S y m p o s i u m on Diff. Geom. Equ., Vol. 1, pp. [38] Hoffman,

481-615,

Science Press, Beijing,

D., and R. Osserman,

D., and R. Osserman,

and Diff. China

1982.

The geometry of the generalized Gauss

map, Memoirs of the Amer. Math. [39] Hoffman,

107-131,

1982.

Soc.,Nov.

1980, Vol.

28, No.

236.

The area of the generalized Gaussian

image and the stability of minimal surfaces

in S n and ~n,

pre-

print. [40] Hoffman,

D.A., R. Osserman,

and R. Schoen,

On the Gauss map of com-

plete surfaces of constant mean curvature in ~ 3 and ~4, Math. [41]

Ishihara,

Helv. 57, T.,

The harmonic

London Math. [42] J~ger,

519-531

Soc. 26,

W., and H. Kaul,

(1982).

Gauss maps in a generalized sense, J.

104-112

(1982).

Uniqueness

and stability of harmonic maps

and their Jacobi fields, m a n u s c r i p t a math. 28, [43] J~ger,

Comment.

W., and H. Kaul, Rotationally

269-291

(1979).

symmetric harmonic maps f~om a

ball into a sphere and the regularity problem for weak solutions of elliptic systems,

J. Reine Angew. Math.

[44] Jost, J., Eine geometrische Bemerkung

Abbildungen, 32,

51-57

343,

146-161

(1983).

zu S~tzen ~ber harmonische

die ein Dirichletproblem

IJsen, m a n u s c r i p t a math.

(1980).

[45] Jost, J., Ein Existenzbeweis

Dirichletproblem

f~r harmonische Abbildungen,

lOsen, mittels der Methode

die ein

des Wdrmeflusses,

115

manuscripta [46] Jost,

math. 34,

J., A maximum principle

Dirichlet problem, [47] Jost,

17-25

J., Existence

(1981).

for harmonic mappings

manuscripta

Math.

Z. 184,

[48] Jost, J., Harmonic maps between

[49] Jost,

1062,

J., Harmonic

mappings

J., and H. Karcher,

4_O0, 27-77

Methoden

functionals,

Math.

H., and J.C.

regularity Ann.

262,

Preprint

H.B.,

30, 509-541

139,

[56] Leichtweiss,

Journ. [58] Lemaire,

K.,

of solutions

[60] Littman,

Geom. 13,

and growth proBonn 1983.

of Euclidean

surface

Oeometrie

harmoniques 51-78

system,

and Acta

in Grassmannschen Mannig-

(1961).

de surfaces

Riemanniennes,

(1978).

W., G. Stampacchia,

equations

Scuola Norm.

Sup. Pisa,

no.

553

fis. mat.

des

Regular points

coefficients,

III.

/'or nonlinear 29, 207-228

et courbure

(1979/80).

and H.F. Weinberger,

Sci.

math.

(1981).

harmoniques

with discontinuous

theorems

manuscripta

187-193

des applications

S~minaire Bourbaki,

M., Liouville

non-uniqueness

and critical points of the energy in dimension

L., Existence

systems,

results

submanifolds

Z. 7~6, 334-366

for elliptic

[61] Meier,

smoothing,

SFB-40 preprint,

to th~ minimal

Zur Riemannschen Math.

L., Minima

vari~t~s,

(1983).

(1977).

L., Applications Diff.

for minima of certain

549-561

Non-existence,

two, Springer Lecture Notes 838, [59] Lemaire,

math.

1984.

1-17

faltigkeiten, [57] Lemaire,

von

manuscripta

(1977).

Non-existence

Wood,

and R. Osserman,

irregularity Math.

zur Gewinnung

center of mass and mollifier

[54] Kasue, A., A gap theorem for minimal

[55] Lawson,

Procee-

1984.

Abbildungen,

perties for harmonic maps and forms,

space,

manifolds,

Canberra

Geometrische

Boundary

and M. Meier,

Pure AppI. Math.

[53] Karcher,

ANU-Press,

f~r harmonische

H., Riemannian

Comm.

Springer Lecture Notes

(1982).

quadratic [52] Karcher,

with the help of

(1983).

between Riemannian

Anal.,

a-priori-Schranken

[51] Jost, J.,

(1982).

(1984).

dings Centre Math. [50] Jost,

489-496

surfaces,

which solve a

129-130

proofs for harmonic mappings

maximum principle,

Nr.

math. 38,

Ser.

elliptic (1979).

Ann.

17, 43-77

equations

(1963).

and

116

[62] Meier,

M., Removable

cation [63] Meier,

to minimal

singularities

of harmonic maps and an appli-

submanifolds,

to appear

M., On quasilinear

elliptic

in Indiana Math.

systems with quadratic

J.

growth,

Preprint 1984. [64] Morrey,

The problem

C.B.,

Ann. of Math.

Springer, [66] Moser,

4_~9, 807-851

S.B., Multiple

[65] Morrey,

R., Minimal

[67] Osserman, 1120

in the calculus

Berlin-Heidelberg-New

York

14,

577-591

varieties,

of variations,

(1968).

theorem for elliptic

Pure AppI. Math.

manifold,

(1948).

integrals

J., On Harnack's

Comm.

of Plateau on a Riemannian

differential

equations,

(1961).

Bull. Amer.

Math.

Soc.

75,

1092-

(1969).

[68] Osserman,

R., Minimal

value estimates

surfaces,

Gauss maps,

and stability,

total curvature,

The Chern Symposium

1979,

eigenSpringer

1980. [69] Price,

P.

math.

A monotonicity 43,

131-166

Eigenfunctions

[70] P o c h o z a e v

Nauk SSSR 165, [71] Ruh,

[72] Ruh,

J. Diff.

E.A.

J.

J.

[75] Schoen,

behaviour

149,

Ann. Math.

Trans.

R., Analytic

manuscripta

Dokl.

Akad.

of non-parametric minimal 509-513

hypersur-

(1970).

The tension field of the Gaus~ map, Trans. 569-573

113,

and K. Uhlenbeck,

surfaces,

fieldsr

of the equation Au+ly(u)=O,

and K. Uhlenbeck,

2-spheres, [74] Sacks,

Soc.

for Yang-Mills

(1965).

Geometry 4,

and J. Vilms,

Amer. Math. [73] Sacks,

33-36

Asymptotic

E.A.

faces,

formula

(1983).

The existence 1-24

of minimal

immersions

of

(1981).

Minimal

Amer. Math.

aspects

(1970).

Soc.

immersions 27),

of closed Riemann

639-652

(1982).

of the harmonic map problem,

Preprint

1984. [76] Schoen,

R., and K. Uhlenbeck,

Journal of Diff. [77] Schoen,

Geometry

R., and K. Uhlenbeck,

results on harmonic maps,

A regularity 1_/7, 307-335

Boundary

theory for harmonic maps, (1982);

regularity

Journal of Diff.

I_88, 329

(1983).

and miscellaneous

Geometry

18, 253-268

(1983) . [78] Schoen,

R., and K. Uhlenbeck , Regularity

maps into

the sphere,

of minimizing

Inventiones math.,

to appear.

harmonic

117

[79] Sperner,

E. jr., A priori gradient estimates for harmonic mappings,

SFB 72-Preprint Nr. 513, Bonn

(1982).

[80] Struwe, M., A counterexample in elliptic regularity theory, manuscripta math. 34,

85-92

(1981).

[81] Struwe, M., Nonuniqueness in the Plateau problem for surfaces of

constant mean curvature, Arch. Rat. Mech. Anal., to appear. [82] Tachikawa,

A., On interior regularity and Liouville's theorem for

harmonic mappings, manuscripta math. 42, 11-40 (1983). [83] Tolksdorf,

P., A strong maximum principle and regularity for harmo-

nic mappings, SFB 72-Preprint Nr. 586, Bonn 1983. [84] Tomi, F., Ein einfacher Beweis eines Regularit~tssatzes fur schwache

LUsungen gewisser elliptischer Systeme, Math.

Z. 112, 214-218

(1969). H., The differential equation Ax=2HxuAX v with vanishing boundary values, Proc. Amer. Math. Soc. 50, 131-137 (1975).

[85] Wente,

[86] Widman,

K.-0.,

The singularity of the Green function for non-uni-

formly elliptic partial differential equations with discontinuous coefficients, Uppsala University

(1970).

[87] Wiegner, M., Ein optimaler Regularitdtssatz fur schwache LUsungen

gewisser elliptischer Systeme, Math. Z. 147, 21-28

(1976).

[88] Wiegner, M., A-priori Schranken f~r LUsungen gewisser elliptischer

Systeme, manuscripta math. [89] Wiegner,

I_88, 279-297

(1976).

M., Das Existenz- und Regularit~tsproblem bei Systemen

nichtlinearer elliptischer Differentialgleichungen, Habilitationsschrift,

Bochum

(1977).

[90] Wiegner, M., On two-dimensional elliptic systems with a one-sided

condition, Math.

Z. 178, 493-500

(1981).

[91] Wolf, J.A., Spaces of constant curvature, New York, Mc Graw-Hill 1966. [92] Wood, J.C.,

Non-existence of solutions to certain Dirichlet problems,

Preprint 1981. [93] Wong, Y.-C., Nat. Acad.

Differential geometry of Grassmann manifolds, Proc. Sci. USA 57,

589-594

(1967).

[94] Wong, Y.-C., Sectional curvatures of Grassmann manifolds, Proc. Nat. Acad.

Sci. USA 6-0, 75-79

(1968).

LECTURES

ON H A R M O N I C

MAPS

(with a p p l i c a t i o n s to conformal m a p p i n g s and minimal surfaces) JOrgen

I. The concept 1.1

of h a r m o n i c

The d e f i n i t i o n

1.2 Some examples 1.3 H a r m o n i c

of h a r m o n i c

mappings

2. E x i s t e n c e

theorems

Nonexistence

123 126

mappings

mappings

for h a r m o n i c

between

maps

128

surfaces

between

131

surfaces

131

results

133

lem~ata

2.3 The e x i s t e n c e

theorem

2.4 The p r i n c i p l e

of adding

3. A v a r i a t i o n a l 3.1

122

and g e o m e t r y

and conformal

2.1

120

mappings mappings

of h a r m o n i c

1.5 Ha r m o n i c

2.2 Some

120

mappings

of h a r m o n i c

1.4 The s i g n i f i c a n c e

Jost

Conformal

3.2 C o n f o r m a l 3.3 Remarks

4. M i n i m a l

method

maps

of Lemaire

139

spheres

that produces

between

136

and Sacks - U h l e n b e c k

conformal

diffeomorphisms

spheres

representation

143

of h y p e r b o l i c

surfaces

147

and e x t e n s i o n s

surfaces

143

150

of higher

topological

structure

in R i e m a n n i a n

152

manifolds 4.1 P r e l i m i n a r i e s

152

4.2 The P l a t e a u - D o u g l a s

problem

for m i n i m a l

surfaces

in R i e m a n n i a n

154

manifolds 4.3 H i s t o r i c a l

5. H a r m o n i c 5.1

Heinz'

remarks

157

diffeomorphisms estimates

5.2 E x i s t e n c e

of the f u n c t i o n a l

of h a r m o n i c

in a convex 5.3 E x i s t e n c e

159 determinant

from b e l o w

diffeomorphisms

if the image

diffeomorphisms

between

is c o n t a i n e d

159 160

disc

of h a r m o n i c

closed

surfaces

162

119

6. G e o m e t r i c almost 6.1

constructions.

linear

The concept

6.2 A p p r o x i m a t e 6.3 A l m o s t

representation

formulae

and

166

functions of c u r v a t u r e fundamental

linear

controlled solutions

constructions

and r e p r e s e n t a t i o n

166 formulae

functions

6.4 An a p p r o x i m a t e

7. The heat

Approximate

into a n o n p o s i t i v e l y 7.1 The e x i s t e n c e

170

representation

flow method.

Existence curved

theorem

167

of

formula

for first d e r i v a t i v e s

and u n i q u e n e s s

of h a r m o n i c

maps

171

172

image

Eells

172

- Sampson

7.2 Proof of the e x i s t e n c e

theorem

173

7.3 The u n i q u e n e s s

of H a r t m a n

178

8 Harmonic

coordinates

8.1 H a r m o n i c 8.2 H i g h e r

theorem

regularity

coordinates

order

Bibliography

and h i g h e r

estimates

for h a r m o n i c

maps

of h a r m o n i c

mappings

179 179 184

186

120

I. The

concept

1.1 T h e

definition

Suppose N, resp., ordinate Let

of h a r m o n i c

of h a r m o n i c

charts =

mappings

that X and Y are

with metric

(yaB)

mappings

x =

tensors (x I, .

Riemannian (y~B)

,x. n)

and

and . .f =

(y~B) -I . If f : X ÷ Y

manifolds (gij), (fl,.

of dimension

resp., .,fN)

is a e l - m a p ,

we

in some

n and

local

on X and Y,

can d e f i n e

co-

resp.

the e n e r g y

density e(f)

where

we

curring

the

twice

I to N) energy

use

and

:= ½ 7 ~ 8 ( x ) g i j ( f ) ~ f i ~f_~j ~x ~ ~x 8

summed

express

summation from

convention

1 to n, w h i l e

everything

in terms

(greek m i n u s c u l e s

latin

of l o c a l

ones

are

oc-

summed

coordinates.

from

Then

the

of f is s i m p l y E(f)

The Euler-

/T

in l o c a l symbols

= JX e(f) d X .

Lagrange I

1.1.1)

~

equations

differential

second

+7~8

nonlinearity

way.

~fj

and

a nonlinear

fk

the

the p r i n c i p a l

system

called

form,

while

the

solution.

of h a r m o n i c

df of f, g i v e n

are

of p a r t i a l

is the L a p l a c e - B e l -

in d i v e r g e n c e of the

the C h r i s t o f f e l

(1.1.1)

part

form

0

F jk i are of

elliptic

in the g r a d i e n t

differential

~

are of the

T

solutions

l o o k at the d e f i n i t i o n

The

df -

maps

in l o c a l

in a m o r e

coordinates

by

dx ~ 3x ~

can be c o n s i d e r e d

e(f)

The

is t h e r e f o r e

is q u a d r a t i c

integral

jk ~x ~

on Y.

where

on X and

Fi

y = det(y~)

constitutes

We can also intrinsic

kind

equations,

operator

the e n e r g y

~x ~fi)

coordinates,where of the

for

(~y~B~

~x ~

harmonic.(1.1.1)

trami

standard

are

~fl as a s e c t i o n

of the b u n d l e

T~X®f-I

TY.

Then

I aS ~f ~f = ~ y < --~x ~ , --~x~ > f _ I T Y I = ~ < df,df

> T ~ X ® f -] TY

i.e.

e(f)

is the

trace

of

the p u l l b a c k

via

f of the m e t r i c

tensor

of Y.

121

In p a r t i c u l a r ,

e(f)

a n d h e n c e a l s o E(f)

of l o c a l c o o r d i n a t e s

a n d thus i n t r i n s i c a l l y

(1 .1 .2) where

T(f)

~(f) = t r a c e

the b u n d l e

Vdf,

(1.1.1)

(df) = V

~/~x B

and

the c o v a r i a n t

(1.1.2)

{ ~fl dx ~

if

derivative

in

dx ~

3x~3x B since

T(f)

k(f)

=

~fi

•

Vdf

F r o m the p r e c e d i n g

B
in f - 1 T Y

gives

symbols

Y k Fij

~fi ~fj ~x ~ Zx 8 '

w e see that the L a p l a c e -

of the c o n n e c t i o n

r i s e to the n o n l i n e a r

in T~X,

while

term involving

Beltrami

the conthe

of the image.

O n e can a l s o u s e the N a s h e m b e d d i n g as a s u b m a n i f o l d

I)

are e q u i v a l e n t .

calculation,

is the c o n t r i b u t i o n

Christoffel

~fi

9X ~

,

~2fk

and

3/~x B

' ' XF ~ dx Y Ofl 3 + YFk ~ fif] 3 f i d x ~ 8y ~x ~ 3fl ij ~ f k

~ ~fl

= trace

(1.1.1)

\

Zx ~

~x~3x B a n d w e see t h a t

[El 4]).

~f~ ]

~/Zx ~ ~fz ~2fi

(cf.

2

~/~x B\ ~x ~

3x 8 \3x ~

nection

f is h a r m o n i c ,

= O ,

and V here denotes

to c o m p a r e

V

operator

defined,

of the c h o i c e

T~X ~f-1TY.

We want

a n d thus,

are i n d e p e n d e n t

of s o m e E u c l i d e a n

theorem and consider

the target Y

s p a c e IRI a n d l o o k for c r i t i c a l p o i n t s

of E among maps f : X ~ ~i satisfying

the n o n l i n e a r

(1.1.3) The E u l e r - L a g r a n g e (1.1.4)

I)

constraint

f(x) 6 y equations

for x 6 X. then assume

Af - D2~(f)

(df,df)

H e r e w e d i s t i n g u i s h the C h r i s t o f f e l s c r i p t X or Y , resp.

the f o r m = O

s y m b o l s of X a n d Y by the s u p e r -

122

where

~ is the L a p l a c e - B e l t r a m i

it is the leading jection, borhood Thus,

•

p a r t of

(I 1 I))

and D2z is the Hessian of Y clRl).

we are

See

looking

tegral on X that

operator

.

[J7]

or

7:

of ~ [J9]

for c r i t i c a l

satisfy

,

~i

on X ~Y

(in local coordinates,

is the n e a r e s t

( note that for m o r e

points

p o i n t pro-

z is smooth

in a neigh-

details.

of the o r d i n a r y

Dirich!et

We can d e f i n e

the S o b o l e v

some constraint.

in-

space I W2(X,Y) Since

I := {f 6 W 2 ( X , I R l ) : f ( x ) 6 Y for a l m o s t

the D i r i c h l e t

integral

is w e a k l y

lower

all x 6 X } I in W 2, we

semicontinuous

obtain LE~MA

1.1.1:

The energy

integral

is lower semicontinuous

w.r,t,

weak

W21 _ convergence. It m i g h t

be w o r t h

ly con t a i n s

pointing

two i n h e r e n t

the equations,

i.e.

arising

the ot h e r one coming self does

not have

of the image. lectures,

we shall

domains.

first

It will

group, also

in h i g h e r

lectures,

serious

maps

defined

difficulties,

on t w o maps,

and this and to see

of energy m i n i m i z i n g

sequences.

arguments

it-

from the global

the second

global

of

and

space

in p a r t i c u l a r

on the o t h e r

regularity

the first nonlinearity.

continuity)

of w e a k

Hildebrandt's

w h i c h do not work

solutions

of h a r m o n i c

The v a r i a t i o n a l such p r o b l e m

problem

hand,

The r e g u l a r i t y of

we shall p r e s e n t

for s o l u t i o n s

(1.1.1)

will

of

some

(1.1.1),

(in p a r t i c u l a r be d i s c u s s e d

i.e.

the H~ider in detail

in

lectures.

1.2 Some e x a m p l e s

examples:

of the image, the t a r g e t

for e n e r g y m i n i m i z i n g

of the image,

the b e h a v i o r

actual-

dimensions.

of the local

many other

at least

problem

the n o n l i n e a r i t y

m o r e on the second n o n l i n e a r i t y

structure

affects

arising

look at h a r m o n i c

enable us to use many

In the last three aspects

geometry

i.e.

does n o t p r e s e n t

a l l o w us to c o n c e n t r a t e

homotopy

treat

structure,

In this case,

the first n o n l i n e a r i t y

how the t o p o l o g i c a l

mapping

one b e i n g

from the local

a linear

In these

will

the h a r m o n i c

from the fact that in general

topolo g y

dimensional

out that

nonlinearities,

problem

one can pose

canonical

maps

for h a r m o n i c

for m a p p i n g s

or n a t u r a l

maps

maps

is the m o s t

between manifolds,

are harmonic.

natural

and t h e r e f o r e

Here are

some

123

isometries

-

-

harmonic

functions

geodesic

as m a p s

minimal

-

conformal

-

holomorphic

-

are

maps

maps

Gauss

maps

complex

of m i n i m a l

K~hler

(in h i g h e r

dimensions,

however)

manifolds

manifolds

are

submanifolds

(holomorphic

in g e n e r a l

of E u c l i d e a n

maps

be-

not harmonic) space,

by a

[RV].

and geometry

to d e r i v e

maps.

and h6 C2(y,~)

surfaces

domains

not harmonic,

between

of R u h - V i l m s

mappings

first want

of h a r m o n i c

minimal

two- dimensional

in g e n e r a l

arbitrary

1.3 H a r m o n i c

manifolds

and parametric

on

tween

theorem

on Riemannian

S 3 ~ S 2 , S 7 ~ S 4 , S 15 ~ S 8

-

they

manifolds

SI ~ M

immersions

Hopf maps

-

We

of Riemannian

an e l e m e n t a r y ,

If u 6 C 2 ( X , Y )

is a f u n c t i o n ,

but useful

is a m a p

then

composition

property

between

Riemannian

manifolds,

the f o l l o w i n g

Riemannian

chain

rule

is v a l i d . (1.3.1)

£(hou)

= D2h(u

~,u e

where

e

a)

+ <(grad

h) ou,

T(U)>y

,

e

is an o r t h o n o r m a l

frame

If u is h a r m o n i c ,

o n X.

i.e.

T (u) = O,

implies

this

(1.3.2)

£(hou)

= D2h(u

~,u e

a) e

Thus LEMMA

1.3.1

If h is a c o n v e x

h o u is a s u b h a r m o n i c We note COROLLARY and on

1.3.1

the

function

following

Suppose

and

u(gX)

on Y a n d u

is h a r m o n i c ,

then

on X. consequence

X is a c o m p a c t

u: X + Y is h a r m o n i c . u(X),

function

If there

is c o n s t a n t

(cf. G o r d o n manifold,

exists

in case

possibly

a strictly ~X ~ 2,

[Go]) : with

convex

boundary,

function

then u is a c o n s t a n t

mapping. PROOF

From

that hou (1.3.2)

the maximum

is c o n s t a n t , implies

principle

and

since

that u itself

for s u b h a r m o n i c

h has

definite

is c o n s t a n t .

functions,

second

it f o l l o w s

fundamental

form,

124

Another COROLLARY

consequence

is

Suppose X is a compact m a n i f o l d with

1.3.2

the sectional curvature of Y is nonpositive. u: X + Y

is constant,

p r o v i d e d u(~X)

PROOF

the homotopy

lifting

map

By

~: X ~ ~

into

convex

function

always

in t h e

is t h e n d 2 ( - , p ) ,

sequel,

d(-,-)

we can

covering where

denotes

and

= O

Then any harmonic map

is c o n s t a n t in case

theorem,

the universal

~1(X)

~X ~ O.

l i f t u to a h a r m o n i c

of Y.

The

required

p is any p o i n t

the distance

i n ~.

function

strictly (Here,

as

on a Rieman-

nian manifold.)

In o r d e r

to i l l u s t r a t e

now want

to use a version

existence

and uniqueness

First

of all,

LEMMA

1.3.2

(1.3.3)

an e a s y

the g e o m e t r i c

aspects

of B o c h n e r ' s results

that will

calculation

(cf.

If

Ae(f)

IVdfl 2 + < d f • R i c X ( e

[J9])

f: X ~ Y is harmonic,

mappings,

in c o m b i n a t i o n

be proved

e.g.

([ES]): =

of h a r m o n i c

argument

we

with

in l e c t u r e

7.

yields

then

),df - c a >

- < R Y ( d f • c a , d r • e B) df • e , d f • e B > (Here,

RY d e n o t e s

sign convention spanned

the curvature

of

by tangent

[ES],

vectors

It is n o t d i f f i c u l t COROLLARY

pact,

1.3.3

Ric X ~0

i.e. v,w

to d e r i v e

([ES])

, and

tensor

that the is g i v e n from

Suppose

this

o f Y,

and we have

sectional by

adopted

curvature

the

of the plane

-)

formula

f: X ~ Y is a harmonic map,

X is com-

the sectional curvature of Y is nonpositive.

Then f is totally g e o d e s i c and has c o n s t a n t

energy density.

the Ricci curvature of X is p o s i t i v e at one point of X at least,

If

then

f is constant. If the sectional c u r v a t u r e of Y is negative,

then f is either

constant or maps X onto a c l o s e d geodesic of Y. We now want basic

existence

Hartman prove

to a p p l y

and uniqueness

(uniqueness)

some well

using harmonic

Cor.

known maps.

which

will

theorems

1.3.3

theorem

in c o n j u n c t i o n

of Eells - Sampson

be proved about

with

in l e c t u r e

nonpositively

the

(existence)

7 in o r d e r

curved

following and

to r e ~

manifolds

by

125

THEOREM

1.3.1

If X a n d Y are compact R i e m a n n i a n m a n i f o l d s and Y has

nonpositive sectional curvature, X to Y contains a harmonic map.

then every

homotopy class of maps from

If the curvature of Y is negative,

then

this harmonic map is unique unless its image is a single point or contained in a closed geodesic in w h i c h case every other homotopic harmonic map can differ from the given one only by a rotation of this closed geodesic. We THEOREM

first

1.3.2

deduce

Preissmann's

Theorem:

If Y is a compact R i e m a n n i a n m a n i f o l d s

tional curvature,

of negative

then every A b e l i a n subgroup of the fundamental

secgroup

is cyclic. Proof

Suppose

between

ab a n d b a a l l o w s

al torus

T 2 i n t o Y.

f: T 2 ÷ Y Cor.

a and b are commuting

I

, and the Hence

both

elements

us to c o n s t r u c t

By Thm.

1.1

o f ~I(Y).

a map

g from

g is h o m o t o p i c

i m a g e o f f is c o n t a i n e d a and b are homotopic

The homotopy

the

twodimension-

to a h a r m o n i c

in a c l o s e d

map

geodesic

to s o m e m u l t i p l e

by

of this

geodesic. Furthermore, Hadamard - Cartan THEOREM

fold,

1.3.3

can prove

the

following

If Y is a n o n p o s i t i v e l y

then all homotopy groups

consequence

of the

curved compact R i e m a n n i a n manivanish for m ~ 2 , i.e. Y is a

~m(Y)

manifold.

K(~,I) Proof

We have

Y is h o m o t o p i c monic

we

theorem.

map

to s h o w t h a t e v e r y m a p

g from a sphere

to a c o n s t a n t .

1.3.1,

f: S m ~ Y

By Thm.

, a n d f is c o n s t a n t

S m, m ~ 2,

g is h o m o t o p i c

b y Cor.

into

to a h a r -

1.3.3. q.e.d.

Finally, THEOREM

1.3.4

we deduce

If Y is a negatively

every isometry of Y homotopic ity, and the isometry Proof metries

This

follows

then

the ident-

group of Y is discrete.

from

are harmonic.

curved R i e m a n n i a n manifold,

to the identity coincides with

the uniqueness

part

of Thm.

1.3.1,

since

iso-

126 1.4 T h e

significance

In the p r e c e d i n g connected

with

manifolds.

between

that harmonic matics, -

and point

for

the

maps

found

problem

From

the point compared

underlying are obtained

of v i e w

hold.

several

applications

for m a p p i n g s

of c o m p l e x

holomorphic

Therefore,

strongly

maps

can pose

with

are

of the

harmonic one

, maps

it is n o t

in o t h e r

analysis,

maps,

that

surprising

branches

of m a t h e -

example: theorems

In o r d e r

to conclude

fold determines a suitable

that

works

type

manifolds:

the t o p o l o g i c a l

its c o m p l e x

manifolds,

Bernstein

for K ~ h l e r

harmonic

approach

K~hler -

structure

results

Rigidity

This

that harmonic

of view,

advantage,

existence

seen

complex

variational

manifolds.

have

general

have

a geometric

natural

maps

we

mappings

the g e o m e t r i c

Riemannian

harmonic rather

sections,

From

f r o m the m o s t

of h a r m o n i c

structure,

map

is a c t u a l l y

for

certain

s e e e.g.

theorems

one

and

for m i n i m a l

attempts

a holomorphic

classes

[Si]

type of a K~hler

mani-

to s h o w t h a t

diffeomorphism.

of n o n p o s i t i v e l y

curved

[JY]. submanifolds

of Euclidean

space: By a t h e o r e m

of Rub- Vilms

manifold

of Euclidean

follows

if o n e m a n a g e s

constant.

This will

[RV]

space

the G a u s s

is h a r m o n i c ,

to s h o w t h a t

be carried

out

this

map of a minimal and a Bernstein harmonic

in m o r e

map

detail

sub-

theorem

has

to b e

in the l e c t u r e s

of H i l d e b r a n d t . Teichm~ller One can

theory:

show with

is c o n t r a c t i b l e -

Topology Eells-

of

Wood

theorem

the h e l p

or even

of h a r m o n i c

that

maps

it is a c e l l

that Teichm[ller

(cf.

space

[EE],[Trl],[J7]).

surfaces: [EWI]

that

discovered

that harmonic

a continous

map

e between

to s a t i s f y

the

inequality

Z I and Z 2 has

maps

closed

can prove

orientable

Kneser's surfaces

IdegSIx(~ 2) ~ X(Z 1) between

its d e g r e e

and

the Euler

characteristics

of

~I a n d E 2,

in c a s e X(Z 2) < 0 . -

Boundary

regularity

In s o m e

cases,

blowing

up p r o c e d u r e

for a c e r t a i n

one

for minima can reduce

of q u a d r a t i c this

regularity

to t h e n o n e x i s t e n c e

Dirichlet

problem

functionals: question

of nontrivial

for h a r m o n i c

maps,

cf.

by a solutions [JM]

and

127

[ S U 2 ]. In the last a p p l i c a t i o n ,

the b l o w i n g up p r o c e d u r e

i n h o m o g e n e o u s terms etc.

and leads

map which

turns out to be harmonic.

harmonic maps mappings

for real analysis,

are the p r o t o t y p e

eliminates

irrelevant

in the limit to some sort of t a n g e n t This

indicates

the s i g n i f i c a n c e of

n a m e l y that the e q u a t i o n s

for h a r m o n i c

for an i m p o r t a n t class of n o n l i n e a r e l l i p t i c

systems. T h e h a r m o n i c m a p p i n g p r o b l e m c o n s t i t u t e s a m o d e l p r o b l e m for n o n l i n e a r elliptic partial differential equations, the fact that the g e o m e t r i c

and a s a l i e n t f e a t u r e rests on

interpretation

to d i s p l a y g l o b a l a s p e c t s and p r o p e r t i e s tems.

of h a r m o n i c m a p s o f t e n a l l o w s

of s o l u t i o n s of n o n l i n e a r

S i n c e t h o s e g l o b a l p r o p e r t i e s of h a r m o n i c m a p s

h i g h l y n o n t r i v i a l nature,

sys-

are o f t e n of a

they s h o u l d serve as a s o u r c e of i n s p i r a t i o n

for the q u a l i t a t i v e b e h a v i o r of s o l u t i o n s of o t h e r n o n l i n e a r problems. We w a n t to m e n t i o n o n l y two e x a m p l e s

at this stage:

The e x i s t e n c e of h a r m o n i c d i f f e o m o r p h i s m s Here,

b e t w e e n c l o s e d surfaces:

a s o l u t i o n of a n o n l i n e a r p r o b l e m d i s p l a y s

to - o n e

correspondance between

w e l l as t o p o l o g i c a l l y ) Nonexistence,

two n o n l i n e a r

regions.

nonuniqueness,

See l e c t u r e

a g l o b a l l y one -

( g e o m e t r i c a l l y as 5 for details.

and n o n r e g u l a r i t y of h a r m o n i c m a p s

into spheres: This w i l l be d i s c u s s e d We a l s o r e f e r Harmonic mappings

in m o r e

detail

in H i l d e b r a n d t ' s

lectures.

to [ B a ] , [ G S ] , [ J ~ K3],[SU3].

h a v e also a c q u i r e d

ics as a p r o t o t y p e of field theories.

some i n t e r e s t They model

n o n l i n e a r i t i e s of the E i n s t e i n e q u a t i o n s , or c o m p l e x p r o j e c t i v e

spaces c o n s t i t u t e

c o n c e i v a b l e n o n l i n e a r f i e l d theories,

in m a t h e m a t i c a l phys-

some a s p e c t s of the

and h a r m o n i c m a p s

into s p h e r e s

s o l u t i o n s of one of the s i m p l e s t

the so - c a l l e d n o n l i n e a r O(n)

~ -

m o d e l w h i c h was e x p l i c i t e l y p r o p o s e d as a m o d e l p r o b l e m for s t u d y i n g the n o n l i n e a r a s p e c t s of q u a n t u m fields.

The

j u s t i f i c a t i o n of such a

m o d e l of c o u r s e u l t i m a t e l y rests on the q u a l i t a t i v e as w e l l as on the p o s s i b i l i t i e s that can be a p p l i e d O n e i n s t a n c e of this

it o f f e r s

later on to m o r e r e a l i s t i c

l a t e d s i n g u l a r i t i e s of s o l u t i o n s of the Y a n g - M i l l s

model

in

it permits,

and m o r e d i f f i c u l t models.

latter a s p e c t is U h l e n b e c k ' s p r o o f [U3]

d i m e n s i o n s are r e m o v a b l e , for h a r m o n i c m a p s

insights

for d e v e l o p p i n g t e c h n i q u e s

equations

that isoin four

employing a technique previously developped

[SkU1 ].

(We r e m a r k h e r e that Y a n g - M i l l s

fields

some o t h e r a s p e c t s of the n o n l i n e a r s t r u c t u r e of the E i n s t e i n

equations.)

128

For a m o r e maps We

detailed

we r e f e r

finally

to

[Mill

want

results

for h a r m o n i c

into

maps

relies

of the

field

a negatively

of

the p h y s i c a l

striking

B u n t i n g [Bu]

to s o l v e

on

some

the u n i q u e n e s s

equations

curved

that

are d i s c u s s e d

in d e t a i l

1.5

:Harmonic

and

mappings

sional

we

use

Riemannian

equipped

with

otherwise Thus,

shall

the w o r d

manifold.

some m e t r i c

of h a r m o n i c

of h a r m o n i c

used

B~cklund

ultimately

lead

space.

approaches

The

by C a r t e r

between

"surface"

In p a r t i c u l a r ,

for type

black trans-

to a h a r m o n i c of Bunting

[Ct].

surfaces

in the m e a n i n g

and not merely

map-

uniqueness

conjecture

nontrivial)

symmetric

Mazur

conformal

application

and M a z u r [ M z ]

(rather

and

Usually,

importance

[Mi2]. a recent

relativity.

The p r o o f

formations map

and

to m e n t i o n

p i n g s to g e n e r a l

holes.

discussion

we a s s u m e

of two - d i m e n -

that

a topological

a surface

object,

is

unless

stated.

let El and

(the e x i s t e n c e

~2 be s u r f a c e s ,

of 2

q

isothermal

with

metrics

parameters

will

given follow

in i s o t h e r m a l from

lecture

charts 3) by

+ iy)

dz d z

(z = x

du du

(u = u I + i u 2) resp.

and 0 For

2

a C 1 -map

f: E1 ÷ Y

:}

, the e n e r g y •

.

,

(ulu3+ gij x x

is t h e n

by g i v e n

by

°

u l u 3 ) d x dy . y y

ThuS LEMMA

1.5.1

If k: ~0 ÷ ZI E(fok)

This m e a n s

that

is g i v e n

by

is c o n f o r m a l l y

the L a p l a c e -

1

a

Beltrami

, and

(1.1.1)

invariant.

operator hence

of

takes

E 1 in o u r the

4d 2 ~z ~z 1 J (where

I

u z := ~

u i_ + % zz ~ 3u

(~

then

= E(f)

the e n e r g y

Moreover,

is a conformal map between surfaces,

~u

- i ~)

Fi uj u k = O jk z ,~

I

, u-z := 2

~u

~u

(~-x + i ~ )

).

form

coordinates

129

In the c a s e

the

image

I

(1.5.1)

of ply

I

-~Uz~ Thus,

the e q u a t i o n

2

Suppose

of u d o e s

Whether

in turn

reads

as

not depend

structure,

o n the

special

metric

simply

multi-

s i n c e w e can

. Hence u:

is harmonic,

21 ÷ Y

conformal map between surfaces. in two dimensions

2 2 , this

zu i = o .

its c o n f o r m a l

by

surface

2Pu

+-y---u

the h a r m o n i c i t y

E I , but only on

LESLMA 1.5.2

is t h e

and

k:

2o ÷ 2 1

Then u o k is also harmonic.

is a

In particular,

conformal mappings are harmonic.

u is h a r m o n i c

does

depend,

however,

on the

image metric,

unless

u is c o n f o r m a l . Moreover, LEMMA

i{ f 6 H 2I N

()~I 'Y)

1.5.3:

(1.5.2) where

Co

E(f)

_> A(f(E1))

denotes

A(f(21))

,

the area of the image of ~I"

Equality holds in

(1.5.2) if and only if f is conformal. Furthermore, LEMMA

1.5.4

If

u:

is a harmonic map between surfaces,

2 1 ÷ 22

=

(lu 12 - lUyl 2 - 2i ) d z

2

then

(~=~+iy)

= 4p2Uz ~Z dz2

is a holomorphic quadratic differential. Proof

Multiplying

(1.5.1)

7(u)

b y the c o n f o r m a l

:= Uzz-

2P u p

+ - - u

z

u~

=

factor

0

2

~ , we obtain

.

Thus, 2 ~z

= 2PPu Uz UZ UZ + 2pp~ U z u z u z + p 2

= p

2

UzzU z + p

-

u z Uz~

- -

(Uz~(U)

+ Uz?(U))

= 0

q.e.d.

130

From

the preceding

with

the p o s s i b l e

the Jacobian to b e o f c l a s s harmonic

proof

see t h a t

if ~ is h o l o m o r p h i c

exception

2

lUz[

-

C 2 or

maps

but which

are neither

it is n e c e s s a r i l y

On the other

~ is h o l o m o r p h i c

of class

Theorem

hand,

i.e.

one

cf.

known

can construct

(and in p a r t i c u l a r

C I nor harmonic,

where

if u is a l r e a d y

[J10]

smooth)

for details.

implies

1.5.5:

a) Any holomorphic quadratic differential 2 S vanishes identically

on the two - sphere

b) Any holomorphic q u a d r a t i c differential on a torus is constant. Lemmata LEMMA

= 0

to b e a d i f f e o m o r p h i s m

for which

Lionsville's

~(u)

=

then

lUzl,

then

lu~I2ovfa npiosihnetss. w hOef r ec o u ri[zl se,

if ~ is h o l o m o r p h i c .

Lipschitz

LEMMA

we

1.5.4

1.5.6:

and

1.5.5

imply

Any harmonic map

u:

S2 ÷ Y

is conformal

~/P

131

2.

Existence

2.1

theore/~s

Nonexistence In t h i s

harmonic

maps

easy,

chapter, between

and

difficulties

for

the

the

image.

the

local

allows

first

us

[LI]

2.t.1

Suppose

spect ~D

theory the

to

the

investigate

situation

are

caused

some

instructive

There is no n o n o o n s t a n t

not

necessarily

being

can

use

a standard

argument

map

But

then

(cf.

extend this

of

maps

lead

is

to a n y

dimensions.

details the

what

global

nonexistence

obstructions topology

of

results.

harmonic map from the unit

but

can

not

D onto a single point°

, where

formal

does

by

for In t h e

showed

u09

one

for m i n i m i z i n g

maps

to v a r i a t i o n s

one

theory

boundary.

in h i g h e r

D ÷ S2

Thus,

with

in m o r e

u are

itself,

existence

image

u:

values

onto

domain

of

the

of

boundary

surfaces

possibly

to

to s t a t e

disc D onto S 2 mapping Proof

between

to p r e s e n t

regularity

of h a r m o n i c

want

L~maire PROPOSITION

maps

surfaces,

geometry

in c o n t r a s t

existence

We

we want

closed

case, the

. This

harmonic

results

two - d i m e n s i o n a l very

for

[LI] it b y

or

is h a r m o n i c constant,

[M3],

reflection

conformal

map

definition,namely

9:

pp.

with

D ÷ D

the

show

Since

on

Hence

point

identity

to

a conformal

9D.

= p 6 S 2 . Since

a critical

map

a curve

it

u

on

that

ZD u

interior

mapping

is a c o n on

the whole

is c o n s t a n t

re-

.

is c o n s t a n t on

the

with

is a d i f f e o m o r p h i s m ,

369-372).

as

is c o n s t a n t on

u(~D)

u is a l s o

to

ZD

of

~2.

its

itself. q.e.d.

PROPOSITION

2.1.2:

a) Let ~ be a closed surface of p o s i t i v e genus, homotopy class of degree

a 6 [E,S 2]

±1 . Then a contains

a

no energy mini-

mizing map. b) Let

%:

~D + S 2 be a p a r a m e t r i z a t i o n

there exist precisely

of a great circle by

arclength.

Then

extensions

u:

particular,

there is no energy m i n i m i z i n g

D -+ S 2 of %

two energy m i n i m i z i n g

These maps have degree ±I.

In

extension of ~ of

higher degree. a)

was

[J6].

proved

by

Lemaire

[LI],

and

b)

by

Brezis-

Coron

[BC2]

and

Jost

132

In Prop.

2.1.2

monic Maps lowing

it is n o t k n o w n

which

results

PROPOSITION

are

not

in g e n e r a l

energy

of E e l i s - W o o d

whether

minimizing.

[EW]

We have,

exist the

har-

fol-

[EL3].

2.1.3

torts and p2

plane

~: S 2 ÷ p2

(with any metrics),

and

be a map of degree

f: T 2 + S 2

topically

could

however,

and Ee!is -Lemaire

Let T 2 be the two - d i m e n s i o n a l

Let

there

nontrivial map.

a)

be the real projective be a covering map.

±I, and let

g:

be a homo-

p2 + S 2

Then

f: T 2 + S 2

b)

~0f:

T 2 ÷ p2

c)

g:

p2 ÷ S 2

d)

Tog:

p2 + p2

are not homotopic

(domain and image may have different metrics)

to any harmonic map.

Proof: a) A s s u m e

that

1.5.5b),

u: T 2 + S 2

in E u c l i d e a n

lUxl

2

-

lUyl

is c o n s t a n t , u is a

2

coordinates

- 2i 

and

where

where

Uy = O,

cannot

have

=:

map

of d e g r e e

ux = 0

we would

a ~O,

a zero

b = O.

homotopic

to f. By L e m m a t a

on T 2 a + ib

a + ib = 0 is n o t p o s s i b l e

± holomorphic

At a point

is h a r m o n i c

±I,

since hence

have

Since

on T2; w . l . o . g ,

that would

mean

that

a diffeomorphism.

a SO,

b = O, a n d

a + ib # O,

either

u x ~ O.

J

If

Ux

denotes

at a p o i n t or

Uy

multi-

p l i c a t i o n by ~--~ in TS 2 (in l o c a l c o o r d i n a t e s J ( u I + iu 2) = 2 - u + iu I ) , u x and Ju x b o t h y i e l d ~nonzero s e c t i o n s of u -I TS 2 • Therefor, Chern

u -I TS 2

class

is the t r i v i a l

c 1 ( u - 1 T S 2)

bundle

vanishes.

0 = ci (u - 1 T S 2)

over

T 2, h e n c e

the f i r s t

Thus

= deg(u) ci (TS 2)

= 2 deg(u) a n d deg(u)

= 0

b) We can a s s u m e c) L e t

P:

(gop)~:

which that

S 2 ÷ p2

contradicts

~ is a local

be again

H 2 ( S 2) ~ H 2 ( S 2)

Hence

deg(gop)

(Lemma

1.5.6),

= O. gop

the a s s u m p t i o n isometry,

a local induced

Since

is o n l y

any

gop

harmonic

homotopic

).

reducing

b)

thereby

isometry. by

(cf. [ E L ~

The

homomorphism

factors selfmap

to c o n s t a n t

to a).

through

of

H2(p 2) = Z 2.

S 2 is c o n f o r m a l

harmonic

maps.

Thus

133

g,

since

homotopically

nontrivial,

cannot

be homotopic

to a h a r m o n i c

map. d)

is a g a i n

reduced

t o c). q.e.d.

2.2. We

Some

first

LEMMA

lemmata derive

2.2.1:

some

useful

Suppose

maximum

principles.

that B O and BI,

B O c B1,

are closed subsets of

some R i e m a n n i a n m a n i f o l d N, and ~ is an open subset of some other R i e m a n n i a n manifold.

Let

: B I + Bo

be the identity on B O and

length decreasing outside B o in the sense

that

(2.2.1)

Id~(v)I < ivl

for every

V ~ O,

v 6 TxN , x 6 BI\B O

(Note that we assume here that ~ is d i f f e r e n t i a b l e

on B ~ X B o.

Lipschitz

continuous maps s a t i s f y i n g a p p r o p r i a t e distance decreasing properties can be easily approximated, If

however).

I ~ + B I is an energy m i n i m i z i n g W 2 mapping with respect

h:

boundary values which are contained in Bo, (2.2.2)

h(~)

c BO

to f i x e d

i.e.

,

then we also have h(~)

= BO

,

if we choose a suitable r e p r e s e n t a n t of the Sobolev mapping h.

Proof : Since

Idz(v) i < ivl

o h6W~(~,N)

,

for

E ( 7 o h) contradicting Thus

we

o n ~, w h i c h

< E(h)

a.e.

conclude easily

on ~

from implies

nonzero

v6 TxN

,

x6

B I\B O,

and

have ,

the minimality

d h = d~ o h

(2.2.2),

every

we would

of

h,

, and

the the

unless since

Poincar~ claim.

dh = O a.e.

h and

~ o h

inequality (cf.

[H2],

on h

agree that

Lemma

-I on

( B I \ B O). ~

z o h = h 6) q.e.d.

by a.e.

134

LEMMA

Suppose

2.2.2:

Riemannian

manifold

3 B O by a unique

every on

9B

that B O and BI,

N, and

geodesic

that every p o i n t

normal

pair of such g e o d e s i c . Then

o

the same

B I , are

BoC

to

to

subsets

of a

in B I \ B o can be j o i n e d

~B o , a n d

normal

conclusion

compact

that

the distance

9B O is in B I \ B O always

as in Lemma

2.2.1

to

between

bigger

than

holds.

Proof: We project

B~B

o along

normal

geodesics

onto

DB o a n d

apply

Lemma

2.2.1.

q.e.d.

LEMMA s

2.2.3:

Suppose

B ° is a g e o d e s i c

, where

~)

curvature

of N and

is energy

minimizing

~2 is an upper

among maps c

h(~) c

(for a suitable

with

which

are

centre

bound for

i(kv) i s the i n j e c t i v i ty

B o , and if h ( D ~ )

g: ~ ÷

ball

radius

p and radius the sectional

of p.

homotopic

s,

If

h: ~ ÷

N

to some map

then also

BO ,

BO

representative

o f h, again).

Proof: By

assumption,

B(pr3s)

we

can

introduce

geodesic

We

define

a map ~

in t h e

following

(Heme,

we

geodesic

have polar

Rauch's

mated

coordinates

(r,~)

on

by

(r, ¢)

if

r -< s

T(r,})

(~--(3s-r),%)

if

s <-r-<3s

if

q EN\B(p,3s)

:

identified

a point

in B ( p , 3 s )

with

its

representation

in

coordinates.)

comparison

a map

way:

~(r, ¢) =

(q) = p

From

polar

(O-< r -<3s).

theorem,

satisfying

the

it is e a s i l y

assumptions

seen

of Lemma

that

~ can

be

approxi-

2.2.1. q.e.d.

We now

state

the well

Suppose manifold LEMMA

for some

E of

2.2.4:

known

~ is a n o p e n class

Let

the c u r v a t u r e

Courant-Lebesgue subset

C 3 , while

u6H12(~,S)

< of E,6

r 6 (6,{-~) for which

of

some

S is a n y

, E(u)

_< D

< min(1,i(E) 2 u I~ B ( x o , r )

Lemma two-

(cf.

e.g.

dimensional

Riemannian , xo 6 Z ~ I/12 )

[ C o 2 ]).

Riemannian

manifold. ,

-I 2

Then

N ~ is a b s o l u t e l y

a lower bound there

exists

continuous

and

135

I d(u(xl) , u(x2)) for all

x I, x 2 6

Finally,

we need

of H i l d e b r a n d t LEMMA

2.2.5:

of B(p,M)

the

two- dimensional

Kaul - W i d m a n

, where

, and

mits an extension Then values

there

exists

map h: ~ + B(p,M)

Vice versa,

of continuity

K, and

E(g)

Proof:

(the i d e a

M<MI<

n/2K

a harmonic

each

the energy such

the modulus is t a k e n

locus

the e n e r g y

sequence

has

a subsequence

continuity

of continuity

from

result

the p r o o f

we conclude

with boundary

map

is harmonic.

in terms

of ~,

of g . o f Thm.

p is a c l o s e d

4.1

set,

a ball.

in its c l a s s integral.

that

in

[HKWI].

w e can

We

find

of

the

Lemma

maps

)

some M I,

take a minimizing

because

Applying

h actually

The

i(2 I) ,M ,

Such

V:=

energy

and ad-

to these boundary

= g] {v 6 H 21 ( ~ , B ( p , M 1 ) ) , v l g ~ I c o n v e r g i n g w e a k l y in H 2, a n d the

in

of the D i r i c h l e t

B I = B(p,MI),

existence

is continuous

minimizing

B ( p , M 1) is s t i l l

for

b y h, m i n i m i z e s

B(p,M)

with respect

energy

of a p o i n t

for w h i c h

÷

of h can be e s t i m a t e d

sequence

noted

local

~

values.

the cut

the

M < i(p) , and g: I (~,B(p,M)) g 6 H2

modulus

and

c a s e of

[HKW 3].

9~ # @, B(p,M) is a ball in some m a n i f o l d N with <2 ~ 0 is an upper bound of the Gauss curvature

g, and h m i n i m i z e s

Since

(log I/~) -~

~B(Xo,r) N

Suppose

radius M < ~

_< 4 z - D ~ -

2.2.2

~ into

limit,

lower

the

a de-

semi-

to B o = B ( p , M ) , smaller

ball

B(p,M). By

[J1]

arc

, every

in B(p,M),

can a p p l y that

the

two p o i n t s and

q 6 B(p,M),

the g e o d e s i c s vectors

this

Rauch

in B(p,M)

arc

is f r e e

comparison

v I and v 2

long

as

Therefore,

c1(t),

every

and

in T q N

Therefore way.

, and

starting

we

Suppose

cI , c2

at q with

are

tangent

~ d(c I (t),c2(t))

~

, min(d(c1(E),c2(s)),

there

exists

so > O

BI:=

B(p,M)

satisfy

q 6 B(p,M)

and

points.

following

geodesic

Then

on B ( p , M ) \ B ( q , s )

Consequently,

in t h e

by a u n i q u e

c2(t) 6 B(p,M)

d(c1(t),c2(t)

n B(p,M)

of c o n j u g a t e

theorem

by a r c l e n g t h

Iv I - v 2 1 K -I sin (tK) as

joined

are u n i t v e c t o r s

parametrized

v I , v 2 resp.

c a n be

and

every

e ~ s°

with

]Vl - v 2 1 K -I sin (2M<))

the property

the a s s u m p t i o n s . Lemma

2.2.4

that

Bo:=

B(q,S)

of Ler~na 2.2.2 then

implies

that

for for

136

each x 6 ~ there

exists

h(~(B(x,p) for s o m e q 6 B(p,M). bounded

Lemma

to see

that

p depends

[HKW3])

lies

h in the d i r e c t i o n follows

[G] a n d

[Hi3]).

lecture

8.

2.3

The

existence

for of

the the

sectional

9:

~ ÷ N

[9]

the class

coincide

with

bound Let

curvature

[LU]

radius

of all =

s

is the

I

~min

nonempty having

as a b o u n d

map with

let

maps ~Z

i(N)

is the

(cf. cf.

boundary,

and

let

lower

the a b s o l u t e

bound

value

energy.

are homotopic

to ~

by

and

. where

injectivity

I/I 2 )

We denote

<2 > 0 radius

(_~2 b e i n g

is an u p p e r

cur-

of N.

a lower

bound

for

the

I (log

exists

of

I/d o)

~ s/2

a finite

for all 2.2.4

rn, I, 6 < rn,1

< s/2

number

the d i s c s

be a c o n t i n u o u s

Lemma

,

9, and

, for w h i c h

w.l.o.g,

Applying

(2.3.3)

that

# ~

<- ~/~o ~ d ( ~ ( X l ) ' ~ ( x 2 ) )

There

un

-< E(¢)

find

details;

a positive

for

finite

(i(Z2) , ~/2<),

energy

. . . . m = m(6)

w e can

regu-

theory

Sacks - U h l e n b e c k

i.e.

in c a s e

and

d(x1'x2)

We E(u n)

so

of E at

Z) s a t i s f y

0 < 6 < 6 O.

i = I,

since

Higher

elliptic

for m o r e

with

I

Let

to v a n i s h .

possibly

continuous

~Z

2n • E(¢)

(2.3.2)

= O,

h(~) c B ( p , M ) ,

linear

geometry, as w e l l

T(h)

it is

curvature.

on N,

E(#)

and

of

is

of g.

the v a r i a t i o n

lectures

of L e m a i r e

surface,

~ on

(2.3.1) where

hand,

and

and classical

See H i l d e b r a n d t ' s

6 0 < m i n (1,i(Z) 2, of

that

(which

of h. F u r t h e r m o r e ,

solution

of V,

of h

of c o n t i n u i t y

continuity

interior

be a c o n t i n u o u s

We c h o o s e vature

in the

of b o u n d e d

injectivity

Let

the m o d u l u s

the

t h a t h is a w e a k

theorem

E be a c o m p a c t

N be a m a n i f o l d

and

in V a n d on the o t h e r

from

also

the p r o p e r t y

the e n e r g y

of any ~ 6 H~ N L ~ ( ~ , Z 2 ) has

e.g.

also

Let

p > O with

o n ~, ~ , i(Z),

implies

minimizing

h actually

larity

of g),

2.2.2

(cf.

h is e n e r g y

small

A ~)) c B ( q , ~ )

by the e n e r g y

Therefore, easy

a sufficiently

n

energy

for X l , X 2 6 ~Z.

of p o i n t s B(xi,6/2)

minimizing

x i 6 Z, cover

Z .

sequence

in [9],

.

and u s i n g < /~ , a n d

(2.3.1) Pn,1

U n ( [ B ( X l , r n , 1 ) ) c B(Pn, l,s)

and

(2.3.2),

6 ~ with

for

every

the property

that

n,

137

where

we define

~B(x,r)

We now have

=

Z(B(x,r) n Z)

two possibilities:

either I) T h e r e some c

exists r

some

6, 0 < @ ~ 6 o

(depending

B(p,s)

for

is h o m o t o p i c

, with

o n x a n d n) w i t h

some

p 6 N

, and every

to t h e s o l u t i o n

g: B ( x , r )

+ B(p,s)

g ~B(x,r)

= Unl~B(x,r)

harmonic

(The e x i s t e n c e this

o f g is e n s u r e d

is n o t n e e d e d

in t h e

that

by Lemma

large

Un(~B(x,r)) n, Unl B(x,r)

problem

and

2.3.5;

following

for a n y x 6 Z,

and with

sufficiently

of the Dirichlet

(2.3.4)

but

the property ~ < r ~ /~

energy minimizing

g is a c t u a l l y

unique,

constructions.)

or

2) P o s s i b l y

choosing

of points

a subsequence

xn 6 Z , and radii

Un(~B(xn,rn) ) cB(Pn~en) but

for w h i c h

Dirichlet

Dirichlet

rn >O

, x n ~ x O 6 Z , r n ÷ O, w i t h

some

Pn6N'

en ÷ O

(using L e m m a

to t h e

solution

2.2.4),

of

the

(2.3.4).

(2.3.4)

and using

t h e u n, w e c a n f i n d a s e q u e n c e

is n o t h o m o t o p i c

I), w e r e p l a c e

problem

rn, I ~ r I

UnIB(xn,rn)

problem

In c a s e

for

of

the

u n o n B ( X l , r n , I) b y

for x = x I interior

the

solution

of t h e

a n d r = rn, I • W e c a n a s s u m e

modulus

of c o n t i n u i t y

estimates

for

t h e s o l u t i o n o f (2.3.4) (cf. L e m m a 2.2.5) t h a t t h e r e p l a c e d m a p s , deI noted by u n , are equicontinuous on B(x1,~-~), for any O < ~ < 6 . By Lemma

2.2.3

(2.3.5) B y the

E(u~) same argument

~ E ( u n)

as above,

we then

find

radii

rn, 2,

~ < rn, 2 < /~

,

with u n1 ( ~ B ( x 2 , r n , 2 ) ) for p o i n t s

Pn,2 6 N

c B(Pn,2,s)

.

Again we replace

I un

on

B ( x 2 , r n , 2)

by the

chlet map

p r o b l e m (2.3.4) for x = x 2 and r = r 2 n,2 by u n Again, w.l.o.g., rn, 2 ~ r 2. If w e t a k e

step, if

(u~)

O < ~ < ~ , we

step are

into consideration

in p a r t i c u l a r see

that,

by

is e q u i c o n t i n u o u s

on

that

equicontinuous

on

the b o u n d a r y

values

the

solution

. We denote

first

of

the Diri-

the new

replacement

B ( x 2 , r 2) n B ( x 1 , ~ - ~ / 2 ) , for o u r

~ B ( x 2 , r n , 2) n B ( X l , 6 - n / 2 )

second

replacement

138

Using

the e s t i m a t e s

u t i o n o f (2.3.4) a t t h e s e 2 u n are equicontinuous on Furthermore,

for

the m o d u l u s

boundary

of c o n t i n u i t y

points

(cf.

Lemana 2.2.5)

B ( X l , 6 - ~) U B ( x 2 , ~ - ~ )

by L e m m a

2.2.3

again

and

for

, if

the

sol-

the m a p s

O < n <6

(2.3.5)

E(U2n ) _< E(uln ) _< E(u n) In this way, m u n =: Vn,

we repeat

the r e p l a c e m e n t

E ( v n)

which

is e q u i c o n t i n u o u s

hence

o n all of

If n o w ~

get

limit

# , then

can

weak

H21

(2.3.6),

bounded

~ E(~)

to u,

convergence u minimizes

it is h a r m o n i c

Observing

that

if ~2(N)

2.3.1:

compact

Riemannian

Suppose

Then any

homotopy

harmonic

map.

2.3.2:

a manifold

a continuous Then

there

minimizes

Finally

Vn

, any

since

we have

is c o m p a c t

subsequence.

our

uI~Z

fixed

in o r d e r

We d e n o t e

= ~I~2

to

sequence

converges

of the

energy

# ~.

also w.r.t.

sequence

by

class.

when

restricted

by Lerama 2.2.3

two m a p s

the

in c a s e ~ Z

are a m i n i m i z i n g

energy

and regular = @

N

semicontinuity

the

, and

to small

and Lemma

from a disc

into

balls,

2.2.5.

N are h o m o -

we obtain

THEOR~

THEOREM

(2.3.6),

lower

m

as well.

to ~ , a n d

u minimizes

and hence

N

case

convergent

since

bounded,

that

e n e r g y in its h o m o t o p y

In p a r t i c u l a r ,

topic,

by

a n d by and

uniformly

in this

, i = I .....

cover

, we a s s u m e

is h o m o t o p i c

E(v n)

B(xi,~/2)

balls

is a l s o

= ~

a uniformly

u then

I H2

in

these

(v n)

select

Since

on all b a l l s

If ~

uniformly

by u.

weakly

w e get a s e q u e n c e

_< E(u n)

~ , since

values.

(v n)

Thus,we

until

with

(2.3.6)

boundary

argument,

is a closed

class

with

of maps from

Suppose

map of finite

the energy

boundary,

N a

surface

with

an energy

n&nempty

with

z2(N)

= ~ , and

~ ÷ N

with

ulDF

minimizing

boundary,

t:

~ ÷ N

energy.

map

u:

= tlDZ

, and

u

in its class.

that

the a c t i o n

without

= 0x

~ to N contains

~ is a c l o s e d

is a harmonic

surface

T2(N)

of b o u n d e d geometry

we observe

not affect

~

manifold

our u~:

replacement ~I(Z)

÷ TI(N)

procedure o n the

on

small

discs

fundamental

does

groups

,

139

although Hence,

it m i g h t

if w e m i n i m i z e

the f u n d a m e n t a l

THEOREM

Let

t:

Then

there

the

of bounded

2 ÷ N be

among

2.3.1

is d u e

[SY3]

and

[J5],

first

obtained

an e x i s t e n c e due

groups such

in

2.4.

again

by ~2" We w a n t

Since E 2 \ B ( p , 2 s n)

[LI]

in

Z -*N

u[~2

on

with

case

~

boundary,

= ~

and

.

energy. with

the

= tI~2

and Sacksproof

does

already

of a d d i n g

with

for

u:

with

If o n e

was

same

if

action

~Z = ~

. u

as

t

minimizes

Uhlenbeck

is t a k e n

not prescribe

proved

by M o r r e y

[SkUI];

from.

Thm.

the h o m o t o p y [M2].

cf.

also

2.3.2 w a s

Thm.

class,

2.3.3

is

spheres

and

image

boundary.

to i n v e s t i g a t e

(2.3.4)

action

[SY3].

B(Pn,En) c B ( p , 2 S n ) of

and

that domain

possibly

compact

the p r e s e n t

[L2].

The p r i n c i p l e

We n o w a s s u m e

prescribed

possibly

of finite

map

to L e m a i r e

to S c h o e n - Y a n

with

maps.

where

result

surface,

map

a harmonic

all

among maps

geometry,

a continuous

exists

class.

we obtain

Z be a c l o s e d

fundamental

energy

T~.

Let

the h o m o t o p y

the e n e r g y

groups,

2.2.3:

N a manifold

on

change

more

and

an

closely s/2

x = x n , r = rn U n l B ( x n , r n)

. If w e

are b o t h

We denote

case

for all

compact

the d o m a i n

surfaces,

by

2) of s e c t i o n n

and

is c o n t a i n e d

is n o t h o m o t o p i c

thus

ZI a n d

2.3. the

in B ( p , 2 E n)

to g,

it has

the d o m a i n the i m a g e

W .l.o.g.

solution by L e m m a to c o v e r

define un

on

~ I\B (x n, r n)

g

on

B (x n , r n)

un =

then w e

see

(2.4.1)

that l i m E ( u n)

-> lim E ( U n [ E 1 \ B ( X n , r n ) ) -> lira E ( ~ n)

since

E(g)

~ O

as

n ~ ~

+ Area(Z 2)

, because

[2T. 2 JO Ige (rn'8) I d8 ÷ O

+ lira E(Unl B ( x n , r n ) ) ,

g 2.2.3.

140

as

n ~ ~ , if w e c h o o s e

[J7] o r

[J9],

e.g.

our radii

as i n t h e p r o o f

by Lemma

(cf.

1 . 5.3

E ( v I B ) -> A r e a ( v ( B ) ) holds

2.2.4

)

(Furthermore,

and equality

of Lemma

if a n d o n l y

,

if v is c o n f o r m a l . )

We now define E for a h o m o t o p y

class

:= i n f { E ( v ) ~

of maps

: v 6 ~}

with

v l~l I = ~

, and

E := rain E

We topy

show

class

e

the existence

(2.4.2)

E

We choose

of a minimizing

a minimizing

< E + A r e a ( Z 2)

.

sequence

in

E ( u n) < E Assuming

that

harmonic

map

in a n y h o m o -

with

2) h o l d s ,

un

e

with

+ A r e a ( l 2)

we define

~n

as above.

Since

clearly

E ( ~ n) >_E , this would as

shown

Now let

contradict

in 2.3., ~

be a homotopy E

and

let

a map

v

(2.4.3)

mal

arguments

2) c a n n o t

harmonic

map

occur

and,

in ~.

= E , map

in s o m e h o m o t o p y

above

show

in ~

, i.e.

class

+ A r e a ( E 2) that we can

E(~)

e # ~

= E.

If w e c a n

with

, find a harmonic

map

of mini-

in ~ .

Drinciple,

tremely useful [BCI]

Therefore,

class with

E(v) < E(~)

energy

This

however.

an energy minimizing

~ an e n e r g y m i n i m i z i n g

construct

then the

(5.4.1),

we obtain

although

apparently

in q u i t e a n u m b e r

, and more

specifically

rather

simple,

of different

for harmonic

has proved

settings,

maps

see

to b e e x -

[Au], [Tb],

[BcC], [BC2],

and

[J6].

141

The

idea

Choose and w:

for a construction

a small

D I along

disc D O on

their

of v satisfying ~I

boundary

S ÷ ~2 t h a t r e p r e s e n t s

, take an to o b t a i n

(2.4.3)

isometric a two-

a generator

is t h e

copy

sphere

o f z2(E2)

following:

D 1 , identify

S,

take

DO

a map

and coincides

with

on Do , and put ~

on

Z I\ D o

on

DO ,

V wlD I One

can

THEOREM

identifying

DO with

D I now.

show

Suppose

2.4.1:

one can construct

v

E(V)

In particular,

if

~2(Z2)

. If

~ O

~

is not a constant map,

then

as above satisfying < E(~)

v

+ A r e a ( Z 2)

.

is not homotopic

an energy m i n i m i z i n g map

to

(in its class)

~

then

v

is homotopic

that is h o m o t o p i c a l l y

to

different

from ~. For

the proof,

we have

one can use

to o m i t

COROLLARY

the details,

2.4.1

3~ I , and

(the s t a n d a r d 2 - sphere), ~I

of

[J6],

for e x a m p l e .

Suppose

is a compact surface,

EI

Z2k is a surface

and

4: ~ I

+ ~2

homeomorphic

~I

two homotopically

to

~2

with finite

different

energy.

and

E2

Let

@:

2.4.2

harmonic maps

Let

([J11]):

be homeomorphic 71(

u:

While that

@

= Z2

with

~I + Z 2

be a nontrivial

than one homotopy

and m i n i m i z e

in Cor.

2.4.1

the overall

contrast)

in t h e

the energy

because

UI~

I = 4,

homotopy classes.

homomorphism

class of maps from

the n o n c o n s t a n c y

situation among

@ is n o n t r i v i a l .

map

~

o f Cor. all maps

of

to

Z2 .

homotopy classes.

the boundary

is n o n c o n s t a n t 2.4.2

~I

that is re-

d i s t i n c t harmonic maps that

the energy in their resp.

minimizing

minimizes

extension

be a compact surface without boundary,

Then there are at least two h o m o t o p i c a l l y induce

not

to the real p r o j e c t i v e plane.

Z I) ÷ ~i(~2)

p r e s e n t e d by more

ZI

S2

Then there are at least

and both mappings m i n i m i z e energy in their respective COROLLARY

to

with

is a continuous map,

onto a single point and a d m i t t i n g a continuous

to a map from

Here,

however.

([BC2],[J6]):

nonempty boundary

mapping

the construction

we

inducing

values

guarantees

(see Prop.

look at the map @ , and

2.1.1 ~

for

that

it is n o n c o n s t a n t

142

For an account face

into

of

the real projective

i.e. m a p p i n g

any oriented

any nonoriented nitely many trivial, these

and hence

two

is a l w a y s classes

mizes

El

energy

the e n e r g y

If

onto of

the

to

if

[EL3].

a trivial ~i (Z2),

on the other

interest

maps

We note onto

o f Cor.

in a t

2.4.2

such a map onto image metric.

loop and

9

least then

two of

it i n d u c e s

for

(this

homotopy

(see

a closed

infi-

is n o n -

nontrivial

a closed

sur-

is o r i e n t e d ,

it i n d u c e s

these

that

9

then

hand,

is m o r e o v e r then all

If

a trivial

then

is a t t a i n e d

9

maps.

contain

an arbitrary

from a closed

E l is n o n o r i e n t a b l e ,

of e n e r g y

and

of maps

we refer

ZI

is o r i e n t a b l e )

clear whether for

in

minimizing

classes

somewhat

it is n o t

plane,

is n o n o r i e n t a b l e , classes,

if

of t h e s e

classification

the generator

the minimum e

the case

decreases

though

If

loop

classes.

homotopy

contain

[Ad]) m o s t fact

one onto

homotopy

classes.

at m o s t

the h o m o t o p y

[EL3]

geodesic. this

and This

c a s e al-

geodesic

mini-

143

3.

A variational

3.1

Conformal

In t h i s

method

maps

section,

to

THEOREM

[J9

;

between

we want

by a variational refer

that

produces

conformal

diffeomorphisms

spheres

to o b t a i n

procedure.

For

conformal

some

diffeomorphisms

details

of

the

proof,

of we

spheres

have

to

5.5].

Let

3.1.1:

E

be a surface homeomorphic

to

with metric

S2

tensor given in local coordinates by b o u n d e d m e a s u r a b l e functions

gij "

satisfying (3.1.1)

2

gllg22

- g12

Then there is a h o m e o m o r p h i s m

almost everywhere.

~ I > 0 h:

satisfying

S2 ÷ ~

the conformality

relations gij

~ h i ~h j 9x ~x - gij

gij

~h i ~h j - O ~x ~y

~h i ~ h j Zy ~y

(3.1.2)

almost everywhere, If

(gij)

satisfying

6 C~

S

h

is a d i f f e o m o r p h i s m

of class

C l'a

everywhere.

(3.1.2)

If

then

,

is of class

C k'a

, C ~ , or C e , resp.,

then so is h.

Proof: We assume follows We

(gi~)

is e v e n

by approximation

choose

points v:

that

three

PI'

S2 + ~

P2'

P3

in

let

LEMMA

in

C 2'a

, since

the general

case

[J9]. Zl, ~

z2,

z 3 in

be the

class

S2 of

and all

three

different

diffeomorphisms

satisfying v(zi) ~

be

Then

(i = of

1,2,3)

,

D

that converges weakly

be a sequence in

(Vn)n6 N

(v n)

= Pi

H 2I _ c l o s u r e

the weak

Let

3.1.1:

H21 .

in

class

points

Z . Let

(3.1.3) and

as

different

of

is equicontinuous

Proof : For each

each n6•

z 6 S2 then

and some

s > 0 r

n

, by Lemma

6 (~,/~)

2.2.4

for which

we

can

find

6 >0

and

for

144

diam(Vn(~B(x,rn)) Here, weaky

@

is

convergent

small

that

Now

independent

sequence

B(z,/~)

enough, contains with

divides

, since

then at

the

vn

one

Vn(B(Z,rn))

into

one

two

the

i.e.

the

v

n

are

, since

of

points

particular

equicontinuous

We

the

parts,

the

can

energy

If

of ~

one

having

P1'

P2'

P3

a

~

so

z 1,

z 2,

z 3.

them

being

is c h o s e n diameter

and

of

choose

points

one

the

hence

small

at most has

to

e ,

coincide

,

diam(Vn(B(z,6)) and

n

bounded.

at most Z

part,

of

o In

and

is a d i f f e o m o r p h i s m .

smaller

most

z

is u n i f o r m l y

contains

Vn(~B(Z,rn))

Vn(B(Z,rn))

of

_< E

_< s ,

as

claimed. q.e.d.

We We

choose

verges

continue an

weakly

ous is

in

by

with

energy

(Vn)n6 ~

Lemma

D

the

vn

in

and

Moreover, from

S2

onto

uniformly

and

~

also

they

the

homotopic

to

if w e

find v

a M~bius

have

a

uniformly vn

sequence

transformation,

E(w n)

if t

(3.1.3)

We

con-

of

the

to

v.

(We c a n

diffeomorphisms

vn

are

equicontinu-

In p a r t i c u l a r ,

of

course

v

assume

that

of

diffeomorphisms

ot:

, with

the

introduce

(3.1.3), then

we

(Wn)n6 N and

converging

still

have

_< E(w) can

always

i.e.

S2 ÷ S2 0

o

=

id

is a

be

achieved

a conformal

by

composing

automorphism

of

uniform local

and

family

of

diffeomorphisms,

S2 ,

depending

, then

d E(voot) d--t

is

then

( c f . LeIrffoa 1 . 5 . 1 ) .

(3. I . 5)

voo t

a sequence

. Since

necessarily satisfying 1 in H2 towards some w,

normalization

on

subsequence

6 D . S i n c e t h e e n e r g y is l o w e r I H 2 convergence (Lemma 1.1.1), v

to

the

. A

Z , not weakly

Hence,

v

in D

homotopic.)

changing

smoothly

3.1.1:

can

converge

E(v)

w with n without

since

some

weakly

(3.1.4) since

Thm.

sequence

to w e a k

. We

converging

are

of

towards respect

3.1.1,

continuous

all

proof

minimizing I H2

in

semicontinuous minimizes

the

energy

weak

coordinates

it = O

= O

,

H 2I _ l i m i t z = x + iy

of

VnOO t on

S2

by

stereo-

145

graphic

projection

and put

E = (~, F, G

iVxi2

are defined

,

F =

,

everywhere,

since

almost

G =

IVy] 2

v 6 H~)

,

°t = ~ + iq (3.1.6) 8a~t t= O = 9 + ie Using

Lemma

1.5.1,

the

energy

.

is g i v e n

by

= ~I J ¢" (E + G) d x d y

E (v)

and

1 [

2 2 ]{ E ( ~ y + n y )

E ( v o O t) = y

Since

Co(Z)

(3.1.5)

= z

then

- 2F(~x~y+qxqy)

and hence

for

+G(~2+h21

t = O

~

-I

dx dy

~x = qy = I , ~y = qx = 0

,

implies J { (E - G) (v x - COy) + 2 F ( ~ y ¢

Putting

] (~xqy-~yqx)

:= E - G - 2iF r | J

Re

Replacing

~ + ie

vanishes,

and thus

by

, this

= O

,

becomes

~(9 + i~)_ d x d y z e - i~

+ e x) } d x d y

, we

= 0

see that

the

imaginary

part

likewise

r (3.1.7)

Jf ~(~ + ie)_ d x d y z

Given

9

and

small

t)

satisfying

e

, we

can a l w a y s

(3.1.6)

ot(z) Hence

(3.1.7)

9_ z

that

~

Since

.

find a family

of d i f f e o m o r p h i s m s

(for

, for e x a m p l e

= x + t~(x,y)

+ i(y + tc~(x,y))

implies

(3.1.8)

i.e.

= O

~ O

,

is h o l o m o r p h i c . ~

represents

a quadratic

differential

on

S 2, in s t e r e o -

146

graphic

projection

we

have

~(~)

: 0

. Hence

t 70 by Lemma

1.5.5

, i.e.

v

IVxl

satisfies 2

the

conformality

relations

2

~ IvyI'

(3.1 .9) almost LEMMA

~ O

everywhere.

The f u n c t i o n a l

3.1.2:

determinant

has

J(v)

the

same

sign

almost

everywhere. For By

the

proof,

(3.1.9),

we

to

is a w e a k

v

equations,

refer

[J97

5.5].

solution

of

the

corresponding

Cauchy-

Riemann

i.e. 2 = -1 v I + k/gv$) Vx -g22(g12 x

(3.1 .I0) 2 : -I ( k / ~ V x I Vy g22 (g = gl] g 2 2 (3.1.10) LEMMA The

idea of

gether See

is a l i n e a r

3.1.3:

proof

5.5]

3.1.4:

the

K

±I

is c o n s t a n t elliptic

for

following

presented

fact

that

Suppose

v

result here

system,

of

can

is t h e

u 6 CI'I (G,IR) lUz~l s K ( l U z l

is a f i x e d

Hartman-

also

uniform

be

for

some

+

, G

a plane

lul)

,

constant.

u(z)

u E N

v

3.1.2

. Since

is r e g u l a r .

Wintner

found

limit

rf (3.1.12)

by Lemma

of

in

[HtW]

[J7]),

(a

to-

diffeomorphisms.

details.

(3.1 . 1 1 ) where

k =

first-order

the version the

and

vI) y

is a d i f f e o m o r p h i s m .

v

is to u s e

with

[J9;

LEMMA

2 g12 ) '

-

g]2

= O(Iz

in a n e i g h b o u r h o o d

- Zo In)

of

zO ,

then

domain,

zO 6 G , and

147

•

-n

lim u z

(z - Zo)

z ~Zo exists.

If

holds

(3.1.12)

/'or all

u 6 ~ , then

I/ =- O

Lemma

3.1.4

roughly

like a complex Lemma

3.1.3

says

.

that a solution

of

(3.1.11)

has

to b e h a v e

polynomial.

completes

the proof

of T h e o r e m

3.1.1. q.e.d.

3.2

Conformal

In t h i s of

section,

surfaces

complex

we

half

free

g ~ 2 by

plane

subgroup

of

surfaces

with

fundamental

the

conformal

regions

H =

{x + iy E f: y > O }

the

isometry

representation

P = H/T

, where

group

+ PL 2 6)

oriented

surface

in t h e

T is a d i s c r e t e

of

H , homomorphic

~I(S).

THEOREM

with

Let

3.2.1:

a metric

functions

Then

be a c o m p a c t

there

exists

local

almost

group

a conformal

map

of i s o m e t r i e s

(gij) 6 C a , then

h

coordinates

of g e n u s

by b o u n d e d

g ~ 2 ,

measurable

everywhere

2 - g12 ~ i >0

homeomorphically

P = H/T

free

in

satisfying

gij

g11g22

region point

S

representable

(3.2.1)

If

of hyperbolic

shall be concerned

S of g e n u s

upper

fixpoint to

representation

is a

h: H + S

onto

mapping

S , where

T

of H , h o m o m o r p h i c

some

fundamental

is a

discrete

to

C 1'a- diffeomorphism

as

fix-

~I(S).

in T h m . 3 . 1 . 1 .

Proof: As

in t h e p r o o f

o f Thm.

3.1.1,

we assume

w.l.o.g,

that

the

gij

are

regular• This

time,

is a s u b g r o u p morphism where

of

w

of H/T

we

let

PL~(~) onto

is t h e w e a k

Vn:

H/T

The

energy

D

be

the c l a s s o f a l l

homomorphic

to

~1(S)

, and

(v,T) v

in t h i s

class

is d e f i n e d

as

where

T

is a d i f f e o -

S . ~ t h e n is t h e c l a s s o f a l l p a i r s I H 2 and uniform limit of diffeomorphism

+ S. integral

pairs

(w,T)

148

I [ = ~ ]z=x+iy6H/T

E(w)

We a g a i n time

choose

an e n e r g y

an a d d i t i o n a l

that we also

have

ficulty

the h e l p

with

achieved

that,

minimizing

to v a r y

(w n)

the g r o u p s

again

of d i f f e o m o r p h i s m s .

not

three

are a l r e a d y

points

3.2.1:

bounded

by

lemma The

some

Matelski

LEMMA Then

3.2.2: there

contains

length

are

from

and

. This

the

fact

overcome

this

[SY3]

. Having

since

all

S ~ 2 , we cannot

now a disc

Let

X

which

3.2.1

of area copy

Yau

the s h o r t e s t

dif-

wn

are

and need

its c o m p l e m e n t

of

[SY3]

on

S

.

closed

geodesic

is i n d e p e n d e n t

is b a s e d

closed

reads

as

geodesic

of

1/sinh(1/2)

around

H/T n

on

is

n.

of

on the c o l l a r

[Hp] w h i c h

lemma

of K e e n

follows. length

1

on

H/T

H/T

¥ , i.e.

the r e g i o n

I
sinh(//2)

_

via

to {re i~/2, l z ÷ ~e z .

that

the p r o o f

corresponds

We note

of

be a s i m p l e

is a c o l l a r

identified

to S c h o e n -

and Halpern

an isometric

y

shall

of S c h o e n - Yau

genus

(Wn,T n)

arises

~ K <

lo > O

of L e m m a

[Ma],

{re i~ 6 H:

where

T n. We

by

.

different.

is due

number

The proof [Ke],

, denoted to 3.1,

to be e q u i c o n t i n u o u s ,

Because

E ( w n)

following

LEMMA

D

3w3 ~ d x d y 3Y /

assume

(3.2.2) The

has

anymore,since

topologically

We can again

In

compared

of an a r g u m e n t

uniform fix

gij(w(z)

complication,

%

( ~w I ~w 3 + ~wl , 3x ~x ~Y

< ~ < ~-arctan

I <_r _< e I }

of L e m m a

{r=1}

, and

3.2.2

given

sinh(//2}}

in

and

[Hp]

{r=e/}

is r a t h e r

elementary. Proof Let

of L e m m a

1

3.2.1:

be the

assume

length

of

that

1

w.l.o.g,

the

shortest

is so small

{ re i@ 6 H : 1 ~ r ~ e l

Let

~ >0

closed

be a l o w e r

curves

on

S

bound

for

. Since

wn

the

closed that

geodesic

on

H/T n contains

, ~~ ~ ~ ~ ~ }

lenghh

as u n i f o r m

H/T n . We can the c o l l a r

.

of h o m o t o p i c a l l y limit

nontrivial

of d i f f e o m o r p h i s m s

149

maps in

the c u r v e

arg z = %

S , w e obtain,

if

onto a homotopically

S

denotes

[el

/

JI

[ gij

nontrivial

the a r c l e n g t h

on

~W i SW j ]½ d x

>

closed curve

arg z =

~

.

(3.2.3)

Now

ds =

I = ~gij

( ~w~xI

(w(z))

the energy density,

a

(3.2.3)

2 _< (lel - i

Integrating

w.r.t.

/

yields

/

\

3~ I4

~ 2 ~-- -< ]~

from

Zy

)

~y

,

z =x

using H61der's

+ iy

l ) = £ lie 2 e (w n) r sin 2 % I

i

,

inequality

dr

2e (wn)

# 6 (~,)

lemma then follows

9w 3 + Zx

l

r

2

(3.2.4)

The

~S

(r sin~) -1 dr , a n d if

e(w) denotes

Ss

dr r sin

, we infer

Z [e dr d~ ] 2e (Wn) ~ ! r sin ~

(3.2.4)

and

2E (wn)

< _

(3.2.2). q.e.d.

We n o w u s e the f o l l o w i n g be p r o v e d L~MMA

in an e l e m e n t a r y

3.2.3:

Let

with nonsingular geodesics

on

a subsequence isomorphic vergence

of

to the

Lemmata

n .

[Mu] w h i c h a l s o can

of isomorphic

H/T n . Suppose

subgroups

converges

to some subgroup

T n . The convergence

3.2.1

which

is

regions.

and 3.2.3 n o w imply t h a t to s o m e s u b g r o u p lim E ( w n)

constant £o " Then

T~PL~(m)

can be interpreted as the con-

normalized fundamental

changing

of PL~(~)

the length of closed

is bounded below by a fixed positive (T n)

converges

Tn . Without

for all

be a sequence

compact quotients

of suitably

a subsequence the

H/T

(Tn)

r e s u l t of M u m f o r d

way.

T

(T n) of

after selection + PL2(~) , i s o m o r p h i c

w e c a n thus a s s u m e

that

of to

Tn = T

150

As mentioned of

before,

a subsequence,

we

In o r d e r

to

suitable

families

more

show

(v,T)

quadratic

to c o n c l u d e

to

for

[J8]

The rest

3.3. By

of

v

THEOREM

and

the

of

the area , the

this,

tat£on.

Identifying a closed . The

is

as

of

P

case case

many

higher

sufficient

nontrivial genus.

holo-

We

refer

each

3.1.1:

[J7]

or

[J9]

elementary

than

3.2,

one

can

homeomorphic

to a two - d i m e n s i o n a l

the a s s u m p t i o n s

lattice

map

of T h e o r e m

C ~ S , mapping

v:

homeomorphically

the same

at

regularity

we

this

onto

and

the f u n d a and

properties

can p r e s c r i b e

point

S,

torus

3.1.1.

as

the image

furthermore

in of a

require

is I . of

a surface

by passing

S'

thus

in

more

a conformal

direction

be

have

an

S

to

with

the

isometric

p 63S S

with

with

theorems

Let

of which

is a c l o s e d

S

is c o v e r e d

i(p)

we

therefore

the

by ~ , if

~

get

can

be

reduced

to

the

double. S

with

the

isometry the

i:

Schottky

reversing

opposite

orien-

S ~ S'

double

isometric

of

S

involution

imply

be a c o m p a c t

surface

of

reversing

an o r i e n t a t i o n

satisfies

boundary

Schottky copy

an o r i e n t a t i o n

the m e t r i c

, ~

but

satisfies

Then

a disc

of

because

is a h o l o m o r p h i c

is no m o r e

exist

that

torus:

S

of some v

preceding

there

there

with

somewhat

as w e l l ,

3.1)

case

is

v

to e x p l o i t

T

in

3.1.

to c o m p o s e

argument have

of

(as

surfaces

As a n o r m a l i z a t i o n ,

surface

COROLLARY

as

on

~

present

satisfying

exists

P

let

We

the

follows

to,

the

tensor

and a tangent

boundariless For

similar

3.1.1

Finally

we

in

extensions

there

region 1 n C a (P,S) V 6 H2

that

again

Suppose

3.3.1:

mental

point

in

that

has

The

since

to v a r i a t i o n s

imply

is c o n f o r m a l

case

has a m e t r i c

Theorem

however,

respect only which

proof

and

the

Then

again

diffeomorphisms.

differentials

considerations treat

one

selection

details.

Remarks

also

i

with alone

that

quadratic

of

in 3.1,

differential

morphic

is e q u i c o n t i n u o u s , and after 1 a uniform and weak H 2 - limit v as is c o n f o r m a l ,

(ot)

v

again

v

than

is c r i t i c a l of

get

that

complicated

variations

(w n)

oriented

surface

with

boundary

same a s s u m p t i o n s

as

(X = S 2 , if

is h o m e o m o r p h i c

to

region,

H

k

S

is an a n n u l a r

in T h e o r e m

and

by

3.1.1.

,

151

in all other cases) with an isometric i n v o l u t i o n

the fixed point set

of which consists of closed geodesic s e p a r a t i n g it into both conformally

equivalent

to

$ . The resp.

m o r p h i s m up to the boundary and has in Theorem 3.1.1.

two components

conformal map is a homeo-

the same r e g u l a r i t y properties as

152

4. M i n i m a l

surfaces

of higher

topological

structure

in R i e m a n n i a n

mani-

folds

4.1. We

Preliminaries

let

means

M be a

that the absolute

bounded, m

(complete)

and

Riemannian

value

the injectivity

manifold

of the radius

of b o u n d e d

sectional

geometry.

curvature

has a positive

of

lower

M

This

is

bound.

:= d i m M .

Let

:=

in

M

If

S

(71,

.... yk ) be a

system

of disjoint

oriented

Jordan

curves

. is a c o m p a c t

boundary maps

curves

ci

denote

oriented

c I,

...

monotonically

surface

,c k

, and

of h:

class

CI

S + M

and orientation

a n d of g e n u s

is a c o n t i n u o u s

preserving

onto

Yi

g

with

map which ' then we

by A(h,S)

the area of

in c a s e

h(S)

this expression

is w e l l

defined,

and

similar-

ly b y E(h,S) the e n e r g y

of

Furthermore, induced

h. let

by

~:

h .

(4.1.1)

Zl(S)

÷ ~I(M)

be the map

on fundamental

groups

We define a(T,g,~)

:= i n f ( l i m inf

A(hn,Sn))

n ~

where

hn

duces

and

~.

Sn

(4.1 .2)

d(y,g,e)

we define take M

some

take

image

under S

latter n

form

In c a s e that

8

loop and

or we

cut

the properties

identify

along case,

which

stated

above,

and

hn

in-

q

q

t h e cut,

is s e p a r a t e d which

loop

interior

S

and we we

the zero of

into of

with

curve

contained

type.

S

and

Ti

the , and In t h e

the t w o c o p i e s

the b o u n d a r y

of

along

a point,

two components). ci

either

element

ci

in

s h r i n k o n e of t h e m shrink

we

S , cut

cut curves

to a n a r c

separate

is n o t o f d i s c

as follows: onto

on a boundary

t h e two p a r t s

t h e cut,

S ~

the resulting two p o i n t

curves, by

of

in t h e

is h o m o t o p i c

(that m i g h t

after

componen%

under

each of

two new boundary S

S'

is m a p p e d

joining

of which

E(hn,Sn))

reduction

as a c l o s e d

an arc e

:= i n f ( l i m inf n ~

a primary

~ 6 ~i(S)

, represent

this

satisfy

Also

of

to a p o i n t . curve of

153

We

note

or

separates

by

1

but

one

but

in t h e

boundary So

cally

that

far,

a primary S

into

curve the

notion

We

a metric

finally

of

define or

reductions

(4.1.2)

and

of

same

being

S

genus

a disc.

a topological will

one,

automati-

applies.

:= ~

d~(y,g,a)

,

if

k =

otherwise

by

over

surfaces

Sn

which

. If

there

no

primary

S

are

I taking

are

the

inf

in

homeomorphic

reductions

of

to S

,

put

4.1.1

:= d ~ ( x , g , a )

a)

([M3]):

Proof:

c)

d(y,g,a)

< d~(y,g,e)

clear to

get

[Na]

that

we

some

a,

d,

a ~, a n d

C I'~

and

d~

we c a n is an

hn

,

d~(x,g,a)

= a~(x,g,e)

9.4.1)

can

choose

immersions,

into

of

C 1'a

= a(x,g,a)

is

in o r d e r

of class

Lemma

.

is o f c l a s s

Sn

d(y,g,~)

[M3],

:= ~

the d e f i n i t i o n

each

b)

(cf.

Theorem

In

that

immersion

It

3.3.1

is

genus

the

one

reductions

resp.

assu~e

a)

primary

the

having

other

of

a~(y,g,e)

LEMMA

Cor.

one

the

reduction

all

:= d ~ ( x , O , e )

a~(y,g,a)

(4.1.1)

again

and

reduces

define

primary we

S

primary

to w h i c h

either

components,

than

below,

a~(y,O,~) and

two

less

considerations

carry

reduction

I~l

we

hn

embed

(with

and

each

fixed

Z

Sn ~Sn

of

class

C I 'a

isometrically

) and

define

by

kn: S n ~

, and Nash's M XlR l

via ki(x) n

= hi(x) n

k m + 3 (x) n

in

b)

then

c)

By

=

(in,

follows

b),

since

e.g.

using

area.

we

...

from

only

n

i 3 (x) n

# ,i~)

1

where

= e

the

have

additional

to

i =

'

is

I

the

show can

..

j =

'

conformal

handles

' "

,m

I,

...

embedding

,L

and

representation

a(y,g,a) always

,

en

~

as

theorems

S a~(y,g,a) be mapped

O

onto

but

n

of this

curves

~

.

lecture

3.

is o b v i o u s without

154

4.2.

The Plateau-

Douglas

problem

for minimal

surfaces

in R i e m a n n i a n

manifolds

THEOREM

Let

4.2.1:

a compact

oriented

o f genus

g

and

monotonically be

the

M

a system

(y1,...,Xk)

be a m a n i f o l d of

k

surface ~:

with

E ÷ M

map

k

oriented

preserving

on f u n d a m e n t a l

geometry,

Jordan

boundary

be a c o n t i n u o u s

and orientation

induced

of b o u n d e d

disjoint

curves

~7'

mapping

onto

yi

y

curves

. . . . ~k

which and

=

in M , E

maps

~:

and ~i

~i (~) ÷ ~I(M)

groups.

If (4.2.1)

then

d(x,g,a)

there

exist

< d*(y,g,a)

a compact

boundary

curves

c I,

induces

the map

~

tonically maps

S

If

by

and

a surface

Plateau-

COROLLARY

4.2.1:

in

X

less

least a r e a

A(h,S)

= a(y,g,a)

surface

We

h

n

assume

that

also

y

be a s y s t e m . If the

h

with

each

ci

mono-

is c o n f o r m a l

topological

k

which

and

class)

.

implies

less

not

(71,

infimum

than

or c o n s i s t i n g does

the

solution

...

of the

of more

exceed

than

g , then

Y , and

this

the proof

of Theorem

of o r i e n t e d

,yk )

of area

the i n f i m u m

spanning

we

Sn ÷ M

be a minimizing

can a s s u m e

immersion

of

the

that

Schottky that

that

class

conformal

H , resp.,

hence

h

g

S ~ M

of surfaces of surfaces

either

one

component

there

exists

surface

is o f

Jordan

of g e n u s

the

sum

a minimal

least

area

ones.

hn:

4.1.1,

is a n

g

to

4.2.1

g 6 N

g

start with

By the

or

Let

of w h i c h

such

Let Lemma

h:

problem.

is s t r i c t l y

o f genus

all

7i

(in its

is h o m o t o p i c

Douglas

than

of the g e n e r a

among

of g e n u s map

and maps

onto

of

h

~m , and

spanning

of genus

groups

preserving

In c a s e M = IRTM , T h e o r e m

g

S

< a*(y,g,a))

X , i.e.

classical

curves

surface

a(y,g,a)

a continuous

on f ~ n d a m e n t a l

= 0 , then

~2(M)

oriented

,c k

and orientation onto

bounded

...

(or e q u i v a l e n t l y

,

each

4.2.1:

sequence S

is o f

n

for class

a(x,g,e) C 1'e

. By

and

each

C l'e

representation double

of

Sn

h n is c o n f o r m a l ,

theorems

of

is e i t h e r that

lecture S2

3, w e c a n

or c o v e r e d

E ( h n , S n) ÷ d ( y , g , a )

by and

155

(4.2.2)

E ( h n , S n) S K We shall r e s t r i c t

where

the d o u b l e of

where

Sn

for

our considerations

Sn

is c o v e r e d by

elementary

it s u f f i c e s hyperbolic

to d e m o n s t r a t e structure

We shall o n l y interior

convergence

to the m o s t d i f f i c u l t

region

the r e m a i n i n g

4.2,1

cases

can be h a n d l e d

then c o n s i s t s

of a s u b s e q u e n c e

by

of

in s h o w i n g

S n . By L e m m a

treat one typical Yn

d o u b l e of

case, of

namely

Sn

cannot

that

3.2.3,

t h a t the l e n g t h s of c l o s e d g e o d e s i c s

on the S c h o t t k y

closed geodesics

case

considerations.

T h e f i r s t p a r t of t h e p r o o f o f Thm. implies

K .

H , since

is e i t h e r a disc or an a n n u l a r

s i m i l a r but m o r e

(4.2.1)

some constant

of the

tend to zero.

s h o w t h a t the l e n g t h s of

(the h y p e r b o l i c

s t r u c t u r e of

)

Sn

c a n n o t t e n d to zero. If

~(yn )

is n o t h o m o t o p i c a l l y

a positive assume

that

lower bound, ~(yn )

(4.2.3)

is h o m o t o p i c a l l y

Z( yn ) =:

By L e m m a

3.2.2,

trivial,

Sn

t h e n t h e l e n g t h of

a n d the a r g u m e n t

en ÷ O

contains

of L e m m a

trivial as

a collar

{re i¢ 6 M : I ~ r ~ e x p ( S n ) ,

3.2.1

hn(Tn)

applies.

and

n ÷ around

arctan

Yn

(sinh

which

is i s o m e t r i c

corresponding

We parametrize and map

to

DD , t h e b o u n d a r y of t h e u n i t d i s c

D, by the a n g l e

it v i a 0 {re i} : 1 ~ r 4 exp(¢ n) We then p u t

(4.2.5)

,

{r exp(iw/2) }

(4.2.4) onto

;2~ ~ 8=0 i

Un(8)

÷

exp

for some

e + i~ / ~

to be c h o s e n

n = hn(eX p (~-~G),~)

[exp (8 n ) u n (8) 12 de = ;r=1

.

l a t e r on

Now (

~ r hn(r'%) 12

) ~Sn

Since r3~/4

(4.2.6)

to

(Sn/2))

< ¢ < w - a r c t a n (sinh (en/2))} Yn

has

We thus

ie x p (Sn)

]~=~/4 ]r=1

i ~ r hn(r'})

2 r dr d~ r2sin~ _< 2 E (hn)

rat

.

0

156

we

can

find

.z 3~. #n C [~,-~)

some

(4.2.7)

for

~ un i~ ~-~

which

if w e

( 8 )I 2 d@

choose

- cIKE n

# = #n

in

(4.2.4)

,

] 8=0 where

LEMMA

K

is g i v e n

We

now

use

in

the

~

there

C2

depends

the i n j e c t i v i t y

only

on

we

satisfying

D + M

can

extend

(4.2.10)

E(Un)

again

being

cut

~0

procedure note

stated of

new

that un

Sn

the

(cf.

[M3],

on

of

Lemma

9B(Xo,R)

n

.

9.4.8b).

has

,

cn + O

I2 ~ ]O

fI~B(Xo,R)

and

= ~

2 i~' (8) i

d@

M , for example

.

on curvature

bounds

the

and

n

curve

curve

two

, E ( h n , S n) implies

is

copies

: I Sr Nexp(en)}

identified

with

of

are

then

genus

of

reduces

D the

BD

. Each via

inserted. Sn

of

by

This

I. W e d e -

S4

We

from

defined

function

. {r e x p ( i ~ n )

cut which by

D as a c o n t i n u o u s

,

of

cutting

3.3..I

comes

with

~D o n t o

. S-' c a n b e c o n f o r m a l l y represented as n define hn' to c o i n c i d e w i t h hn on that part Sn

before

, and

put

hn'

(4.2.10)

E(h_'n ' Sn ') ~ E ( h n ' S n )

, (4.2.11)

< ~

of

from

n

_< c 2 ~ n

,

surface

was

(4.2.11)

As

of

+ i~n

in C o r .

S n'

where

along

is a p r i m a r y

the

u

independant

Sn

copies

O + exp

lemme

independent

radius

Un:

two

is

continuous

-< c 2

the geometry

Therefore,

th~

is a b s o l u t e l y

E(f'B(Xo'R))

then

likewise

f 6 H 2I N C ° ( B ( X o , R ) , M )

exists

and

We

cI

elementary

[27 2 ]0 i~' (e) I d@

(4.2.9)

c2

and

M , and

(4.2.8)

Then

(4.2.2)

following

Suppose

4.2.1:

values

from

+ d(y,g,~)

.

= Un

on

(4.2.10)

each

inserted

disc,

implies

+ C2£n

, and

S ' n

is a p r i m a r y

reduction

of

157

d~(y,g,e) contradicting Thus,we cannot

(S n)

[JS].

somewhat

(hnlDS)

of i n t e r i o r

length

the r e m a i n i n g 3.2.3

surface

then

S

for all

similar

equicontinuous

the

For

Lemma

to some Sn = S

,

.

that

to zero.

to

assume By

shown

tend

we refer of

(4.2.1)

have

< d(y,g,e)

of

cases

implies

the

same

closed

that

geodesics

have

of S n

to be c o n s i d e r e d ,

convergence

of a s u b s e q u e n c e

topological

type,

and w e m a y

n .

sonsiderations,

and h e n c e

that

converges uniformly

one

(after

also

shows

selection

to some m a p

h:

that

(hnl~S)

is

of a subsequence)

~S + y

. See a g a i n

[JS]

for d e t a i l s . We now replace Unl~S map

= hnl~S

and

hn

Un~

on the f u n d a m e n t a l

in c a s e

z2(M)

= O

(Unl~S)

(after

selection

(4.2.12)

converges

(4.2.2)

E

is

lower

u = limu

,

n

d(y,g,~)

area

1.5.3

that

uniformly,

also

also by the

has

semicontinuous

÷ nl(M)

un

is t h e

is h o m o t o p i c

that

u

(u n)

converges

estimates

to c o n v e r g e w.r.t,

~ liminfE(u n~

implies

completes

Historical

After

Un: :S ~ M

with

induced to

hn

n)

weak

l e c t u r e 8. By I w e a k l y in H 2 , and 1 H 2 convergence, with

~ l i m E ( h n) n~

is c o n f o r m a l

uniformly

and

of

= d(y,g,~)

that

u(S)

has

least

in its class. This

4.3

~1(S)

2.3.2

map

~ E(hn,S)

(u n)

~ E(u)

then

minimizing

Un~:

By Thm.

of a s u b s e q u e n c e )

and

Lemma

= ~ , where groups.

E(Un,S)

Since

energy

. By c o n s t r u c t i o n

(4.2.12)

since

by the

J.

a minimal

a n d T.

surface

of

to i n v e s t i g a t e

under also

curves

Rad6

in s h o w i n g

of

such

of T h e o r e m

4.2.1.

remarks

Douglas

consists

the p r o o f

the

that

independantly

a Jordan

topological

which spans

had

type

conditions a minimal

curve

solved

Platea~'s

in E u c l i d e a n

of the disc,

a Jordan

surface

curve

of h i g h e r

space

Douglas

problem spans suggested

or a c o n f i g u r a t i o n topological

type.

158

More in s h o w i n ~

precisely,

the existence

connectivity Euclidean

spanning

space,

topological

type

of s u r f a c e s

of

in p a p e r s

the

is s t r i c t l y

lower

genus [Doll,

but

- that

the moduli

he d i d n o t

of

lower

of

the c a s e of h i g h e r

about

genus

- and

and

likewise

that

Riemannian

Tromba

which

[Lu]

is e q u i v a l e n t

to s u r f a c e s

manifold

was

of prescribed

to r e p l a c e C.B.

surfaces

Morrey

his

by T r o m b a

hand

nonorientable

The considerations

surfaces, of

we refer

the present

[M2],

he s o l v e d

to

as-

of a minimizing We

were

should extended

by a general

connectivity,

the corresponding but genus

[Dol].

chapter

proof.

a priori

only. For

a complete

consider-

curvature.

space

and

sur-

to be too

statement.

Euclidean

of a r b i t r a r y

of

as a v a l i d

and Shiffman

mean

topo-

criticised

provided

to t h e c o m p a c t n e s s

of Courant

sequence,

consists

zero,

the o t h e r

a weaker

Recently, of D o u g l a s

of the considered

Courant

judged

this

treated

[Sf].

is t h e p o i n t

to b e a c c e p t e d

could only prove

to p r o p o s e

for m i n i m a l

were

was

of the p r o o f s

but genus

in [ T r l ] - o n

the considerations

Shiffman

surfaces

Whereas

genus

of

and

in s o m e

combinations

problem

compactification

in o r d e r

genus

surfaces over

consists

curves

of a m i n i m i z i n o

this

connectivity

enough

therefore

first man

problem

of this

the c a s e of h i g h e r

sumed a condition

The

space of

or connectivity.

and not detailed

by Luckhaus

and

the v a l i d i t y

show - and

the b o u n d a r y

faces

note

[Coi],

the c o m p a c t n e s s

proof

also

This

to g e t

type,

sequence

Courant

Jordan

infimum

In o r d e r

compactified

Shiffman

the

about

logical

ations

than

or c o n n e c t i v i t y .

problem

of prescribed

of area over

doubts

Douglas

vague

surface

of oriented

infimum less

of Plateau's

expressed

Courant.

by Tromba

of a minimal

a configuration

provided

by D o u g l a s

Tromba[Trl] and

his generalization

are based

on

[J8].

zero

159

5. H a r m o n i c

5.1.

diffeomorphisms

Heinz'

In l e c t u r e

estimates

of

the functional

3, w e o b t a i n e d

conformal

faces

by allowing

the conformal

wise,

we obtained

conformal

a free boundary indispensable formally mit

value

as two

Hence,

and

if w e f i x t h e c o n f o r m a l data

to o b t a i n

harmonic

on a compact

be demonstrated

the energy

feomorphisms

(this

stronger

feomorphism

[JK]].

of

limit.

D

with

M < ~/2K

Then,

for

are

data

on

by

sur-

Like-

solving

was of course

in g e n e r a l the disk

to o m i t

Assume

the J a c o b i a n

this

not

con-

do n o t ad-

is i n d e e d

again

is an u p p e r , on

shall

of)

dif-

time,

(after

we get a dif-

result

of t h e based

on

is t a k e n

from

proof.

is an

B(p,M)

we

This

estimates

The present

D ~ B(p,M)

J(u(x))

to 2.3)

diffeomorphisms,

difficult

and

[JS]).

to s h o w t h a t

the deep

harmonic

[HzS].

in

expect

the case will

the global result, I H 2 - limits

similar

are

or prescribe

can at best

and weak

however,

tools

u:

disc

For

sequence

e.g.

that

the u n i t 2 where K

That

we

first originated

the r a t h e r

is

boundary

tools,

These

cf.

of t h e d o m a i n

(uniform

for u n i v a l e n t

of E. H e i n z ,

5.1.1:

where

analytic

the minimizing

in t h e

We have

THEOREM map

genus

closed

to vary.

disks

freedom

with

lecture.

amongst

idea actually

from below

ideas

structure

diffeomorphisms.

we need much

Jacobian

between of

Dirichlet

surface

in t h e p r e s e n t

again minimize

the

of p o s i t i v e

between

the domain

extension.

Dirichlet

modifications

of

This degree

general

from below

diffeomorphisms

structure

diffeomorphisms

problem. surfaces

equivalent,

a cenformal

determinant

injective

harmonic

is a d i s c

on some

curvature

bound.

each disc

B(O,r),

surface

O < r < 1, we

have

(5.1.1)

IJ(u(x)) I >_ 6 1 > O depends

where on

M , r,

meas

and a suitable points

of

(5.1 .2)

and

and

one

interior assume

lower

either

normalization

If we f u r t h e r m o r e g (~D)

on u p p e r u(D)

for

on

x 6 B(O,r)

curvature the

like f i x i n g

bounds

Ca-norm

of

the i m a g e s

for

the image,

ulZD of three

or on

E(u)

boundary

point. that

g ": ul ~D

is a C 2

diffeomorphism

satisfying 0 < b -< i ~

I

for a l l

t 6 ~D

onto

160

and if

g(~D)

(5.1.3) for

is strictly

convex

w.r.t,

0 < a I -< ~g(g(~D)) (g(t))

the geodesic

curvature

u(D)

-< a 2

for all

62

as well

5.2.

depends

as on

of h a r m o n i c

in a c o n v e x

with

5.2.1

C2

~(~) 2 K ~0

([J2]):

that

~

is an upper C2

diffeomorphisms

~

the image

if the

on some maps

in some

is a compact

surface, ~

i~age

disc

curvature

there

exists

~I~

, and

this map

a diffeomorphism Moreover,

a harmonic

B(p,M)

b o u n d on

and on

M ,

is c o n t a i n e d

that

~

with

connected is another

onto

radius

B(p,M)

and

domain

its

M < ~/2~

that

surface.

image,

that

(where

the curve

~(~)

~(~)

mapping

u:

is a h o m e o m o r p h i s m

~ ÷ B(p,M)

between

~

with boundary and its image,

values and

in the interior. if

up

and

simply

homeomorphically

and convex w.r.t.

Then

diffeomorphism

for

,

Iglc1, a

Suppose

~: ~ + ~

is c o n t a i n e d

is of class

bounds

x 6D

disc

boundary

We assume

for all

on curvature

a l , a 2 , b , and

Existence

THEOREM

again

~ E~D

IJ(u(x)) I _> 6 2 > 0 where

, with

~I~

is even a

C2-diffeomorphism

then

u

is a

to the boundary.

Proof: First

of all,

the u n i t

disc

Approximation ~:

~D ~ ~(~D)

that the

~(~) following

(5.2.1)

and

b y Cor.

3.1.1

and Lemma

1.5.1,

w e can

assume

that

~

is

D. arguments, is a

cf.

e.g.

[J2]

C2-diffeomorphism

is n o t o n l y

convex,

quantitative

-d~2

but

, justify between

strictly

the a s s u m p t i o n

curves

of c l a s s

convex,

and

for

~ 6 ~D

bounds

~(~)

< bI

that C 2'e

,

that we have

161

i~ ~(~)

(5.2.2)

i

~ b21

and (5.2.3)

0 < a I <
k:

D ÷ ~(D)

this

map

Without

into

metrization

scribed

by

curve

:=y(iy-1(k(~))

we may of

of

of

k

assume

~D 9(D)

+ (1 _ I)~

values

is a c o n f o r m a l

values,

. Now

map

we now want

diffeomorphism

the orientation

the b o u n d a r y

u

to d e f o r m

.

that

the boundary

let

y

by arc

be a para-

length.

I (~(~))) ,

We

set

~ 6 ~D , I 6 [0,1].

i n t o the b o u n d a r y

values

pre-

~ .

Since we assumed

~(~D) 6 C 2,e , imply

a harmonic

of the boundary

~(~,i)

there

of boundary

loss of generality,

value map preserves

deforms

again,

. By a v a r i a t i o n

conformal

(5.2.4)

3.1.1

o f Cor.

_< a 2 .

well-

that

(5.2.1)

known

regularity

,

e(¢,l)

and

(5.2.2)

properties

hold

and

that

of c o n f o r m a l

maps

that

(5.2.5)

~(~,I)

are continuous

(5.2.6)

functions

Let now boundary

values

and

let

By t h e

for

any

uI

denote

e(',l) in6

[J~KI],

I ,

uI

and

I £ [0,1]

the harmonic

map

estimates and

from

of

be a sequence

uI

converging

for h a r m o n i c

the uniqueness

to the h a r m o n i c

D

map

to

follows

maps

(cf.

I 6 [0,1]

lecture

of J ~ g e r -

in the

with

from Lemma

to s o m e

theorem uI

B(p,M)

8)

continuously

on

1

C1'~-topology,

J(ul) (x) I

(J(ul)

denotes

the Jacobian

,

Kaul

In p a r t i c u l a r , p(1) : = i n f xED

depends

e(%,l)

and

, (the e x i s t e n c e

Theorem

converges n

} 6 ~D

[O,1]

a- priori

the Arzela -Ascoli

o < ~ < ~

of

~2 "~2

and

~--~ ~ ( ~ , I )

does not vanish

2.2.5)

-~

of

ul).

We

162

define

L :=

conformal (5.2.3)

{I 6 [O,1]:

map

and

the e x t e n t

k ) , and

(5.2.2),

.

By Cot.

therefore

which

L

implied

3.1.1

is not

(5.2.6)

, 0 6L

empty.

we can

(uo

Since

apply

is the

we a s s u m e d

Thm.

5.1.1

to

that

(5.2.7)

p(1)

Since

p(1)

Thus,

uI

>_ Po > O

depends

of

D

homotopy

for

continuously

is a local

boundaries by the

p(1) > 0 }

on

I,

diffeomorphism

and

u I (D)

lifting

, and

I 6 L (5.2.7)

and

. implies

L =

a diffeomorphism

consequently

[0,1]

between

a global

the

diffeomorphism

theorem. q.e.d.

5.3

Existence

THEOREM

5.3.1

boundary,

[Sh]

This

result

is t a k e n

this

of T h e o r e m and

(and h e n c e

Z2

a disc

handled

w.l.o.g, We

curve

and

that

let to

D

and

argument

the o t h e r

in the p r o o f

result

The

of

surfaces Then

to

without

there

exists

9 • Furthermore,

homotopic

earlier

surfaces

to

9 •

attempt

by S h i b a t a

of the m i s t a k e s

[J10] m i g h t

give

u

could

some

be

explana-

not work.

is r e s t r i c t e d and

to h a v e n o n p o s i t i v e

Sampson

z2(Zi)

[JS]) to

S 2 , then we can homotopic

class

two

(i = 1,2)

= O

higher

3.1.1.

projective

from

different

connectivity.

gives

a conformal

by Thm.

coverings.

a homotopically

topologically

3.2.1

~

find

Thus

The

space w e can

.

of diffeomorphisms

one having of Thm.

to

to the r e a l

to two - s h e e t e d

= 0

z2(E2)

curvature

[Sa].

are h o m e o m o r p h i c

by p a s s i n g

into

closed

compact

homotopic

[JS].

diffeomorphism Z2

are

(but n o t all)

(following

be the

Z2

from

homeomorphic

% . Since

separates

~I + Z 2

The

[SY2]

5.3.1 are

ZI

is s i m i l a r l y

homotopic

Yau

~2

diffeomorphisms

could

image

harmonic)

case where

assume

the

and

between

is a d i f f e o m o r p h i s m .

some

[Se].

approach

to S c h o e n -

ZI

all

although

case where

Proof

u:

among

by S e a l e y

EI

~I ÷ E 2

energy

was wrong,

is due

that

9:

least

tion why

If

that

diffeomorphisms

diffeomorphism

corrected

The

Suppose

and

a harmonic is of

of h a r m o n i c

ZI

trivial parts,

onto

one being

Therefore,

equicontinuity

~2

Jordan

the

of a w e a k l y

163

convergent

sequence

We energy

towards

topic

%.

sequence

weak

I H2-closure

in

~

subsequence

un

. A

, and

also

to

uniformly The

the

uo 6 ~ We

I H2

in

also

to

be

again.

weakly

converge

~

some

semicontinuity verging

9.

let

minimizing

I H2

in

again

in

uo

to

uo

can

, since

D

the hence

converging

un

in

by

lower

~

con-

and

homo-

in

equicontinuous,

is c o n t i n u o u s

weakly,

an

weakly

~

(Un)n6 N

are

uo

choose

converges

energy

a sequence

, and

, and

then

minimizes

find

. Since

uo

of

have

uniformly

they

bounded

energy, E(Un)

We sider

an

want

to

_< K

show

arbitrary

,

that

say

u

is

o

Xo £ Z1

point

.

a harmonic

and

diffeomorphism.

We

con-

define

O

B o :=

i.e.

the

open

We smaller where

disc

in

restrict

than the 2 < again

w.l.o.g.,

we

can

uniformly

to

uo

B(Uo(Xo),0)

~2

ourselves

a conformal

follows

the

Since

Fn

is

bound

an

upper

~ o :=

uol(B

~ n :=

Un

assume D

:=

UnOF

maps

ex£Sts

n a unique

values

prescribed

class

and In

of

Z~n

the

radius

to v a l u e s and

of

smaller

curvature

q.

o

which

than

of

Z2

the

un

are

w/2< . We

,

define

(n 6 N)

for

the

all

unit

maps

Cor.

is

n

, since

disc

in

the

complex

converge plane

and

0

to

x O.

(The

existence

of

Fn

3.1.1).

a Jordan

curve

of

a homeomorphism

mapping

UnOF n

is a d i f f e o m o r p h i s m . particular

for

homeomorphically

harmonic by

~2

with

D -~ ~ n

is

Fn ~D

sequel of

(B0 )

be

which

proof

U o ( X o)

) G

xo 6 an

. Let

is a d i f f e o m o r p h i s m ) , fore

the

radius

mapping

from

in

at

injectivity

Fn :

be

centred

, and

Vn: vn

onto D * Bq

class

CI

of

D

~B

. By

which

minimizes

(because

onto

~n

Thm.

assumes

energy

in

' and

5.2.J the its

un therethere

boundary homotopy

164

(5.3.1)

E D ( V n)

_< E D ( U n O F n ) = EQ

(u n) _< K n

by L e m m a

1.5.1

(Es(f)

Since

the

stays

in an a r b i t r a r i l y

can a g a i n are

apply

the

UnOFnl~D

lecture

8), w e

D

D

we

see

argument on

can

uniformly small

D.

, are

to a m a p

5.1.1, of

converge

equicentinuous

namely

on

un

is the e n e r g y to

the m a p p i n g

uo

of

section

equicontinuous.

which

furthermore

, we

neighbourhood

assume

over

assume

U o ( X o)

to s h o w

Using

that

vO

f

the

vn

that

set

of

converge

UnOF n ,

(see

uniformly

of

is a d i f f e o m o r p h i s m

we

vn

estimates

interior

S).

UnOFn(O)

the m a p s

values

a-priori

in the

the

. Therefore,

that

the b o u n d a r y

is h a r m o n i c that

can

of

3.1

In p a r t i c u l a r ,

therefore

v°

of

D

. Using

Thm.

in the

interior

E ( ~ n) ~ K

. We c a n

. We define

now VnOFn I

in

~n

un

in

~1\~n

and

in

I H2

and

that

the

=

n

Clearly, also

~n

is a L i p s c h i t z

assume

Then,

for

bounded

w.!.o.g.

each

from

n

, the f u n c t i o n a l

below

approximation

on

I H2

and

formly

on

compact

but

, and

that

0

does

UoOF to

D

UoOF

subsets , F

UnOF n

coincide

by u n i q u e n e s s

conformal

(5.3.2)

maps

that

~o 6 D

the

and

xo

. F

is d e f i n e d

n It is e a s i l y

u n converge

that

the

Fn

is n o t n e c e s s a r i l y

the f o l l o w i n g limit vn

there, energy

E D ( V o) Since

u

by an

of

ZI

UnOF n

since

minimizing

energy

_< E ~ ( u o)

.

weakly uni-

E D ( F n) = some

open

smooth

on

set ~D

,

arguments.

coincide and

on

on

and

thus

extends

~D , it f o l l o w s vo

is h a r m o n i c

in its h o m o t o p y

continuously

that

and

by L e m m a

1.5.1,

this

also

therefore

class,

-< ED(UoOF)

preserve

E ~ ( ~ o)

seen

C Ixe and

converge

to a c o n f o r m a l m a p F . Since 9 D diffeomorphically onto

to

and

([J~KI])

of

a r e of c l a s s

.

w.l.o.g,

is t h e u n i f o r m

. Since

5.1.1.

un

maps

not a f f e c t

and

determinant

~n 6 D

to a m a p

is m a p p e d

vo

lies

by Thm.

that

again

uniformly

A r e a ( ~ n) ~ A r e a ( Z I) cZ1

Z1\~n

argument

We can assume in

map

(by a p p r o x i m a t i o n )

implies

165

One

finally

(5.3.3)

shows

(cf.

EE1\~(Uo)

[J5]

for details)

that

= E~IA~(Uo)

Since

uo

is e n e r g y m i n i m i z i n g ,

(5.3.2)

and

ergies

of

uo

on

and hence

O n the o t h e r

and hand,

same boundary and

is

coincide

vo

values

hence

J~ger - Kaul, and hence

uo

was on

harmonic vo

also

neighbourhood

energy minimizing,

~D

Uol~

and

by Lemma u ° oF

and

EI

the

2.2.5.

is a h a r m o n i c

and

point

D

This

on

the en-

.

UoOF

By the uniqueness on

that

has

the

it a l s o m i n i m i z e s

diffeomorphism.

chosen

imply

also on ~ since

same energy,

then coincide

of an arbitrarily

(5.3.3)

result

shows

Since

ZI

of

that

~

UoOF

was a

, this proves

Thm.

5.3.1 . q.e.d.

In a s i m i l a r Dirichlet as

manner,

problem

also

a corresponding

a size restriction

on

the

result image,

for t h e

due

to

[J5]

well.

THEOREM

Let

c

~1

be a two - d i m e n s i o n a l

consisting

of

C2

5.3.2

boundary

~

morphism

of

are o f class Then

~

there

among

boundary

~

and all

image

convex

exists

least

energy

its

and

C2

to

same

~

onto

is h o m o t o p i c

the

one obtains

without

curves, ~(~)

with

respect

let

~:

suppose to

diffeomorphism

satisfies

u = ~

on

with

~ ÷ ~2

that

nonempty

be a homeo-

the curves

~(~)

~(~)

a harmonic

diffeomorphisms

values.

and

, and

domain

~

homotopic

u:

~ ~ ~(~)

. Moreover, to

~

u

which is of

and assuming

166

6. G e o m e t r i c almost

6.1

The

constructions.

linear

concept

In p o t e n t i a l (6.1.1)

n ~3

if o n e m

formulae

and

fixes

constructions

observation for

them

that

x 6~n\[o]

s a k e of e x p o s i t i o n )

the G r e e n M,

a point

one

m 6M

has m a n y

representation cannot

expect

important

formula. an identity

a n d d e n o t e s by

d(m,.)

con-

If o n e w o r k s like

(6.1.1),

the distance

, then £ d ( m , x ) 2-n N d ( m , x ) 2-n

least

order

in s o m e n e i g h b o r h o o d

ixl - n

cancel

cancel

only

orders

of m a g n i t u d e

allows

to derive

with

controlled

simple

manifold

(6.1.2) at

the

for t h e

amongst

on a R i e m a n n i a n

from

of curvature

theory,

(we a s s u m e

representation

functions.

g ixi 2-n = O

sequences,

but

Approximate

almost,

an error

lower order

each other i.e.

better

that

m

. Whereas

out,

the

up to an e r r o r than

approximate

term

than

of

corresponding

terms

terms.

of G r e e n ' s

is n e g l i g i b l e

the o t h e r

(6.1.])

term which,

the o r i g i n a l

versions

in

in m o s t

already

all

terms

terms

in

(6.1.2)

however,

is t w o

And

, (6.1.2)

thus

representation

applications

appearing

of

formula,

since

of

in the E u c l i d e a n

version. It is m o r e formula

for

achieved folds. which

with

These

is o n l y b a s e d that

the help

lead

Ix-pl 2

almost, linear

to the normal

(p = -q) i.e.

were

leads

idea

poor

functions in

[JKI].

but does

regularity

representation

can nevertheless

estimates

function

is t o u s e

the Euclidean

as a d e f i n i t i o n . error

in E u c l i d e a n second

after

second

term.

We

remark

that all and

Almost

term,

space,

on Riemannian Their

decisive

is t h a t not

use

be manifeature

the c o n s t r u c t i o n any a n g u l a r

properties

lower

the e r r o r bounds

or

of,

terms

e.g.,

that

the

2<x,p-q>

functions

=

first

Ix-ql

satisfy

characterizations

the

the T a y l o r

terms

for

identity

linear

the u s u a l

e.g.

ones v a n i s h ,

the

by upper

linear

this

coordinates.

are constant,the

trolled

but

introduced

comparatively

up to a s m a l l

functions

an a p p r o x i m a t e

as w e l l ,

to the d e s i r e d

on the d i s t a n c e

The basic -

to o b t a i n

of a l m o s t

functions

eventually

Riemannian

difficult

first derivatives

of

derivatives

expansion

terminates

discussed

above

can be con-

sectional

curvature,

a lower

167

bound

for

the

under

consideration, The

presented

6.2 We

injectivity

results

start with

LEMMA

6.2.1

fold

M

for

of

this

in

define

r(x)

the K

2

We

:=

curvature from

function

derivatives.

[JKI]

and

are re-

cut

locus

of

formulae

on R i e m a n n i a n

{x 6 M : d(m,x) ~ Q}

to

curvatures

-~

of the m a n i f o l d

and r e p r e s e n t a t i o n

distance

:=

disjoint

sectional

any

originated

solutions

the

B(m,p)

is

the d i m e n s i o n

involve

chapter

fundamental

Let

and

[J9!.

estimating

which

the

b u t do n o t

in d e t a i l

Approximate

radius,

be its

a ball centre

in m

manifolds.

some . We

maniassume

B(m,g)

in

2 <_ K

< <

d(x,m)

and

h(x) :=

and

P < ~2-~

½d(x,m) 2

Then

h6 C2(B(m,p),~)

and

(6.2.1)

[grad Kr(x)

(6.2.2)

h(x) I = r(x)

ctg(
<_ er(x)

for

x 6 B(m,p)

Proof:

grad Let

V6TxM

and

h(x)

q(t)

c(s,t)

Then

gradh(q(t))

-

coth(er(x))

• ]vl 2

.

-1 = - eXPx m

be a c u r v e

IvI 2 <-D2 h(v,v)

in

which M

with

implies q(O)

(6.2.1).

= x

and

-I = e X p q ( t ) (s e X p q ( t ) m )

~s c(s 't) I s=O

D v grad

h(x)

~-

~t

, and

hence

D ~s ~ c (s ' t) Is = O , t = O St

c (s t)

q(O)

= v

and

168

For fixed

t , Jt (s) = 7t c(s,t)

desic from

m

to

q(t) with

Jt(O)

D v grad h(x) = Djo(O ) grad h(x) D 2 h(v,v) (6.2.2) Jt(7)

is the Jacobi field along the geo= q(t)

=

follows from standard Jacobi

= O , Jt(1)

and

for the upper bound

in

We apply Lemma

J~(1)

Jr(1)

= 0 6 T m M . Hence

Since

= -<Jo(O),Jo(O)> field estimates,

, cf.

[BK]

are linearly dependant which

(6.2.2) 6.2.1

and

= -J'o(O)

(since

is needed

)

to construct approximate

fundamental

sol-

utions of the Laplace and the heat equation on manifolds. LEMMA 6.2.2 and

let

A

B(m,p)

Let be

the

be

Laplace-

as

in

6.2.1,

Lemma

Beltrami

operator

A 2 := max(<2,e 2) ,

on

M

, and

n = dim M ,

h(x) := ~I d(x,m) 2

(6.2.3)

A log r(x)

(6.2.4)

A r(x)2-n I _< n2--~2 A 2 r2-n(x)

(A-

(6.2.5)

_<

for

x

for

~ m

if

n =

2

x # m

if

n >_ 3

.~t) ( t -n/2 exp ( - --~-t-]h(x)~) 2A 2 ~

The proof follows Lemma

_< 2A 2

(_

exp

fo, r

through a straightforward

(x,t) ~

computation

(re,O)

from

6.2.1 q.e.d

In the same way as for example the identity is used to derive Green's r e p r e s e n t a t i o n to derive approximate

representation

£1xl 2-n = 0

for

x61Rn\{o}

formula we can use Lemma

6.2.2

formulae for solutions of the

Laplace and heat equation on R i e m a n n i a n manifolds.

In the elliptic case,

we have LEMMA 6.2.4 the

unit

B(m,p)

Let

sphere

in

~n

be

. If

as

above.

Let

~ 6C2(B(m,p),~)

denote

n

then

the

volume

of

169

(6.2.6)

if

n = 2

[~2 %(m) +

[ ~|B(m,p) g% " log r P(x)

_ 1 r P ]gB(m,p)

(6.2.7)

if

n _> 3

(~I

[ (n-2)c0n <~(m) +

I

_
B(m,p)

~B(m,p)

(n-2)n_1 I ~I p ~B (m, p)

I~I

r(x)n-2

-< n2~2 A2 I

pn

I%I B (m, p) r(x) n-2

We note that the error term is of lower order than the other two terms which are the same as in the Euclidean version of the Green representation formula. In the parabolic case, the corresponding LEMMA

6.2.5

Let

B(m,p)

B(m,p,to,t)

version is

be as above,

:= { (x,T)6 B ( m , p ) x

t(-,T) 6 C2(B(m,p),~)

[to,t]}

,

, %(x,') 6C1([to,t],~)

Then

(6.2.8)

I (7~) n }(re,t) +

[ (A] B (m, P,to,t) < exp<

c

<_

n

P n+2

f

]B(m,p,t O ,t)

+

(t-t°)-n/2

+

2A 2

I~

~B(m, p

[ ]B (m, p, t o, t)

+

P

~t ) %(x,~{) (t-T) -n/2

r2(x))) - (t-T 4

_p2 , exp( 4 (--~-7))) dx dT]

Cn I n+-----~

I} (x, t O)

r (x)=P to_
r 2 (x) (x,T) i ~-~_-~ (t-r) -n/2 exp (

r 2 (x) 4 (t-T)) "

170

6.3

Almost

linear

We n o w of

functions

introduce

almost

linear

functions,

one of the m a i n

tools

[JKI ]. Let

manifold vature

B(m,g) M

be a g a i n

which

a ball

is d i s j o i n t

in some

to the cut

n- dimensional

locus

of

m

Riemannian

, and

assume

cur-

bounds 2

~2

2

and

P
r(x)

DEFINITION p(x)

= d(m,x)

6.3.1

U6TmM

Let

= eXPm(r(x)u)

, q(x) £(x)

is c a l l e d

almost

an

We observe cisely more

the

note

•

i.e.

_ d(x,p(x) ) 2)

in the E u c l i d e a n because

case,

this

notion

of P y t h a g o r a s '

-r(x)

of

[JKI]

-< £(x)

the a s s o c i a t e d vector

field

(6.3.2)

grad

(6.3.3)

D 2 £(x)l

(6.3.4)

l£(x)-
For

pre-

We f u r t h e r -

for a l m o s t

-< r(x)

linear

functions

are c o n t a i n e d

T H E O R E M 6.3.1 Suppose B(m,p) is d i s j o i n t to the cut 2 <2 A2 7r -~ < K < , [KI -< on B(m,p) , a n d p <~ Let £(x)

yields

theorem.

that

estimates

parallel

lul = I, a n d p u t

. Then

function.

functions,

(6.3.1)

The

vector,

I 4r(x) (d(x'q(x))2

linear

that

linear

_

be a u n i t

= eXPm(-r(x)u)

the p r o o f

£(x)

of Thm.

almost on

linear function,

B(m,p)

- u(x) I _< 2
6.3.1,

U(X)

and

= u

u6TmS

.

of

m ,

, lul = I,

the r a d i a l l y

Then

sinh(2Ar)sin~ " r2(x)

9
£(x),-exPxl

u(m)

with

locus

in

m>l

we refer

er c t g h ( e r )

-<

(9

to

r(x)

sinh (2Ar) sin(2~r)

[JKI]

or

[J9].

er c t g h ( e r ) )

r3(x)

171

6.4

An approximate

We f i r s t c o n s t r u c t

representation a Riemannian

formula for first derivatives

analogue

of a d e r i v a t i v e

of G r e e n ' s

f~nction LE~tMA 6.4.1:

B(m,p)

Let

be

as

before.

x 6 B(m,p)

For

, x # m

, we

define

a(x) l (x)

where

is

:= £(x) (r(x)-n - p-n) an

almost

linear

,

function.

Then

i£al

(6.4.1)

Lemma

< 9n2
6.4.1

the E u c l i d e a n

sinh(2Ar) sin(2Kr)

implies

same

6.4.2:

t h a t the g r a d i e n t

c a s e by d i f f e r e n t i a t i n g

a g a i n h o l d s on R i e m a n n i a n

LEMMA

er ctgh({~r) r

as

bound

Green's

for

error

B(m,p)

where

x # m

term.

satisfies

the

before.

n ~°n ] g r a d g ( m ) i - < ~ +

[ ]ZB(m,p)

f ] B (m, p) r n-1

[g(.)

c A2 I

- g(m) l + |

Ig(')- g(m)I

]B(m,p)

Here

o

is

a constant

which

depends

only

rn-1 (.)

on

n

and

Ap

.

in

formula,

Then

(6.4.2)

.

that is o b t a i n e d

representation

up to a small

g6C2(B(m,p),IR),

Suppose

assumptions

manifolds

-n+1

172

7.

The heat

flow method.

into a nonpositively

7.1

The

existence

Let

g: X ~ Y

curved

theorem

fundamental

question

is h o m o t o p i c

In t h e p r e s e n t existence

and uniqueness

maps

Sampson

mapping

in t h e t h e o r y

chapter,

of harmonic

image

of E e l l s -

be a c o n t i n u o u s

whether

g

Existence

between

Riemannian

of harmonic

manifolds.

mappings

The

is to f i n d o u t

to a h a r m o n i c m a p . we

shall

discuss

the heat

flow approach

to t h i s

question.

Here, g: X ~ Y

,

in o r d e r one

to f i n d a h a r m o n i c

investigates

9u(x,t) ~t

map

the parabolic

- • (u(x,t))

homotopic

to a g i v e n m a p

system

for

x 6 X

for

x 6 X

and

t ~ 0

(7.1.1) u(x,O) and one and We

tries

that

for

assume

that

X

for a n e x i s t e n c e

it is o p e n

theorem

[Ht]).

THEOREM

7.1.1:

(7.1.1)

Suppose

subset

for

of

uniformly

Y all

Besides

ideas of

Wahl

[vW]

have

to r e f e r

and Jost to

compact

value

whether lecture

Sampson

[ES]

[ES]

[J3].

[J9].

map

if

as

[Ht],

For many

for all

t ~0

t ~

and have

no b o u n d a r y

case,

the we

and

solution

shall

[Hm]

we

Y t ~ ~

of

obtain

(cf.

and

[J3]

(7.1.1) the following

improvements

curved.

Then

. I f the s o l u t i o n

to a h a r m o n i c

and

as

(with t h e

is n o n p o s i t i v e l y t 6 [0, ~)

exists

u

problem).

Y , in p a r t i c u l a r

is h o m o t o p i c

the

are

in g e n e r a l

to a h a r m o n i c

g 6 C°(X,Y)

(7.1.1) map

in t h e h o n c o m p a c t

in the p r e s e n t

of Eells-

exists

Y

boundary

time,

Hartman

verges

and

for all

of

to a h a r m o n i c

result

Whereas

existence

bounded

that a solution

converges

the corresponding

exists

of

to p r o v e

u(-,t)

shall

[SY1]

= g(x)

the s o l u t i o n

remains

is c o m p a c t ,

by

then

In p a r t i c u l a r ,

in a it c o n any m a p

map.

shall

also

of the details

use of

arguments

the proof,

of v o n we again

173

7.2.

Proof

of t h e e x i s t e n c e

Linearizing for

(7.1.1)

linear

LEMMA

only

and using

parabolic

on the geometry

Thus we obtain restriction

COROLLARY

exists

X

small

on

Y

Y

function

there

and on

for

existence

theorem

and results

is some g

with

s

>0

depending

the property

that

0 ~t < e .

for

(7.1.1)

even without

a curvature

for which

the solution

of

.

The set of

7.2.1

. Then

and

u(x,t)

time

implicit

one obtains

g 6 (X,Y)

of

has a s o l u t i o n

(7.1.1)

the

equations,

Suppose

7.2.1:

theorem

for

t £ [O,T]

This

follows

T 6 (0,~)

(7.1.1)

is open.

by

taking

u(-,t)

as

initial

values

in L e m m a

7.2.1.

q.e.d.

We now of

show

that

E(u(.,t))

the energy

is a d e c r e a s i n g

function

t . For,

d

(7.2.1)

since

d--t E ( u ( . ,

u

satisfies

Furthermore, (7.2.2)

one

be(u)

e

again

is t h e e n e r g y In o r d e r

- ~

X

=

of

frame on

of

I
u,du >

~-{ u = T(u)

(7.1.1)) ),du • e >

X , and

e(u)

of course

u. (7.2.2),

we

shall

is c o m p a c t

and,

f o r ~example,

(7.2.2)

i.e.

=

• e ,du - e B) d u • e a , d u - e~ >

t h e sectional

and

du,du >

l?dul 2 + < d u • R i c X ( e

that

is b o u n d e d ,

<

(for a s o l u t i o n

is a n o r t h o n o r m a l

u s e of

=

(7.1.1),

chapter

rest of this Since

e(u)

density

to m a k e

the equation

calculates

-

where

t) =d~ ~lJ

therefore

assume

curvature of class

implies

f r o m n o w o n for

of

Y

the

is nonpositive.

C 3 , its R i c c i

tensor

174

(7.2.3)

£e(u)

If we now a p p l y

(6.2.8)

exp(-y) we o b t a i n

- ~ t e(u)

from

_> - c e(u)

and use

-< c y

for

y > 0 ,

e 2_ 0 ,

(7.2.3)

I (7.2.4)

e(u) (m,t)

C r n ~ n+2 JB(m, P P , to,t)

+

+

Here, of

cI

_< c I ~ J B ( m , p , t o , t)

(t-to)-n/2

depends

oll

n

P

A2

2 r(x)

!f ~r (x) ~p to-< T-< t

-n+1

dx dT

e(u]

e(u) (x,t O) dx

, a bound

for the s e c t i o n a l

curvature

X.

For a s u i t a b l e r(x)

= p

choice

of

into a v o l u m e

(7.2.5)

p, one can absorb term,

]r(x

For example,

~

e(u)

min ( i ( x ) , ~ )

If we iterate the first

integral

(7.2.4),

in

-< --2 ] 0 B(m,p,to,t )

e(u)

__s ,

LEMMA

c3(t-t°)-n/2

integral

with

t

after

integral

for

e(u) (x,7)

a finite

number

at

t - n ( t - t o)

dx

because

(7.2.1))

t I >_ 0 , we o b t a i n

e(u) (m,t)

~ c4((t-t])

in

of steps

from

(7.2.6),

using

(7.2.1)

again

7.2.2

(7.2.7)

at

e(u)

IX e(u) (x,t-n(t-to))

m a y be l o c a t e d

by

again

[ ]B(m,np,t_n(t_to),t)

e(u) (re,t) < c 2 p

+

(7.2.4)

we o b t a i n

-n-2 (7.2.6)

the b o u n d a r y

namely

)=0 to-< T-
If

e (u) + ~ C n

[ JB(m,p)

and

e(u) (x,T) (t-T)

p

-n-2+-n/2)

I

e(u) (x,t 1) dx X

the

175

We now want u(x,t,s)

to d e r i v e

be a smooth

parameter

s ,

LEM~. 7 . 2 . 3 sectional

Hartman's

stability

lelmaa. F o r

family

solutions

of

O < s _<so

of

, with

[Ht])

(Hartman

initial

Suppose

(7.1.1),

values

again,

this purpose, depending

u(x,O,s)

that

Y

let

on a

= g(x,s)

has n o n p o s i t i v e

curvature.

For

every

s 6 [O,So] sup x6x

is n o n i n c r e a s i n g

in

t.

(u(x,t,s))

-

gij

~s

Hence also

x6X, s6 [O, s o ]

is a n o n i n e r e a s i n g

Proof:

One

function

calculates

of

t .

in n o r m a l

Zu i 8U j ) -

~2ui

~2uJ

3S

~xaSs

$Xa~S

nonpositive

sectional

~s

and since

Y

has

(cf.

coordinates

(7.2.2)) Zu i 9u k ~U j ~u £

Rikjl

curvature,

9s

~x a

~s

~x ~

hence

~u~ ~u J The

lemma

then

follows

from the maximum

principle

for p a r a b o l i c

equations.

q.e.d.

We now assume X

to

ul (x)

Y

that

, and

h: X x

, h(x,1)

= u2(x)

Since necting

h(x,s)

u I (x)

[0,1]

u 2 (x)

from

parametrized

proportionally

curved,

geodesic

We define

u I (x)

arc

d ( u I (x),u2(x))

We then put [O9])

and ~ Y

is s m o o t h

and

the geodesic

this

uI

u2

in

x

u 2(x) to a r c

and

which length.

= g(x,s)

length

homotopy

on

with

x

. We

is h o m o t o p i c Since

and hence

to b e t h e

homotopic

s , the curve

smoothly

is u n i q u e

u(x,O,s)

smooth

is a s m o o t h

depends to

are

Y

and derive

from

h(x,O)

=

h(x,-)

let to

con-

g (x, " )

be

h(x,.)

and

is n o n p o s i t i v e l y

depends

of this

maps

smoothly

geodesic

from Lemma

on

x

.

arc. 7.2.3

(cf.

176

COROLLARY Assume and

7.2.2

that

Suppose,

as before,

the solution

t 6 [O,T]

that

u(x,t,s)

of

Y

(7.1.1)

is n o n i n c r e a s i n g

in

t

use L e m m a

of a s o l u t i o n

of

LEMMA

Suppose

7.2.4

has no n p o s i t i v e

for

7.2.3

S 6 [O,1]

This

I))

t 6 [O,T].

to o b t a i n

u(x,t)

a bound

for t h e

time

derivative

follows

for

t 6 [O,T]

and

Then for all

t 6 [O,T]

and

x 6X

and

Y

solves

curvature.

311(x,t) ~t

< sup x6X

(7.1.1)

Lemma

Y

~ t u(x,O)

by p u t t i n g u(x,t,s)

and applying

for all

(7.1.1).

sectional

(7.2.9)

Proof:

exists

curved.

. Then sup ~ ( u ( x , t , O ) , u { x , t , xEX

We c a n a l s o

is n o n p o s i t i v e l y

7.2.3

at

s = O

= u(x,t+s)

. q.e.d.

LEMMA

7.2.5

Suppose

has no n p o s i t i v e

(7.2.10) c5

sectional

depends

on

Proof:

We w r i t e

radii

(7.1.1)

and

The proof

then

argument,

see

statement will

follows

(7.1.1)

for

Then for

t 6 [O,T]

every

~ 6 (0,1)

[ _< c5 . ~~u (.,t) ic~(x,y)

values

g(x)

dimensions

of

= u(x,O) X

and

, and on curvature Y .

as

I ~ (/~ya~ /7 ~ x~

(7.2.11)

Y

curvature.

~ , the initial

injectivity

and

solves

lu(.,t) Ic2+~(x,y) +

bounds,

The

u(x,t)

~u i ) = _y~B ~i ~xS

~u j ~u k

jk

from Lemmata

7.2.2

~u i

~x ~ ~x ~ +

and

7.2.4

....... 3t

a n d a n easy

bootstrap

[J9]. concerning be p r o v e d

the d e p e n d a n c e in l e c t u r e

of

c5

on

the g e o m e t r y

8. q.e.d.

of

X

177

LEMMA has

7.2.6

The

nonpositive

Proof: that

solution sectional

Lamina 7.2.1

the s o l u t i o n

Lemma

7.2.5

exists

for

t 6 [O,~)

all

, if

curvature.

shows

exists

implies

(7.1.1)

of

that for

that

the

all

set of

T 6 [O,~)

t 6 [O,T]

is o p e n

it is also

with

the p r o p e r t y

and nonempty,

while

closed. q.e.d.

If w e use

the e n e r g y ~ E(u(. ~-~

observe

that

E(u(.,t))

time independant

LEMMA

7.2.7

t))

formula

= -

I

"

exists

a

I~I,

namely

2

~t

definition

for

(7.2.1),

I 9u(x't) X

is by

Ca-bound

There

decay

dX

always

nonnegative,

w e obtain,

(t n)

sequence

,

since

, tn ~ ~

Y

as

and

use the

is c o m p a c t

n ~

~ , for

8u

~-~ (x,t n)

which

Now assume,

using

converges

the

by p o s s i b l y

to a h a r m o n i c

may

because

map

of L~mna

convergence,

continuous

in

of

(7.1.1),

This

u(x,t)

finishes

u(x,O,s

selection

uniformly

the p r o o f

=

can

converges

7.2.2

which

we

u(x)

is a time t

of the

of Thin. 7.1.1

we

.

all,

since

u(x,t)

is

.

all

converges

u ( x , t n)

~

put

. Cor.

to

independent 7.2.2 for

< d(u(x,t n),u(x))

the

n ~

of Lamina 7.2.5, that

(and h e n c e u

map

for

as

= u ( x , t n)

o)

to

as a h a r m o n i c

that

then

u ( . , t n)

u ( x , t , s O) = u(x)

it f o l l o w s

and that

=

x 6X

t n ~ ~ . In Cor.

= u(x,O,O)

some

d(u(X,tn+t),u(x)) Hence

we

in

u(-,t)

as

7.2.6,

t ) are homotopic u(x)

for

u(x)

g(x,so) By u n i f o r m

uniformly

to a s u b s e q u e n c e ,

g(x,O)

Since

zero

C2+e-bounds passing

uniformly apply

to

all

subsequence u(x)

as

then

solution implies

t ~ O

.

is n o t n e c e s s a r y

t ~ ~ .

178

7.3

The uniqueness

The following [SY4]

and

THEOREM

uniqueness

[ B u ])

is b a s e d

X,

let

results

7.2.3

(Hartman

harmonic maps from fixed

of Hartman

. We again

on L e m m a

7.3.1

theorem

X

u(x,s)

refer

are due

to H a r t m a n

to

for details

[Jg]

[Ht]

of t h e p r o o f w h i c h

.

Let

[Ht])

into

u1(x)

be two homotopic

, u2(x)

the n o n p o s i t i v e l y

curved m a n i f o l d

be the unique geodesic from

u1(x)

in the homotopy class d e t e r m i n e d by the homotopy between and let the p a r a m e t e r

s 6 [0,1]

Then, for every E(u(-,s)

S £ [O,I]

= E ( u I) = E ( u 2)

is independent of

u(x,')

(cf. a l s o

be p r o p o r t i o n a l , u(',S)

. Furthermore,

Y . For

to uI

u2(x)

and

u2

to arc length.

is a harmonic map with

the length of the g e o d e s i c

x .

Hence any two harmonic maps can be joined by a parallel family of harmonic maps with equal energy. If u:

X ~ Y

maps

X

Y

has negative sectional

curvature,

is unique in its homotopy class, onto a closed geodesic.

then a harmonic map

unless

it is constant or

In the latter case,

only occur by rotations of this geodesic.

nonuniqueness

can

179

8.

Harmonic

8.1. The

coordinates

Harmonic purpose

Riemannian nate

of

this

section

manifolds,

functions

are

i.e.

We

almost

linear

functions.

have

of

best

a

on

curvature

fixed

above.

Thus,

p

if

is

orthonormal the

We

then

as

ii

the

look on

at

g 1

some

Also,

size

is

(e I . . . . .

e n)

almost harmonic

ball

shall

and

on

the

Riemannian of

i.e.

(n=dim

on

hi

with

M

, we and

Def.

the

of

the

coordinates

quantities

M)

cf.

can

in t e r m s

dimension

harmonic

on

coordi-

we

symbols

manifold

TpM

based

harmonic

example,

geometric

functions,

we

the

in g e o -

argument

that

For

construct

coordi-

construction

Christoffel

radius can

function

B(p,R),

see

properties.

we

linear

this

on

individual

choose

let

1 I,...,I n I)

6.3.1.

same

an

boundary

values

solve B(p,R)

)

aim

is

to

injective,

Let

mappings

coordinates

the

a perturbation

we

depending

hil ~ B ( p , R ) Our

harmonic

harmonic

control by

injectivity

some

the

to

£h I = O

(8.1

of

for w h i c h

corresponding

in

corresponding

construct

regularity

a point

base

and

the

only.

their

to

Eventually,

for

bounds,

balls,

is

achieve this

possible

manifold

mentioned

be

shall

Ca-bound

underlying

regularity

coordinates

harmonic,

terms.

achieve

higher

coordinates

metric

nates

and

for

= ill ~B(p:R)

show

that

i.e.

a coordinate

i 2 = max(IK 1) , where

suitably

K

chosen

R

,

H =

(h I

,h n)

map. is t h e

sectional

curvature

of

M

, and

R < -2A We put k i = hi - ii Our then

second

(8.1.2)

We want

I)

derivative

implies

estimate

because ]£kil

to u s e

~

of

for

All the following particular choice

almost

linear

I .....

n)

functions,

(6.3.3),

(8.1.1)

9nA 2 • A d ( x , p )

(8.1.2)

(i =

ctgh(Ad(x,p))

in c o m b i n a t i o n

with

sinh(Ad(x,p • sin(Ad(x,p)

the maximum

constructions can be made of base by averaging over

)

• d(x,p)

princlple,

in

independant of t h i s all orthonormal bases

180

order

to control

ki

and choose

therefore

~(x) = Co A2(d 3(x,p) - R 3) By Lemma

6.2. I &~(x)

_> Co A2(3d 2(x,p) (n-l) A ctg(Ad(x,p))

For given R -< R o < ~ , we can calculate c0 k i + ~ is sub- or superharmonic, resp. Since maximum principle

Co(A-Ro,n) for which k I ± ~I~B(p,R) = O , the

implies

(8.1.3)

[ki(x)[-<

and for

+ 6d)

x I 6 ~B(p,R)

I~(x)I

_< Co A2 R 3

, x2 6B(p,R)

[ki(Xl ) - ki(x2) 1 (8.1.4)

Col~(x2)l

3Co A2 R 2

-<

tx I - x21

Ix I - x21

or Iki(x2)l

(8.1.5) Let

x 6B(p,R)

,

_< 3Co A2 R 2 d(x2,3B(p,R))

p := d(x, BB(p,R))

. Lemma

6.4.2,

applied on

B(x,p)

,

yields n J[ ki(x) l _< --~ p ~B ( X , p )

enlgrad

[ JB(x,p)

and hence with

(8.1.2)

(8.1.6) Here

c 2 = c2(ARo,n)

(8.1.7) Here,

transport

and

(8.1.5)

remains yields ~ ~n

IdH-id[

the map ei(q)

[ ]ki(y)-ki(x)l JB(x,p) d(x,y) n-1

IAki(y) l dy + Cl(AP,n) d (x,y) n-1

Igrad ki(x) l < c2 A2 R 2

Using (6.3.2), (8.1.6) (h I, ... ,h n) : B(p,R)

via

[k i (y) - k i (x) I dy

id

where

bounded

for fixed

n

and

for our wanted coordinate

< c3(ARo,n)

/n A 2 R 2

from identifying

TqM

ei(q)

is obtained

e i = ei(p)

along radial

geodesics

map

H =

on B(p,R)

arises

from

RO ~ O .

with

~n

(q6 B(p,R))

via parallel

dv

181

(8.1.7)

implies

enough.

Thus

THEOREM

8.1.1:

only

that

H

is injective

For each p 6 M A 2 = max(IK I ) ( K is

on

(the i n j e c t i v i t y

radius

B(p,R)

that o n

there

of

provided

there the

exists

some

sectional

is chosen

harmonic

small

R > 0 , depending

curvature

of

n = dim M , w i t h

p ) , and

exist

R

M ) , i(p)

the p r o p e r t y

coordinates.

We let gjk =
(8.1.6)

(8.1.8)

hk>

tensor of those harmonic

coordinates. From

(6.3.2)

again Ig i k - 6ik I = l
hk>-
h k - uk>I

(2 + c3 A2 R2) c3 A2 R2 = c4 A2 R2 (8.1.8)

implies I

llgikll~

-< I - c 4 n A ~ R

and hence (8 1 9) " "

igik

6ik I

c4 A2 R 2

-

~

llgiklI~

We now want to estimate coordinates.

A calculation

LEMMA 8.1.1: (e i)

Let

the Christoffel

is a n o r t h o n o r m a l

symbols

for harmonic

yields

(h I, ... ,h n)

H =

c4A2 R2 ~ I - c 4 n A ~c---~z RZ

frame,

be

harmonic

satisfying

V i(ej)

coordinates.

= O

Then,

if

at

x

on

n, A 2 , i(p),

e

(8.1.10)

Ag ik = A
e where

Rml

LEMMA 8.1.2: with

is

the R i c c i

There

the p r o p e r t y

coordinates

tensor

exists

that f o r

the m e t r i c

tensor

e3e £ M

RO >0 RfR o

all g

+ 2 hi •

e of

some

hk> hk ~Je £~

'

.

, depending on

of which

B(p,R)

only there

satisfies

exist

harmonic

182

(8.1.11)

c 5A 2 R 2 -< d(x,ZB(p,R) i

idg(x)l

for

x 6 B(p,R)

,

C 5 = c 5 ( n , A R O)

where

Proof:

Since

(8.1.12)

e I
in n o r m a l

coordinates,

hi,grad

(8.1.7)

i h k • + h i h~. h k > = hej # e3 e] 3e l

and

(8.1.10)

imply

for the m e t r i c

tensor

g = (gij) 9 2R2 ) 2 lag I S 2 !IRicll (1 + c 3 A 2 R 2 ) 2 + ~(I + e 3 A Idgl

(8.1.13)

We now use a m e t h o d Let Then

there

p ::

max x6B (p,R)

is some

(8.1.14) Let

of Heinz

[Hzl]

d(x,~B(p,Ro))

x I 6 B ( p , R o)

d(Xl,3B(p,Ro)),

By Lemma

6.4.2

applied

~ -< dnen

idg(Xl)~

~ = ]dg(Xl)b B(xl,de)

,

Ig(x) - g(xl)l

O < 8 < I ,

+ c6 ~ B (x I ,de )

+

c7A

ig(x)

2 [ ] B (x I ,de )

By

iAg~x) J d(X,Xl )n-1

]

d (x,x I )=de

=:

.

with

on

[

c5

(8.1.15)

i.e.

(8.1.11)

Idg(x)l

p = d(x1,~B(p,Ro))

d :=

to o b t a i n

I + II + iII

- g(xl)l

d (x,x I )n-1

.

(8.1.8) c8A 2R2 i <_

by

'

(8.1.13)

2

ii < % if we choose _> d

de

and by

de(11~ic11

I 8 _< -~ , since

(8.1.8)

again

then

+ Idgl 2) -< c 9 11RicIlde + 2c 9 de ~ for

x £ B(Xl,de)

d(x,3B(p,Ro))

,

>- d(1-e)

183

III S clO A4 R 2 d0 Hence I 2R2 d282 R2 2 2 ~ ~(csA + c 9 ][Ric]l + C i o A4 d2~ ) + 2c9g~

(8.1 .16)

~I

=:

2 a A 2 R 2 + b8 2

a

and

b

d e p e n d o n l y on

We n o w c h o o s e

Ro

(8.1.17)

AR o

so s m a l l

(for

R ~ R o)

that

that

(8.1.18)

If

and

abA2R 2 < I o

We c l a i m

which

n

~ < 2aA2R 2

then implies

(8.1.18)

would

(8.1.11)

by d e f i n i t i o n

not hold, w e c o u l d

e

aA 2R2

of

choose

Z . e

in

(8.1.16)

as

_
-

2 and o b t a i n I abA2R 2 ( 2 + 2

~ < contradicting

(8.1.17) q.e.d.

We n o w w r i t e

the L a p l a c e -

It then b e c o m e s (by L e m m a by L e m m a

8.1.2).

THEOREM

8.1.2

the i n j e c t i v i t y fold

X

property

for

tenser

M

in h a r m o n i c with

the r i g h t h a n d side of

and hence

p 6 X

radius

and bounds

the m e t r i c

on

operator

elliptic

regularity

coordinates.

C1-coefficients

(8.1.17)

theory

is b o u n d e d ,

implies

g . We obtain

Let

that f o r

operator

type elliptic

Furthermore,

8.1.2 again,

C1'e-estimates

Beltrami

a divergence

of

for R

g= dglc~

. There

exists

the s e c t i o n a l

~ R

e

(gij)

Ro >O

p , the d i m e n s i o n

there

curvature

exist

of which

~ c(ARo,n,a,~)

harmonic

satisfies

A2R 2

, depending

n

of on

B ( p , R O)

coordinates on

solely

the c o n s i d e r e d

each

ball

for any

with on

on manithe

B(p,R)

B(p, (I-~)R)

~ 6 (O,1)

184

In particular, are b o u n d e d

the H~lder

in terms

A corresponding as

the

norms

of

AR 0

estimate

for

example

of

following

o f the c o r r e s p o n d i n g

and

symbols

n .

Riemannian [JEll

normal

coordinates

cannot

hold

shows:

G2(r,~)

ds 2 = dr 2 +

Christoffel

d~ 2

with

G2(r,~)

For

this

That

harmonic

~ < ~ < 2~

- ~ 6

sina9 I + r2sin~

for

0 < % < -

0

for

z < ~ < 2~

K

I

_

is H 6 1 d e r

coordinates

observed of

Higher

bounds

for

the metric

this

order

best

possible

Kuzdan

[dTK]

section

were

supplied

in

posses

for

harmonic

maps

estimates

and

8.2.1: Let X 2 2 - e X -< K X -< K X

.

regularity , and

the

is n o t

even

X~

Y

< rain (i (p), ~ y

is harmonic

then for all

R
and c B(p,~

,

o

d ( u ( x ) , u ( x o) ) I V u ( x O) I -< c O

C o = Co(Ro,eX,~x,dimX,

con-

be R i e m a n n i a n m a n i f o l d s with c u r v a t u r e 2 2 - e y -< K y < ~ y , resp. , x O £ X , p 6 Y

and

p

properties

explicit

[JKI].

Y

u(B(Xo,Ro))

(8.2.3)

.

tensor

R O < m i n (i (xO) ' ~ < X

(8.2.2)

U:

continuous,

by de Turck-

(8.2.1)

If

(0 < a < I)

continuous.

first

THEOREM

r

G rr G

although

structions

8.2.

0 _< ~ _< ~

metric

Lipschitz

was

for

=

K -

Here,

r2(I + r 2 s i n a ~ ) 2

max x £ B (x o, R) p,ey,
R .

185

The proof is similar to the proof of L e m m a 8.1.2, u s i n g instead of (8.1.8) the estimates for the m o d u l u s of c o n t i n u i t y of v a l i d b e c a u s e of a s s u m p t i o n

(8.2.2), cf. H i l d e b r a n d t ' s

u

that are

lectures.

We now c o n s i d e r the system I

~

(/~ y a ~

/7 ~x ~

~

u I)" + y aS F i

~--~

~uJ ~uk

=

O

jk ~x a ~x 8

in h a r m o n i c c o o r d i n a t e s on domain and image. Thus (8.1.2) and

(8.2.1)

then imply in c o n j u n c t i o n w i t h standard e l l i p t i c r e g u l a r i t y theory T H E O R E M 8.2.2: Under

the a s s u m p t i o n s

o f Thm.

8.2.1

IUlc2,a(B(Xo,6Ro)) and

~ 6 (0,1)

any

where

cI

depends

~ eI

for

any

a 6

(O,1)

, on

the

same

The r e m a r k a b l e feature of Thm.

quantities

as

co , and

8.2.2 is that even the

on

~

and

~ .

C 2 ' a - n o r m of

harmonic m a p s can be c o n t r o l l e d w i t h o u t using any c u r v a t u r e derivatives. In a similar way

THEOREM Under

8.2.3:

the

assumptions

o f Thm.

8.2.1

IUIH~(B(Xo, gRo)) and and

6 £ (0,1) , w h e r e on

p

and

6

c2

again

S c2 depends

any

p 6 (I,~)

same

quantities

for on

the

as

C0 •

.

The results of this section are again due to J o s t - K a r c h e r can also be found in [J9]). Results similar to Thm.

[JK1]

(details

8.2.1 w e r e also

o b t a i n e d by Choi [Ci], G i a q u i n t a - H i l d e b r a n d t

[GH], and Sperner

C o r r e s p o n d i n g e s t i m a t e s hold at the boundary,

cf.

[Sp].

[JK1] or [J9] again.

186

Bibliography

[Ad ]

Adams, J.F., M a p s f r o m a surface to the p r o j e c t i v e plane, Bull. L o n d o n Math. Soc. 14 (1982), 533 - 534

[AB ]

Ahlfors, metrics,

[Au ]

Aubin, Th., Equations d i f f 6 r e n t i e l l e s non lin~aires et p r o b l @ m e de Yamabe c o n c e r n a n t la courbure scalaire, J. Math. Pures Appl. 55 (1976) 2 6 9 - 296

[Ba]

Baldes, A., Stability and u n i q u e n e s s p r o p e r t i e s of the equator map from a ball into an ellipsoid, Mz 185 (1984), 5 0 5 - 516

[BcC]

Benci, V., and J. M. Coron, The D i r i c h l e t p r o b l e m for h a r m o n i c m a p s from the disk into the E u c l i d e a n n-sphere, P r e p r i n t

[BJS]

Bers, L., F. John, and M. Schechter, Partial d i f f e r e n t i a l equations, Interscience, New York, 1964

[BI]

Blaschke, W., V o r l e s u n g e n Hber D i f f e r e n t i a l g e o m e t r i e , Springer, Berlin, 1945

[BCI]

Brezis, H., and J.M. Coron, M u l t i p l e solutions of H - systems and Rellich's conjecture, Com/~. Pure Appl. Math., to appear

[BC2]

Brezis, H., and J.M. Coron, two dimensions, Comm. Math.

[Bu]

Bunting, G., Proof of the u n i q u e n e s s c o n j e c t u r e for black holes, Thesis, Univ. of New England, 1983

[BK]

Buser, P., and H. Karcher, A s t ~ r i s q u e 81 (1981)

[Ca]

Carter, B., The Bunting identity and a g e n e r a l i z e d M a z u r identity for non - linear e l l i p t i c systems including the black hole equil i b r i u m p r o b l e m and gauge coupled d-models, P r e p r i n t

[ci]

Choi, H.J., On the L i o n v i l l e theorem for h a r m o n i c maps, P r e p r i n t

[coi]

Courant, R., The e x i s t e n c e of m i n i m a l surfaces of given topological type, Acta Math. 72 (1940), 51 - 98

[co2]

Courant, R., Dirichlet's principle, conformal mapping, surfaces, Interscience, New York, 1950

[dTK]

De Turck, D., and J. Kazdan, Some r e g u l a r i t y theorems in R i e m a n n i a n geometry, Ann. Sc. Ec. N. Sup. Paris

[Do1]

Douglas, J., One sided m i n i m a l surfaces w i t h a given boundary, TranS. AMS 34 (1932), 731 - 756

[Do2]

Douglas, J., M i n i m a l surfaces of higher t o p o l o g i c a l Ann. Math. 40 (1939), 2 0 5 - 298

[EE]

Earle, C.J., and J. Eells, A fibre bundle d e s c r i p t i o n of Teichm H l l e r theory, J. Diff. Geom. 3 (1969), 1 9 - 4 3

[ELI]

Eells, J., and L. Lemaire, A report on h a r m o n i c maps, L o n d o n Math. Soc. 10 (1978), I - 6 8

L., and L. Bers, Riemann's m a p p i n g theorem for v a r i a b l e Ann. Math. 72 (1960), 3 8 5 - 404

part I,

Large solutions for h a r m o n i c m a p s in Phys., to appear

G r o m o v ' s A l m o s t Flat Manifolds,

and m i n i m a l

structure,

Bull.

187

[EL2]

Eells, J., and L. Lemaire, D e f o r m a t i o n s of m e t r i c s and a s s o c i a t e d harmonic maps, Patodi Mem. Vol. G e o m e t r y and Analysis, Tata Inst., 1980, 3 3 - 45

[EL3]

Eells, J., and L. Lemaire, On the c o n s t r u c t i o n of h a r m o n i c and h o l o m o r p h i c m a p s b e t w e e n surfaces, Mat~. Ann. 252 (1980), 27 - 52

[EL4]

Eells, J., and L. Lemaire, Selected topics in harmonic maps, Regional Conf. Ser. 50 (1983)

[EL5]

Eells, J., and L. Lemaire, Examples of h a r m o n i c m a p s from disks to hemispheres, M Z 1 8 5 (1984), 517 - 519

[ES]

Eells, J., and J.H. Sampson, Harmonic m a p p i n g s of R i e m a n n i a n manifolds, Am. J. Math. 86 (1964), 109 - 1 6 0

[EWI]

Eells, J., and J.C. Wood, R e s t r i c t i o n s on harmonic maps of surfaces, Top. 15(1976), 2 6 3 - 266

[EW2]

Eells, J., and J.C. Wood, Maps of m i n i m u m energy, J. London Math. Soc. (2) 23 (1991), 30! - 31~

CBMS

[GRSB] Garber, W., S. Ruijsenaars, E. Seller, and D. Burns, On finite action solutions of the nonlinear d model, Ann. Phys. 119 (1979), 3 0 5 - 325 [G]

Giaquinta, M., M u l t i p l e integrals in the calculus of v a r i a t i o n s and n o n l i n e a r elliptic systems, Ann. Math. Studies, Princeton, to appear

[GGI]

Giaquinta, M., and Giusti, E., On the r e g u l a r i t y of the m i n i m a of v a r i a t i o n a l integrals, Acta Math. 148 (9982), 31 - 46

[GG2]

Giaquinta, M., and Giusti, E., The singular set of the minima of certain q u a d r a t i c functionals, to a p p e a r in Analysis.

[GH]

Giaquinta, M., and S. Hildebrandt, A priori estimates for harmonic mappings, J. Reine Angew. Math. 336 (1982), 1 2 4 - 1 6 4

[GS]

Giaquinta, M., and J. Sou~ek, Preprint

[GT]

Gilbarg, D., and N.S. Trudinger, Elliptic partial d i f f e r e n t i a l equations of second order, Springer, G r u n d l e h r e n 224, Berlin, Heidelberg, New York, 1977

[Go]

Gordon, W., Convex functions and harmonic mappings, Math. Soc. 33 (1933) 433 - 437

[GKM]

Gromoll, D., W. Klingenberg, and W. Meyer, R i e m a n n s c h e G e o m e t r i e im Grossen, L.N.M. 55, Springer, Berlin, Heidelberg, New York,21975

[GW]

GrHter, M., K.-O. Widman, The Green f u n c t i o n for u n i f o r m l y elliptic equations, Man. Math. 37 (1982), 303 - 342

[Hp]

Halpern, N., A proof of the collar lemma, Bull. London. Math. 13 (1981), 141 144

Harmonic m a p s

into a hemisphere,

Proc. ~ e r .

Soc.

-

[H m]

Hamilton, Springer,

R., H a r m o n i c maps of m a n i f o l d s with boundary, L.N.M. Berlin, Heidelberg, New York, 1975

471,

188

[Ht]

Hartman, P., On h o m o t o p i c h a r m o n i c maps, 673 - 687

[HtW]

Hart/nan, P., and A. Wintner, On the local b e h a v i o r of solutions of n o n p a r a b o l i c partial d i f f e r e n t i a l equations, Amer. J. Math. 75 (1953), 449 - 476

[Hzl]

Heinz, E., On certain n o n l i n e a r elliptic d i f f e r e n t i a l equations and u n i v a l e n t mappings, Journ. d'Anal. 5 (1956/57), 197 - 272

[Hz2]

Heinz, E., Neue a - p r i o r i A b s c h ~ t z u n g e n fHr den O r t s v e k t o r einer F l ~ c h e p o s i t i v e r G a u s s s c h e r Kr%~mmung d u r c h ihr Linienelement, Math. Z. 74 (1960), 1 2 9 - 157

[Hz3]

Heinz, E., E x i s t e n c e theorems for o n e - t o - o n e m a p p i n g s a s s o c i a t e d w i t h elliptic systems of second order, part I, Journ. d'Anal. 15 (1965), 3 2 5 - 353

[Hz4]

Heinz, E., E x i s t e n c e theorems for o n e - t o ~ o n e m a p p i n g s a s s o c i a t e d w i t h e l l i p t i c systems of second order, part II, Journ. d'Anal. 17 (1965), 1 4 5 - 1 8 4

[Hz5]

Heinz, E., Zur A b s c h ~ t z u n g der F u n k t i o n a l d e t e r m i n a n t e bei einer Klasse t o p o l o g i s c h e r Abbildungen, Nachr. Akad. Wiss. G~tt. (1968), 1 8 3 - 197

[HH]

Heinz, E., and S. Hildebrandt, Some remarks on m i n i m a l surfaces in R i e m a n n i a n manifolds, C P A M 23 (1970), 371 - 377

[Hil]

Hildebrandt, S., B o u n d a r y b e h a v i o r of m i n i m a l surfaces, Arch. Mech. Anal. 35 (1969), 47 - 81

[Hi2]

Hildebrandt, S., On the P l a t e a u p r o b l e m for surfaces of c o n s t a n t m e a n curvature, Comm. Pure Appl. Math. 23(1970), 97 - 114

[Hi3]

Hildebrandt, S., N o n l i n e a r e l l i p t i c systems and h a r m o n i c mappings, Proc. Beijing Symp. Diff. Geom. & Diff. Eq. 1980, Science Press, Beijing, 1982, also in SFB 72, V o r l e s u n g s r e i h e No. 4, Bonn, 1980

[HJW]

Hildebrandt, S., J. Jost, and K.-O. Widman, H a r m o n i c m a p p i n g s and m i n i m a l submanifolds, Inv. Math. 62 (1980), 2 6 9 - 298

[HKWI]

Hildebrandt, S., H. Kaul, and K.-O. Widman, H a r m o n i c m a p p i n g s into R i e m a n n i a n m a n i f o l d s w i t h n o n - p o s i t i v e sectional curvature, Math. scand. 37 (1975), 2 5 7 - 263

[HKW2]

Hildebrandt, S., H. Kaul, and K.-O. Widman, D i r i c h l e t ' s b o u n d a r y v a l u e p r o b l e m for harmonic m a p p i n g s of R i e m a n n i a n manifolds, Math. Z. 147 (1976), 2 2 5 - 236

[HKW3]

Hildebrandt, S°, H. Kaul, and K.-O. Widman, An e x i s t e n c e t h e o r e m for h a r m o n i c m a p p i n g s of R i e m a n n i a n manifolds, A c t a Math. 138 (1977), 1 - 16

[HWI]

Hildebrandt, S., and K.-O. Widman, Some r e g u l a r i t y results for q u a s i l i n e a r e l l i p t i c systems of second order, Math. Z. 142 (1976), 67 - 86

[HW2]

Hildebrandt, S., and K.-O. Widman, On the H61der c o n t i n u i t y of weak solutions of q u a s i l i n e a r e l l i p t i c systems of second order, Ann. Sc. N. Sup. Pisa IV (1977), 1 4 5 - 1 7 8

Can. J. Math.

19

(1967)

Rat.

189

[HOS]

Hoffman, D.A., R. Osserman. and R. Schoen, On the3Gauss g a d of c o m p l e t e surfaces ef c o n s t a n t m e a n c u r v a t u r e in ~{ and ~ , Comm. ~ t h . Helv. 57 (1952), 5 1 9 - 531

~J~KI] J~ger, W., and H. Kaul, U n i q u e n e s s and s t a b i l i t y of h a r m o n i c m a p s and their J a c o b i fields, Man. Math. 28 (1979), 2 6 9 - 293 [J~K2] J~ger, W., and H. Kaul, R o t a t i o n a l l y symmetric h a r m o n i c maps from a ball into a sphere and the r e g u l a r i t y p r o b l e m for w e a k solutions of e l l i p t i c systems, J. Reine Angew. Math. 343 (1983), 146 - 161 [J1 ]

Jost, J., Eine g e o m e t r i s c h e B e m e r k u n g zu S~tzen ~ber h a r m o n i s c h e Abbildungen, die ein D i r i c h l e t p r o b l e m l~sen, Man. Math. 32 (1980), 51 - 57

[J2]

Jost, J°, U n i v a l e n c y of h a r m o n i c m a p p i n g s b e t w e e n surfaces, J. Reine Angew. Math. 324 (1981), 141 - 1 5 3

[J3]

Jost, J., Eine E x i s t e n z b e w e i s fur h a r m o n i s c h e Abbildungen, die ein D i r i c h l e t p r o b l e m 16sen, m i t t e l s der M e t h o d e des W~rmeflusses, Man. Math. 34 (1981), 1 7 - 25

[J4]

Jost, J., A m a x i m u m p r i n c i p l e for h a r m o n i c m a p p i n g s w h i c h solve a D i r i c h l e t problem, Man. Math. 38 (1982), 129 - 1 3 0

[J5]

Jost, J., E x i s t e n c e proofs for h a r m o n i c m a p p i n g s w i t h the help of m a x i m u m principle, Math. Z. 184 (1983), 4 8 9 - 496

[J6]

Jost, J., The D i r i c h l e t p r o b l e m for h a r m o n i c maps from a surface w i t h b o u n d a r y onto a 2-sphere w i t h n o n - c o n s t a n t b o u n d a r y values, J. Diff. Geom., to a p p e a r

[J7]

Jost, J., H a r m o n i c m a p s b e t w e e n surfaces, in Math., 1062 (1984)

[J8]

Jost, J., conformal m a p p i n g s and the Preprint

[J9]

Jost, J., Harmonic m a p p i n g s b e t w e e n R i e m a n n i a n manifolds, CMA, Vol. 4, ANU-Press, Canberra, 1984

[JIO]

~OSt, J., A note on h a r m o n i c m a p s b e t w e e n surfaces,

[J11]

Jost, J., On the e x i s t e n c e of h a r m o n i c maps into the real projective plane, P r e p r i n t

[JK1]

Jost, J., and H. Karcher, G e o m e t r i s c h e M e t h o d e n zur G e w i n n u n g von a - p r i o r i - S c h r a n k e n fur h a r m o n i s c h e Abbildungen, Man. Math. 40 (1982), 27 - 77

[JK2]

Jost, J., and H. Karcher, A l m o s t linear f u n c t i o n s and a - p r i o r i e s t i m a t e s for h a r m o n i c maps, Proc. D u r h a m Conf. 1982

EJM]

Jost, J., and M. Meier, B o u n d a r y r e g u l a r i t y for m i n i m a of certain q u a d r a t i c functionals, Math. Ann. 262 (1983), 5 4 9 - 561

[JS]

Jost, J., and R. Schoen, On the e x i s t e n c e of harmonic diffeom o r p h i s m s between surfaces, Inv. Math. 66 (1982), 353 - 359

[JY]

Jost, J., and S. T. Yau, H a r m o n i c m a p p i n g s and K~hler manifolds, Math. Ann. 262 (1983), 1 4 5 - 166

Springer Lecture Notes

P l a t e a u - D o u g l a s problem,

Proc.

Preprint

190

[KW]

Karcher, H., and J.C. Wood, N o n - e x i s t e n c e results and growth properties for h a r m o n i c m a p s and forms, Preprint, SFB 40, Bonn, 1983

[~I]

Kaul, H., S c h r a n k e n fHr die C h r i s t o f f e l s y m b o l e , (1976) 261 - 2 7 3

[Ke]

Keen, L., Collars on Riemann surfaces, (1974), 263 - 268

[LU]

Lady~enskaja, O.A., and N.N. Ural'ceva, Equations aux d&riv~es p a r t i e l l e s de type elliptique, Dunod, Paris, 1968

[Lv]

Lavrent'ev, M.A., Sur une c l a s s e des r @ p r ~ s e n t a t i o n s continues, Math. Sb. (1935), 407 - 4 3 4

[LI]

Lemaire, L., A p p l i c a t i o n s h a r m o n i q u e s de surfaces Riemanniennes, J. Diff. Geom. 13 (1978), 51 - 78

[L2]

Lemaire, L., B o u n d a r y v a l u e p r o b l e m s for h a r m o n i c and minimal maps of surfaces into manifolds, Ann. Sc. Norm. Sup. Pisa (4) 8 (1982), 91 - 1 0 3

[L3]

Lemaire, L., A p p l i c a t i o n s h a r m o n i q u e s de v a r i & t & s produits, Math. Helv. 52 (1977), 11 - 24

[Li]

Lichtenstein, L., Zur T h e o r i e der k o n f o r m e n Abbildung. K o n f o r m e Abb i l d u n g n i c h t a n a l y t i s c h e r s i n g u l a r i t ~ t e n f r e i e r F l ~ c h e n s t H c k e auf ebene Gebiete, Bull. Acad. Sci. Cracovie, CI. Sci. Mat. Nat. A (1916), 1 9 2 - 217

[Lu]

Luckhaus, S., The D o u g l a s - p r o b l e m f o r surfaces of p r e s c r i b e d m e a n curvature, SFB 72 - P r e p r i n t 234, Bonn, 1978

[Ma]

Matelski, J., A c o m p a c t n e s s t h e o r e m for F u c h s i a n groups of the second kind, Duke Math. J. 43 (1976), 829 - 840

[Mz]

Mazur,

[MO]

M i n - 0o, Maps of m i n i m u m energy from compact s i m p l y - c o n n e c t e d Lie groups, P r e p r i n t

[Mil]

Misner, C.W., H a r m o n i c maps as m o d e l s for p h y s i c a l theories, Rev. D 18 (12) (1978), 4 5 1 0 - 4524

[Mi2]

Misner, C.W., N o n l i n e a r field theories based on h a r m o n i c mappings, in: Spacetime and Geometry, The A l f r e d Schild lectures, Univ. of Texas Press, Austin, 1982

[MI]

Morrey, C.B., On the solutions of q u a s i - l i n e a r elliptic partial d i f f e r e n t i a l equations, Trans. A.M.S. 43 (1938), 126 - 1 6 6

IN2]

Morrey, C.B., The p r o b l e m of P l a t e a u on a R i e m a n n i a n manifold, of Math. 49 (1948), 807 - 8 5 1

[M3]

Morrey, C.BI, M u l t i p l e integrals in the calculus of variations, Springer: Berlin, Heidelberg, New York, 1966

[Mu]

Mumford, D., A remark on M a h l e r ' s c o m p a c t n e s s 28 (1971), 288 - 294

P.O., J. Phys.

A 15

(1982),

man. math.

Ann. Math.

19

Studies 79

Comm.

3173

theorem,

Phys.

Ann.

Proc. A.M.S.

191

[Na]

Nash, Math.

[RV]

Ruh, E.A., and J. Vilms, The tension A.M.S. 149 (1970),569 - 573

[SkU1]

Sacks, J., and K. Uhlenbeck, The e x i s t e n c e of m i n i m a l of 2-spheres, Ann. Math. 113 (1981), I - 24

[SkU2]

Sacks, J., and K. Uhlenbeck, M i n i m a l i m m e r s i o n s surfaces, Trans. A.M.S. 271 (1982), 639 - 652

[sa]

Sampson, J.H., mappings, Ann.

[Sn]

Sanini, A., A p p l i c a z i o n i tra v a r i e t a r i e m a n n i a n e con e n e r g i a critica r i s p e t t o a d e f o r m a z i o n i di m e t r i c h e , R e n d i c o n t i di Mat. (VII) 3 (1983),53 - 64

[SUI]

Schoen, R., and K. Uhlenbeck, A r e g u l a r i t y theory fom h a r m o n i c maps, J. Diff. Geom. 17 (1982), 307 - 335 see also a c o r r e c t i o n in J. Diff. Geom. 18 (1983), 329

[su2]

Schoen, laneous 268

[SU3]

Schoen, R., and K. Uhlenbeck, R e g u l a r i t y m a p s into the sphere, P r e p r i n t

[SYI]

Schoen, R., and S.T. Yau, H a r m o n i c m a p s and the t o p o l o g y of stable h y p e r s u r f a c e s and m a n i f o l d s of n o n n e g a t i v e Ricci curvature, Comm. Math, Helv. 39 (1976), 333 - 341

[SY2]

Schoen, R., and S.T. Yau, On u n i v a l e n t h a r m o n i c faces, Inv. math. 44 (1978), 265 - 278

[SY3]

Schoen, R., and S.T. Yau, E x i s t e n c e of i n c o m p r e s s i b l e m i n i m a l surfaces and the topology of three d i m e n s i o n a l m a n i f o l d s with n e g a t i v e scalar curvature, Ann. Math. 110 (1979), 127 - 142

J., The e m b e d d i n g p r o b l e m 63 (1956), 20 - 63

for R i e m a n n i a n manifolds,

Ann.

field of the Gauss map,

Trans.

immersions

of closed R i e m a n n

Some p r o p e r t i e s and a p p l i c a t i o n s of h a r m o n i c Sc. Ec. Sup. 11 (1978), 211 - 228

R., and K. Uhlenbeck, B o u n d a r y r e g u l a r i t y and miscelr e s u l t s on h a r m o n i c maps, J. Diff. Geom. 18 (1983), 253 -

of m i n i m i z i n g

harmonic

maps b e t w e e n

sur-

non-

[sY4]

Schoen, R., and S.T. Yau, C o m p a c t group actions and the topology of m a n i f o l d s with n o n - p o s i t i v e curvature, Top. 18 (1979), 361 - 3 8 0

[se]

Sealey, H., Some p r o p e r t i e s Warwick, 1980

[Sh]

Shibata, K., On the e x i s t e n c e Math. 15 (1963), 173 - 211

[Sf]

S h i f f m a n , M., The P l a t e a u p r o b l e m for m i n i m a l surfaces of a r b i t r a r y t o p o l o g i c a l structure, Amer. J. Math. 61 (1939), 853 - 882

[Si]

Siu, Y.T., The complex a n a l y t i c i t y of harmonic maps and the strong r i g i d i t y of compact KMhler manifolds, Ann. Math. 112 (1980), 73 111.

[sm]

Smith, R., H a r m o n i c 364 - 385

[sp]

Sperner, E., A priori g r a d i e n t e s t i m a t e s SFB 72 - P r e p r i n t 513, Bonn, 1982

mappings

of h a r m o n i c

mappings,

of a h a r m o n i c

of spheres,

Thesis,

mapping,

Amer.

Univ.

Osaka J.,

J. Math.

97

(1975),

for h a r m o n i c mappings,

192

[Tb]

Taubes , C. , The e x i s t e n c e of a R o n - m i n i m a l s o l u t i o n Yang - M i l l s - Higgs e q u a t i o n s on IK3, Co~m. Math. Phys. part I: 257 - 298, p a r t II: 299 - 320

[To]

Toledo, D., H a r m o n i c m a p s from surfaces Math. Scand. 45 (1978), 13 - 26

[Trl]

Tromba,

[Tr2]

Tromba, A., On P l a t e a u ' s p r o b l e m for m i n i m a l surfaces genus in IRn, SFB 72 ~ P r e p r i n t 580, Bonn, 1983

Jut]

Uhlenbeck, K., H a r m o n i c maps: variations, Bull. Amer. Math.

[U2]

Uhlenbeck, K., M o r s e theory by p e r t u r b a t i o n m e t h o d s with to h a r m o n i c maps, Trans. A.M.S. 267 (1981), 569 - 583

loB]

Uhlenbeck, K., R e m o v a b l e s i n g u l a r i t i e s Math. Phys. 83 (1982), 11 - 29

[U4]

Uhlenbeck, K., V a r i a t i o n a l p r o b l e m s for g a u g e fields, in: Seminar on Diff. Geom., ed. S.T. Yau, A n n a l s Math. Studies 102 (1982), 455 - 464, P r i n c e t o n

[vw]

von Wahl, W., K l a s s i s c h e L ~ s b a r k e i t im G r o s s e n f0r n i c h t l i n e a r e p a r a b o l i s c h e Systeme und das V e r h a l t e n der L S s u n g e n fHr t~ ~, Nachr. Akad. Wiss° G6ttingeno

[Wt]

Wente, H., The D i f f e r e n t i a l e q u a t i o n Ax = 2Hx. A x. b o u n d a r y values, Proc. Amer. Math. Soc. 50 (I~75)~

[w1]

Wood, J.C., S i n g u l a r i t i e s of h a r m o n i c m a p s and a p p l i c a t i o n s of the G a u s s - B o n n e t formula, Amer. J. Math. 99 (1977), 1329 - 1344.

[W2]

Wood, J.C., N o n - e x i s t e n c e Preprint, Leeds, 1981

A.,

A new proof

to certain

that T e i c h m H l l e r

space

A direct method Soc. 76 (1970),

of s o l u t i o n s

to the SU(2) 86 (1982)

K~hler manifolds,

is a cell,

Preprint

of higher

in the c a l c u l u s 1082 - 1087

in Y a n g - M i l l s

to c e r t a i n

of

applications

fields,

Coram.

with vanishing 131 - 137

Dirichlet

problems,

HARMONIC MAPS

IN K A H L E R G E O M E T R Y

J. H. S a m p s o n

§ I.

Introduction

In t o d a y ' s talk I w o u l d

like to d e s c r i b e

some g e o m e t r i c a p p l i c a t i o n s

of h a r m o n i c m a p p i n g s w h i c h are not c o v e r e d e l s e w h e r e C o r s o Estivo.

I shall give a b r i e f a c c o u n t of h a r m o n i c m a p s

of r i g i d i t y of K ~ h l e r m a n i f o l d s , to Y.-T.

Siu.

due,

although

However

e.g.

form,

in a d i f f e r e n t

the i m p o s s i b i l i t y of m i n i -

i m m e r s i o n s of c o m p a c t K ~ h l e r m a n i f o l d s of d i m e n s i o n

of c e r t a i n types,

in the t h e o r y

in r a t h e r d i f f e r e n t

I shall t h e n s h o w how a s i m i l a r a n a l y s i s

s e t t i n g leads to some new r e s u l t s c o n c e r n i n g mal

in this e x t e n s i v e

> I

in spaces

those of c o n s t a n t n e g a t i v e c u r v a t u r e .

I shall not shrink here f r o m t e l l i n g you that our h a r m o n i c

Gradus ad P a r n a s s u m

lies s t r e w n w i t h

failures,

whose

ruins r e m i n d us

that we still do not have a good u n d e r s t a n d i n g of n o n l i n e a r p a r t i a l differential equations,

even of the r e l a t i v e l y simple form of the h a r m o n i c

m a p p i n g and heat e q u a t i o n s . My own w o r k

in the s u b j e c t i n d e e d s t a r t e d w i t h

failure.

Beginning

in 1952 I sought to a p p l y the ideas of the then still r a t h e r r e c e n t Hodge t h e o r y to the f u n d a m e n t a l l y n o n - l i n e a r p r o b l e m s of h o m o t o p y theory. One of the v e r y l i v e l y topics

in those b y - g o n e days was the c o m p u t a t i o n

of the h o m o t o p y g r o u p s of the spheres,

~(S)

of a p p l y i n g h a r m o n i c m a p p i n g s

led far a f i e l d in a p r o g r a m

to

~(S)

for short.

L u r i d dreams

w h i c h w a s to prove abortive. Let us b e a r in m i n d that h o m o t o p y g r o u p s and k i n d r e d c o m b i n a t o r i a l o b j e c t s are e s s e n t i a l l y non c o m p u t a b l e , t r a d i s t i n c t i o n w i t h c o h o m o l o g y groups, can be r e d u c e d to c o m p u t a t i o n s

e v e n for

~1(X),

in sharp con-

w h i c h at least for c o m p a c t spaces

in l i n e a r algebra.

194

Since to m a n y nation

the e a r l y

of of

that we

the p r o b l e m s ingenuity

still

seem

by a n a l y t i c a l theorems. harmonic

theory

class

where

equations

explicit such

things

maps

which

ford

solutions At

point

delle

Ricerche. invitation

K~hler

tions

Riemann

Cauchy

I should

with

made

my warm

maps,

there

in t h i s

are

of

in f a v o u r

of

counterparts

in

explicit

some

of the

harmonic

of w h i c h

also

and

af-

cases. sincere

appreciation

to m e b y P r o f e s s o r the C o n s i g l i o

to P r o f e s s o r

C.I.M.E.

some

especially

that harmonic the c l o s e complex

Riemann

long

notable

to s u r f a c e

maps

have

connexion

variable

E. V e s e n Nazionale

E. G i u s t i

for h i s

series.

theory.

found

between

having

successes

It is of c o u r s e

congenial

harmonic

in a p p l i c a -

quarters

among

and analytic

been decisively

exploited

by

ago.

dimensions

harmonic

objects

there and

still

complex

is m u c h m o r e

difficult

to e x p l o i t .

setting,

therefore

I shall

features.

thanks

~(S)

in g e n e r a l

structure

other

my most

di Pisa,

as

to a m u c h m o r e

to t h a t

harmonic

in s p e c i a l

available

such

development

set a s i d e

years,

to r e c o r d

distressing

today.

and many

equations

I express

failures

of o n e

and

their

combi-

and uniqueness

them

the g e n e r a l

in r e c e n t

like

to p a r t i c i p a t e

the g r a d u a l

treat

answers

manifolds

In h i g h e r between

discovered

existence

temporarily

o-model

very

problems

us to a p p l y

have

Superiore

surfaces,

and

These

produced

by a d a z z l i n g

us c a n be l i k e n e d

Normale

surprise

functions

that

than we can

at l e a s t

opportunity

of h a r m o n i c

no g r e a t

to a t t a c k

to u n d e r s t a n d

solutions.

splendid

Along

been

has

It is t h e r e f o r e

soon enable

to the E i n s t e i n

the S c u o l a

§ 2.

will

then,

adequate

I am hopeful

efforts

have been

tini,

kind

lack both

as the n o n - l i n e a r

this

for the

We

topology

pressing

able

that confronts

have

special

were

from being

of p r o b l e m s

situation

relativity, field

far

Nevertheless

extensive

which

subject,

and new methods.

methods.

maps

The

d a y s of o u r

take

persists function

K~hler

a certain theory.

manifolds

a moment

But

form

to r e c a l l

affinity in g e n e r a l the n a t u r a l

certain

formal

it

195

Such hates

z

I

a manifold

, ...,

z

n

M

can

has

be

a metric

expressed

in

which the

in

local

complex

coordi-

form

ds 2 = g ~ dz e d [ B

with

the

hermitian

condition

that

for

a tangent

vector

Io~1 : Io1.1<1

The

~

K~hler

g~8

= gB[

and

a complex

condition

"

So

can

far

this

number

c

merely we

be w r i t t e n

ensures

shall

have

as

~gv~ 3z 7

It

is n o t

red

difficult

in o r d e r

of v e c t o r s be

having gent are

Par

: T

o

an

linear

complex

spanned

only

p

can

by

or ple,

or

ds 2

parallel cement follows

being that

consequence, also

be

expressed

in

not

merely

to

(1,1).

the

latter

the

Christoffel with

requi-

operation

our

local

the

over

the

~

T*(M) G

q

shall

. tensors

tangent

and

to

to also r

tensor-type coordinates:

dz ~

,

complex

local

K~hler

complex

symbols

and

and .

cotan-

These

by

in t h e i r

equivalent

p

to c o n s i d e r

T(M)

resp.

Under the

linear

replacing

according

occur

points

essential

~ ........

dz ~

of

joining

system

tensors

essentially

terms

amounts

z-coordinate

preserves

compatible

o

complexifications,

,

transport

curve

but

which

type

condition

+ Tq(M)

indeed

dz ~

the

transport

natural

their

is of

is p r e c i s e l y

and

distinguish

~

this

isomorphism

values,

in o u r

then

(M)

that

arbitrary

=

We

show

parallel

is n o t

bundles

to

that

along

a complex It

~z a

dz ~

type,

expressions.

condition,

tensor

i.e.

types.

but

how many

For

exam-

only

Parallel

then,

displa-

covariant

differentiation,

preserves

complex

types.

As

a

Riemann

tensor

R

must

and

the

classification. The

only

This

non-zero

it

is e a s i l y F'S

are

196

of

the form

F ~~Y

and

are of the form no c o u n t e r p a r t

R~

F~ ~ ,

the only n o n - z e r o

;

R~,

R~,

In [33 it is shown

M ÷ M'

is harmonic.

of local

application

that a h o l o m o r p h i c

In fact,

coordinates

(I)

.

These

components

symmetries

have

in the real case.

Let us now turn to a simple on.

R ~

curvature

z

on

w h i c h will be of user

later

m a p of K~hler m a n i f o l d s

the h a r m o n i c equation, a M and w on M' is

expressed

in terms

gab w a

where

the

w~l ~

are the c o m p o n e n t s

of a c e r t a i n

covariant

derivative,

namely

wa = ~ w ~I~

(2)

We o b s e r v e

here

and

M

duced m e t r i c subvariet~, This

F 'a of M' bc is a f u n d a m e n t a l point

is h o l o m o r p h i c ,

a complex

is also K~hler) again

result

as shown

then

submanifold is a h a r m o n i c

of harmonic

map theory

rather

indirectly,

§ 3.

Strong

rigidit [

in

in all

for all

[I0],

local

immersion

in nature.

are global

topological

Y.-T.

of a K~hler m a n i f o l d

that

a,

8,

Siu gave an i m p o r t a n t

geometry.

It will be useful

results,

following

the m e t h o d

of

F i r s t of all, According

let

a minimal

denote

interesting

appli-

involve,although

application to e x p l a i n

[7], w h i c h will

of an e n t i r e l y X

Many

and t h e r e f o r e

to K ~ h l e r

us to some new results

- i.e.

(the in-

considerations.

mappings

g a 2.

w~ = O

occur

in [3, § 2].

is of course

often

In [9],

This

symbols

holds.

In particular,

cations

do not.

If the m a p p i n g

(I) c l e a r l y

b c + F'abe w w~

that the C h r i s t o f f e l

(2), but those of follows.

a

different

a closed

to the u n i f o r m i z a t i o n

quite

briefly

naturally

Siu's

lead

sort.

Riemann

theory

of h a r m o n i c

surface

of Poincar~

of genus and Koebe,

197

the u n i v e r s a l half-plane

covering

H

the Poincar~

(or u n i t disc)

F

a subject

number

depends

of g r e a t

= 3g - 3. nature,

Teichm~ller

(ca.

n

and

Such manifolds both

compact

(ca.

1890),

using

to c o n s t r u c t

F

continuously?

The

For bounded

table

in

D,

D', I

they are

with upon

the

under

have

they are

in

investigations

global

treatment

were

to p a s s

were

by

from

where

varieties.

first given

imaginary

D

is a

of

D.

Important

in

~2

quadratic

fundamental

exam-

by E.Picard

number

results

can we

"vary"

(negative)

n > I.

come

fields from Poin-

again,

was

were

domains,

M = F \ D

or

2

essentially

local

rigidity

the

real

and

b y E. C a l a b i

a n d E.

produced

and

Spencer. a striking

by M o s t o w

,

with

s e n s e o f De R h a m ' s i.e.

mentioned

above.

[5],

bounded

and do not contain

phenomenon

[8]

impor-

M' = F' \ D'

This

F

Selberg

and hermitian,

identical,

of

time very

of K o d a i r a

was

M

group

by A.

same

obtained

problem

(in the

given

theory

both

analogue

the d i s c r e t e

At nearly

of the metrics.

local

splitting

isometric

after

is s t r o n g Mostow's

sui-

rigidity, results

methods. by a signal

of e x i s t e n c e

the circumstances

dimensional

are diffeomorphic

Lie group

the p r e s e n c e

of

to the m o d u l i

Here we are confronted despite

being

which

of automorphisms

automorphiq

the d e f o r m a t i o n

if c o m p a c t

normalization

lean heavily

that

M = F \ D,

in t h e h i g h e r

negative

symmetric

domains

then

~n,

using

answer

that

in c o n t r a s t

F

etc.

answer

of dimension

theorem),

the u p p e r

X ~ F \ H,

showed

subsequent

group

Further

first

[2],

factors

F .

I.e.,

Vesentini

and definitive

Riemann

o f the t y p e

called

X ?

results,

symmetric

with

or moduli,

generalization

is a d i s c r e t e

have moduli

D = unit ball

showed

follows

successful

natural

the a r i t h m e t i c

tant general

who

identified

parameters,

and

and non-compact,

surface

in

There

since

first

manifolds

Bianchi,

still

the R i e m a n n

for

a very

the group

Fubini, Do w e

6.

account

the

are often

ples,

car~,

c a n be

1941).

to c o m p a c t

in

in

interest

until

It is o f c o u r s e

domain

M

on certain

Riemann's

of a l o c a l

X = F \ H

of

group.

The group been

surface

failure

and uniqueness

just described,

M

and

of harmonic theorems. M'

have

map

theory,

Indeed, negative

198

sectional neral

curvature. of

[3],

n e s s of h a r m o n i c

maps

can

results

If

infer

that

h : g

normalization). not

part D

f: M ÷ M'

[4] g u a r a n t e e g ~ f -I

and

are diffeomorphisms.

Hence

the

and

§ 4.

D'

results

of

negative

L e t us c o n s i d e r

a

Siu's

it is n o t e v e n

which

unique-

results

(with

show

that

map

we

suitable

known whether

by harmonic

symmetric

the g e -

essential

isometries

results,

hermitian

and

then

From Hostow's

are

c a n be o b t a i n e d

are classical

Strongly

these

however,

g, h

importance

the e x i s t e n c e h ~ f-1.

and that

At p r e s e n t ,

of M o s t o w ' s

is a d i f f e o m o r p h i s m ,

or

the m a i n

theory

when

manifolds

with

spaces.

curvature

C2

map

f: M ÷ M'

of K ~ h l e r

metrics

ds 2 : g ~ dz ~d[8

referred splits

to l o c a l

into

coordinates

a sum

ds '2 = g a,~ d W a dw-b

resp.

z

~ + ~ + ~

wa

resp.

of t y p e s

,

f, (ds' 2)

The pull-back

(2,0),

(1,1),

(0,2).

In p a t t i -

(@@8)

is n e i t h e r

cular,

a-b @ = @~@ ~ = g a'b w w~ dz~dz

(3)

where tric

w a@ = ~@w

a

,

nor anti-symmetric.

clearly

e = 0.

If

elusive,

except

for

Whether

holomorphic

quadratic theory type

-b

w@ : 28

(1,1).

M = Riemann rentials.

sphere,

is h o l o m o r p h i c harmonic,

d i m M = I,

in w h i c h

@

These

is k n o w n play

developments.

It is e a s i l y

assuming

f

is o n l y

o r not,

subsequent

In g e n e r a l

If

differential.

and

df # 0),

f

~b

dz~dz 6

still which

seen that d i m M : I. has

our

(or a n t i h o l o m o r p h i c ) ,

tensor

case

e

an i m p o r t a n t

f

e = 0 ,

~

is m u c h

surface

r61e then

no n o n - z e r o

is a l w a y s holomorphic

theory

as a

in T e i c h m H l l e r f * ( d s '2) = @ =

is t h e n c o n f o r m a l That

more

is h o l o m o r p h i c .

in R i e m a n n

If

symme-

(wherever

the c a s e e.g. quadratic

if

diffe-

199 Going

n o w to the g e n e r a l

situation,

we

introduce

the

(1,0)-form

by

B~ (covariant

From

(3)

it is e a s i l y

= g

seen

~

gab

'

If

f

is h a r m o n i c ,

that

( ,_ w a

= gab(W

from

wa The

next

step

ga~

w a~ w~

(I) we

IF)g see

that

this

simplifies

to

-b g ~

is to c o m p u t e

{ =

-b

~ wB) '

~ IT w}~ +

then

derivative).

~

the d i v e r g e n c e

_

d,~

As

in all

sists

applications

of a p o s i t i v e

term,

a n d of a c u r v a t u r e

gB~

term whose

here

(cf.

it c o n t a i n s

only

the c u r v a t u r e

a general

vature

term

prevented case,

us

of

be p o s i t i v e

contains

from

following

curvature

§ 5 below).

Riemannian

also

drawing

which

also,

whence

r e m then

yields

~

= 0.

vanish.

In p a r t i c u l a r ,

the

result

complicated

expression

indicated

in c o n n e x i o n

with

that

If

As

of

M',

but

not

manifold,

we

the R i c c i

curvature

useful

Siu we c a n

M',

method,

con-

g~

rather

require

were

vanishing

namely

g,_ w a _ -b ab ~IY w s l ~

(4)

not

of B o c h n e r ' s

conclusions.

introduce

ensures

from

For

our

of

found M,

compact

find

M

both

that

M. that

which

In our

curvature

a consequence, (4) we

have

the c o n d i t i o n

that

6~ a 0. As

should

of

of

we

shall

cur-

would

have

K~hler

strongly in

M

the

present

term

(I),

negative

6~

the d i v e r g e n c e

will theo-

p a r t s of ~ must a all w i~ = 0 and that

200

our

(2,0)-form

term

Siu has

~

shown

is h o l o m o r p h i c .

From

that,

further

under

some

the v a n i s h i n g technical

of

the c u r v a t u r e

conditions

a

which all

we n e e d a,

e.

not

I.e.

tarry

here,

the m a p p i n g

we

shall

f

must

have

over

a

w- = 0

(or

be h o l o m o r p h i c

w

= 0)

for

(or a n t i h o l o m o r -

phic) .

§ 5.

Mappings

We

shall

into

now

above

will

lead

kinds

of m i n i m a l

Riemannian

manifolds

show how a rather

to n e w

results

immersions.

similar

bearing

upon

In p a r t i c u l a r ,

argument the

to that

described

impossibility

we

shall

obtain

of c e r t a i n the

follo-

wing

Theorem:

A compact

immersed

K~hler

as a m i n i m a l

manifold

subvariety

of c o m p l e x

in a space

dimension

of c o n s t a n t

> 1 cannot negative

be

curva-

ture.

It m a y cording

suggestive

to w h i c h

constant direct

be

a K~hler

curvature.

bearing

Now

on the

M

into

again

consider

on

Y.

This

(2,0),

(1,1)

~

manifold

the p u l l - b a c k

tensor and

~P =

With

stated

a mapping

a Riemannian

we a s s o c i a t e

local

of B o c h n e r

> I cannot

result

however

[I],

ac-

be of n o n - z e r o and

has

no

above. f: M ÷ Y Y.

f*(ds '2)

of a c o m p a c t

Following

K~hler

the m e t h o d

of

mani§ 4 we

of the m e t r i c

dy3' dyk

decomposes

(0,2),

a theorem

of d i m e n s i o n

is a p u r e l y

theorem

ds ,2 = gjk ,

to r e c a l l

manifold

This

let us c o n s i d e r

fold

here

into

three

terms

~ + %0 + ~

of t y p e s

where

dzC~ dzB the

,

(Pc~[{ = g'jk yJc~ Y~

(1,0)-form

6

defined

(YJ = ~yJ/~z ~)

by

201

The harmonic mapping equations in our local coordinate systems resp

yJ

z

are

g~B

J

YO~IB

=

0

,

where j

(Here

y~ = ~ y Z

yj

= y~ ). ,

,j

k

Z

Consequently, if

f

is harmonic we obtain

k gB~

j

~

= gjk Yel~ YB

~

= g6~

we compute ~

~,~

Using covariant differentiation in the bundles (T0,P M)~ ~ f-1 TY ,

following the pattern of (5) above, we find ,

j

k

j

k

) B~

The Ricci identity yields y~ _ _ _ yj = _ R, j k m IYI~ ~I~l~ k£m Y~ Y~ Y~ Therefore we obtain, using the harmonic mapping equation again: ,

j

k

gB~

-

, j k ~ m gB~ - Rjk~m YB Y~ Y~ Y~ g~

202

The complex

conjugate

o f the

last term

Rik~m Y~ Yk Yy Y~ g

is

g ,

m

= RZmjk

and therefore

the c u r v a t u r e

~j £

(7)

8~ =

Clearly

(~JZ)

g

The

(8)

R' ~j£ km jk£m ~

led

strongl~

can hold

Theorem M

curvature

hermitian

If

of

Write

now

( > 0)

hermitian

matrix.

Y

be called

hermitian

negative

curvature

will

will

matrix

negative only

f: M ÷ Y

into a manifold

is real.

if

< 0

hermitian

at t h a t p o i n t

I.

(6)

~@ g

j y_~

is a p o s i t i v e

positive

in

gy~

k

Y~

Definition.

for e v e r y

term

j

Y$ Y~ YZ Y[

for

(~J£) .

at a p o i n t (~J£)

of

is a h a r m o n i c

of hermitian

The

of

Y

rank

if e q u a l i t y

in

(8)

< 1

m a p of a c o m p a c t

negative

be c a l -

curvature,

K~hler

t h e n the

manifold

(2,0)

form

is h o l o m o r p h i c .

Proof. theorem of

on

M

(6) m u s t

that

Our assumptions then gives

vanish,

that

@~ z 0 , a n d

6~ = 0 . B o t h

and consequently

terms

all

yJ

the d i v e r g e n c e

in the r i g h t - h a n d

member

= 0

clear

It is t h e n

~ ~,~ = 0 .

Remark.

If

d i m M = I,

J Y I I T = 0,

and

holds with

no c u r v a t u r e

Theorem

2.

hermitian rank

imply

s 2.

homotopie

therefore

Under

Y

the h a r m o n i c

~11;T

everywhere,

is c o m p a c t ,

to a h a r m o n i c

= 0 .

restrictions,

the c o n d i t i o n s

curvature If

then

map

I.e.,

equation

I, if

then every

then every ~ 2.

reduces

the c o n c l u s i o n

as is w e l l

of Th.

of r a n k

map

Y

to

o f Th.

I

known.

has

strongly

negative

harmonic

map

f : M ÷ Y

has

continuous

map

f

is

o

: M ÷ Y

203

Proof. ce

theorem

sly

f

The of

is 0

in

that

w

has

first

[3].

point

If

a constant

rank

the

last

an

open

rank

row

set

w

is

the

first

of

(7)

has

~

and

this

(cf.

in

from

matrix

map;

I

of

the

follows

an

for

we

[63, § 2). open,

non-zero.

need

The

dense Then

only

We

rows

from

rank

existen-

then

obviou-

that

w

alternative

can are

0,

the

assume

remaining

set. all

and

assume

is

that

proportional

has

(say) to

the

first:

3 k•

=

w Ik

i

(j =

2, .... m;

k :

I .... ,m)

~ = ~'

+ iw",

where

,

3

where

m = dim

follows by

the

one f

or

that

all

row

two.

If m o r e o v e r

M

onto

Hermitian

a point

of

that

operator

w',

the

negative

to

our

in

w".

a closed

w' Hence

rank

geodesic

is

and

w"

the

real

equal

in

Y

to

w',

are

in

~"

are

the

space

matrix one

real.

(w',w")

in

an

open

spanned has

set,

rank then

([6], § 4).

curvature

definition,

point

for

sends

w jZ

any

This

new

ture

operator

w km

Y

has

hermitian

negative

curvature

matrix

to

the

hermitian new

matrix

matrix

w

R'w

.

with

The

curvature

components

~j~

R'w R'

s 0

positive

(w 3k)-

R ,'k'm

be

is

also

negative

hermitian, on

the

and

cone

of

we

require

positive

that

the

hermitian

ces: (~,

If w e

It

if

R' jk£m

at

vectors

rows

According at

Write

first

maps

§ 6.

Y.

decompose

R'~)

w

~ 0

into

real

and

imaginary

parts

~',

~"

,

then

curvamatri-

204

R'w

where

now

= R'W'

R'~'

orthogonal

is s y m m e t r i c

in t h e

usual

(~,

Hermitian

Theorem has

R'~)

negative

3.

If

strongly

Proof.

=

has

hermitian

and

inner

(~',

curvature

Y

We

+ iR'~"

R'w"

product,

R'w')

thus

-

is s k e w - s y m m e t r i c . and

(w",

means

negative

negative

curvature.

R'~")

(w',

is t h e

let

in

such

At

then

it

any

a way

as

point

w

p

to h a v e

of , gj~

,

Y =

we

6j~

can at

choose p.

the

Further,

us w r i t e

w j£ = b jZ + ic j£

where tric.

B = At

(b JZ) ~ p

we

shall

then

eigenvalues

matrix

and

C =

(c~i)i

is

skew-symme-

have

= K [ ( t r B ) 2 - Z ( b J £ ) 2 - E(cJ£) 2 ] = K[(trw)

the

,

is a s y m m e t r i c

(~,R'w)

Let

R'w").

km

= K(g~^3~ g'km - g~3m gk~' )w

curvature.

y3

coordinates

( < 0),

curvature

j£

<

R'w') ~ (w",

have

(~,R'w)

where

are

so

that

constant

These

of

w

be

2

- tr~

2

] .

1 1 , 1 2 , . . . , I m.

The

last

quantity

is t h e n

= <[Zljl k - SI~3 (9) : 2<

Since

ljl k

I. are all ~ 0, t h i s s u m is o f c o u r s e < 0. If m o r e o v e r 3 K < 0, t h e n (9) is < 0 unless all the i. vanish except at most one 3 of them. I . e . , if (w,R'~) = 0, then the matrix ~ can have rank at most

the

[ j
one.

205

Combining

Theorem

4.

A harmonic

of c o n s t a n t there

negative

cannot

dimension

this w i t h T h e o r e m

exist

m a p of a c o m p a c t

curvature

a minimal

> I in a m a n i f o l d

cited

only ones question (m,R'e)

in § 5.

having

elsewhere, = 0

results

Normale

in a v a r i e t y

Superiore

negative

of c o n s t a n t

K~hler m a n i f o l d

of

curvature.

negative

curvature.

here w i t h

image of the g i v e n h a r m o n i c

Scuola

of a c o m p a c t

In particular,

as a kind of g e n e r a l i z a t i o n

negative

closing

into a m a n i f o l d

~ 2 everywhere.

of c o n s t a n t

Spaces

hermitian

we o b t a i n

K~hler manifold

has rank

immersion

This may be looked upon theorem

2 above,

curvature

We shall

the remark

that

of i n t e r e s t i n g

of B o c h n e r ' s are not the

return

to this

the c o n d i t i o n

restrictions

on the

map.

di Pisa,

luglio

1984.

REFERENCES

[I] S. Bochner, C u r v a t u r e 53 (19~7), 179-195.

in h e r m i t i a n

metric,

Bull.

[23 E. C a l a b i and E. Vesentini, On compact, locally m a n i f o l d s , Ann. Math. 71 (1960), 472-507.

Amer.

Math.

symmetric

K~hler

[33 J. Eells and J.H. Sampson, H a r m o n i c m a p p i n g s folds, Amer. J. Math. 86 (1964), 109-160.

of R i e m a n n i a n

[4] P. Hartman, 673-687.

J. Math.

On homotopic

harmonic

[5] G.D. Mostow, Strong R i g i d i t y Math. Studies 78, Princeton,

maps,

of L o c a l l y 1973.

Can.

Symmetric

19

On h a r m o n i c

mappings,

Symp.

Math.

Spaces,

XXVI

mani-

(1967),

[6] J.H. Sampson, Some p r o p e r t i e s and a p p l i c a t i o n s of h a r m o n i c pings, Ann. S c i e n t . E c o l e Norm. Sup., 4e S~rie, 11 (1978), [7] J.H. Sampson, 197-210.

Soc.

Ann.

of

map211-228.

(1982),

[8] A. Selberg, On d i s c o n t i n u o u s groups in h i g h e r - d i m e n s i o n a l symmetric spaces, C o n t r i b u t i o n s to F u n c t i o n Theory, Bombay (1960), 147-164. [9] Y.-T. Siu, of complex

C o m p l e x a n a l y t i c i t y of h a r m o n i c maps and strong r i g i d i t y K~hler manifolds, Ann. Math. 112 (1980), 73-111.

[10] Y.-T. Siu, C o m p l e x a n a l y t i c i t y of h a r m o n i c maps, v a n i s h i n g L e f s c h e t z theorems, J. Diff. Geom. 17 (1982), 55-138.

and

ISOLATED

SINGULARITIES

GEOMETRIC

We of

shall

here

functionals

tic

of

behaviour

points.

We

harmonic We ×

be

of

are

"geometric"

especially

(0,~),

with where

Z

of

form

(*)

F(u)

=

where

m

nential

is

D(w) =

singularities

principal

approach

to

in

such

this

to

for

interest isolated

for

extrema

is

asympto-

singular

minimal

surfaces

and

a singular

with

the

most

r 2 I Imwl

,

of

have

implicit

+ ~

manifold.

t

on t

it

point

point,

can

for

extrema

Specifically,

we

look

at

+ E(~,t,u,Au,~u/~t))d~dt

as

problems,

trivial

and

+ ~ turns

< be

E ,

,

u

can

out

that

with

reduced of

where

~ to

consider

the

ball

be

has

the

asymptotic

study as

the

expo-

vector-valued.

possibly,

functionals

example,

E

in

beha-

but of

asympto-

(*) .

Dirichlet

not

To

il-

integral

n where

variable

Ipl 2 + lql 2

to

cylindrical domains

on

V = gradient

a given

t

of

functionals

a compact

geometric

as

takes

We

is

respect

approach

behaviour

ly

with

behaviour

lustrate

of

a constant,

necessarily,

change

Our

r I e-mt(F(~,u,Vu,~u/~t) JZ

various on

:

the r~ I J0

decay

For

D(w)

isolated

interested

a class

functionals

tic

in

maps. deal

viour,

Simon

on

OF

PROBLEMS

Leon

type.

extrema

EXTREMA

VARIATIONAL

interested

the

OF

the

form

, and u(~,t) two

t = (*)

is

-loglxl, with

behaviour as

principal

function

~

t

of

in

~ : x/Ixl, n-1 Z = S , m w(x)

as

.

u(~,t)

= w(r~),

= n-2,

Ixl

If w e

and

+ 0

is

make

the

then

F(w,z,p,q)

exactly

given

= by

+

aims:

theorem)

unit

in

Part

method

for

I we

give

a general

constructing

examples

(essentialof

extre-

207

ma

(of f u n c t i o n a l s

t-independent maps

point.

This

with

detailed

In p a r t

II w e

t + ~

The main given

tic b e h a v i o u r maps

near

veloped in the

geometric

obviously

were

the p r i n c i p a l

natural

with

the q u e s t i o n

of w h e n

mass minimizing at l e a s t

of P a r t

form)

are

applications in

isolated

near

good

of w h e n

a given limit

modified

information

The general and

theorem

the a p p l i c a t i o n we r e f e r

the

of

as

version

about

asympto-

harmonic

of

there may well

the

II

§ I

is in

be applications

the t e c h n i q u e s

reader

de-

to the d i s c u s s i o n

The

singular

examples

for the c o d i m e n s i o n

surfaces

existence

proposed

in

[CHS]

points.

of m i n i m a l

singular I case.

abstract

to be f o u n d

to m i n i m a l

[SLI ].

isolated

(in r a t h e r m o r e

essentially

of the m e t h o d

submanifolds

question

[SLI].

established

extension

For

problems,

results

applicable

already

of

§ 7.

large

at a g i v e n

and energy minimizing

context

problems.

introduction principal

points.

a general

to p a r a b o l i c

in

type

an appropriately

submanifolds

singular

to c o n s t r u c t

specified

to g i v e

to a g i v e n

to h a r m o n i c

(*) h a s a n a s y m p t o t i c

(or r a t h e r

specializes

of m i n i m a l

in q u i t e

here

The

tainly

§ 5)

t + ~

specializes

of

interesting form

as

it p o s s i b l e

are given

of t h e

theorem

isolated

fact valid to o t h e r

in

result

making

take up the

of a functional

it,

general

singularities

results

extremal

of

(*)) w h i c h are a s y m p t o t i c

submanifolds,

of e x t r e m a The

in

extremal.

and minimal

families

as

points

in

and

[SLI]

less

- cer-

and harmonic

theory

of P a r t

for g e n e r a t i n g

A detailed

I is a minimal

discussion

submanifolds

are

is to be f o u n d

maps

of

actually in

[HS],

208

PART

GENERAL

The

principal

including

We

specified general

decay

in

to

other

functionals

§ I.

SUBMANIFOLDS

We ~P

consider

; a typical

dard

unit

for

first

§ 3)

which

of

infinity.

the of

area

the

our

form

EXAMPLES

is

key

to

existence

of

(*)

up

maps

give

the

the

terminology,

energy and area

minimal

and

for

submani-

solutions

with

applications

of

(and

to m a n y

functionals in

basic the

theorem

§ 7 we

energy

set

examples,

harmonic

In

and

AND

part

give

§ 6 a general

near

theorem

this

(in

extrema prove

PRELIMINARIES

of

a discussion

functionals,

folds.

aim

I

in

fact

this

introduction).

P 19

OF

a compact example

(n-1)-dimensional

will

be

in

19n

on

E

(n-1)-sphere

P = n

submanifold n-1 ~ = S ,

and

S

~ n-1

embedded

in

the

stan-

on

~

I For be

defined

where of

C

function

as

usual

T1,...,Tn_ Z

tiation the

a

at of

~

, f

I

take

Vf

, gradient

of

f

, to

by n-1 [ i=I

Vf(~)

=

is

orthonormal

and

an

f Ti in d i r e c t i o n

alternative

we

D

just T

1

(D

T.

f)~

i

'

basis

denotes

for

the

ordinary

. Notice

that

(D T f ) e T e.

,

tangent

space

directional

we

can

T

differen-

represent

Vf

in

form

1.1

Vf(~)

=

P [ i=1

1

where

e,~,...,ep

is

denotes the orthogonal p lq Notice that

the

standard

projection

orthonormal of

e. l

onto

basis the

for

19P

subspace

and T

Z

T e. l

of

209

1.1'

Vf(~)

=

(V1f(~) ..... Vpf(~)),

V f(~)

P [ j=1

=

1

This (i.e.

extends

to the v e c t o r - v a l u e d

f e C I (E;]R N) ),

in w h i c h

case

c a s e we

f

eT-eT(D 1

(fl

=

]

T f) e. 3 N

,fN)

,...

:

E

+

IR

define

P

I .2

Vf(~)

= i=I

eT ® D f 1 T e. 1

(D T f = ei the

are

p x N

matrix

In some

cases

"sections

that

of

~ = {V }

(rife)

we

=I

,...,N i=l,...,P

shall

E

is a s m o o t h l y

dimension

of

~N

we d e f i n e

V=f

(often

V

f(~)

abbreviated

E " ,

family E V

Vf

f

functions

over

varying

with

represented

by

as in 1.1'

V. 1

vector

bundle

is n a t u r a l l y

Vf(~)

'

consider

some v e c t o r

NI V N)

arise)

sO that

(D T f l , . . . , D TfN)), ei ei

of V ~

: ~ ÷ N

which

which

just m e a n s

subspaces ~ E .

(of f i x e d

In this

if no a m b i g u i t y

case

is l i k e l y

to

by

V

P

v=f(~)

1.3

=

e .r ® ~ (o T f )

[

1

i=1

[o

e. 1

so that V = V f(~)

1.3'

where

~

=

P ~ i=I

e

T

V ® V~f,

1

V V~f =

1

is the o r t h o g o n a l

1

projection

P ~ j=1 ~N

e

T 1

T

-e. 1r (D T f) 3 W e 3

÷ V

,

,

~ ~ ~ ,

which,

by

assumption,

v a r i e s s m o o t h l y in ~ . In this case we w r i t e f ~ CI (Z;~). V Notice that V=f(~) is n a t u r a l l y r e p r e s e n t e d by the P × N matrix V= I N (e~.Vif(~)) = I where e ..... e is the s t a n d a r d o r t h o n o r m a l ,...,N ' i=1,...,P basis We spaces

for

N

shall

use

similar

of f u n c t i o n s .

For

notation

for the v e c t o r

example

L 2 ( Z ; ~ N)

denotes N

functions

on

E

with

inner

product

(f,g =

bundle the

case

of o t h e r 2 space of L

r

[

~

~=1

J~

fa(~)g~(~)d~

,

and,

210

for

any

vector

bundle

{f { L 2 ( Z ; ] R N) : f(~) C k (Qa,b;V)

to

(oo,t) ~ Q a , b

"

§ 2.

where

we

usual

P

product

Q

for

any

fixed

u(~,t)) u(~,t) u(t)

as

metric

H n-1

a.e.

c k (Q a ,b;IqN)

the

Q =

Z × I9

in

§ I;

E x

denotes

~ c Z}

We

shall

also

functions

f

with

f(~,t)

with

cross-section

~

< ]R P+I

Q

is

from

(0,~)

L2(Z;V)

write c Vw,

DOMAINS

cylinder

induced

:

above,

]R

,

:

,

assumed P+I

Qa,b

to We

be

also

equipped

with

the

write

E x

(a,b)

we

define

- ~_< a< b < +

Given any

} E~

CYLINDRICAL

as

Q+

{V

for

be

is

=

the

OF

let

Z c ]R

c V

denote

FUNCTIONALS

Here

V

_

a function t ~ E

u = u(~,t)

, by

u(t) (~)

~u(~,t)~t

Notice

cV

for ~ c Z , w c C I (Z;~) for each

usually

denoted

writing

on

= u(~,t)

that

if

a,b and

V= =

t

{V

}

~(~,t) c V , V , then V = u ( t ) (~)

by

V-u(w,t)

;

when

no

ambiguity

again is

u(t)

~(t) (~))

then

simply

Vu(e,t)

Q

Z

as

in

(defined

likely

to

§ I , and

Also

often

Z

(sometimes

~ ~ ~

we

on

as

We

denoted if

if in

§ I)

abbreviate

arise.

, for

also

is

this use

by the

notation V D u(t0,t)

2.1

in

for fact

abbreviated

Du(im,t))

=

(V

(~,t),u(~,t))

u ~ C I (Qa,b;~)

case We

V (often

consider

this

has

consider

functionals

already

been

functionals

of

a

briefly of

the

special

form

discussed form

in

on

0 ; the motivation -a,b the introduction. W e in

211

2.2

F

(u)

a,b

= I J

-mt

F(~,u,Vu,u)e

d~dt

,

Qa,b where

m

u ~ C I (Q

is a c o n s t a n t ,

b ; ~ N ) (or a, to b e a s m o o t h

V Vu Z

= V u). ×

aN

F

is a l w a y s

PN

x

×

N

~

,

assumed

and

F = F(w,z,p,q),

and of

we write

IRN, 2.3

u ~ C 1 ( Q a , b ;V) = function on all

~ e >],

PN e ]19

~

z e

_m =

(. m i )6=I ..... N i:1,...,P

q (

N

~

thus

Subscripts

F

= ~F/~z ~

denote

,

F ~

So as n o t

of

'

with

G a b(U)

to

z,p,q:

etc.

important

consider

= F

,

there

examples

somewhat

more

(see

§ 3 below)

general

are

a,b

(u)

+

I

e

-mt

it is e s -

functionals

E(~,t,u,Vu,6)dmdt

Ga, b

c = c(z,p,q)

and

g0

> 0

(IEI + IDEI + ID2EI) ( ~ , z , p , q )

where

such

_< c e

D

d e n o t e s t h e f u l l g r a d i e n t of E PN aN x ~ × ~ x . Thus E is " l o w e r

E x ~ t + ~

for We

let

bounded M,

z,p,q

N

be

respectively.

e

,

JQa,b -s

2.5

Ga, b

respect

form

2.4

where

to e x c l u d e

to a c t u a l l y

the

= ~F/~Pi

derivatives

Pi

z

sential

partial

t

the

order"

product

space

asymptotically

as

.

mt

Thus

on

0

that

times for

u

the

Euler-Lagrange

operators

e C 2 ( Q a , b ; a N) , M , N

are

of

F

a,b ' characterized

by

(e-mtM(u),{)

d ds

(e-mtN(u)'~)

= - d ds

Ea,b(U+S%)

Is:0

2.6

for

~ ~ C cI ( Q a , b ; ] R N )

,

where

the

inner

Ga,b

(u+s~)is_0

product

is t h e

usual inner product

212

for

L 2 (Qa,b; ~ N ) . In t h e v e c t o r b u n d l e c a s e , w h e n u(t) E C 2 (E;~) V Vu = V = u , w e f u r t h e r s t i p u l a t e that M(u) (t) , N(u) (t) e C 0 ( Z ; ~ ) each

t e (a,b) Thus

- grad

in t h e

G(u)

in terms

and

of

F

Frechet

FE

sense

this,

on

2.7

the

FE(u)

u ~ C I (Z;V) The

we

=

We

derive

note

that

V Vu = V = u

,

where

the

inner

duct.

In the vector

product

c c 0 ( E ; V=) now

least

, N(u)

are

for

~ e C ~ ( Q a , b ; ~)

just

an explicit

there E

I F(~,u,Vu,0) JE

in t h e M

operator

is,

of

- grad

F(u)

expression

and

for

M(u)

for

This

to

F,

is d e f i n e d

by

a

u ~ C I (Z;IR N)

,

vector

E

corresponding

Q.

bundle

the

d F E ( u + s ~ ) is=0 = - d-~

(Mz(u),~)

We

M(u)

at

case). F

functional

is t h e n

by

2.8

ME(u)

hold

cross-section

Euler-Lagrange

characterized

should

for

below.

doing

functional

(and

2.6

respectively.

Before

MEu

that

and

and

want

on

bundle 2.8

the

left

is

case

we

have

is r e q u i r e d

to d e r i v e

the

usual

~ e C I ( z ; ] R N) '

L 2 ( E ; ~ N)

u e C 2 (E;V)

to hold

explicit

'

for

expressions

,

inner

Vu = V - u

pro-

,

~ ~ C I (~;~)

all for

the

operators

Mu

,

. First

2.9

notice

that

(H(u)

by direct

-|

computation

e

F

a,b where

+ F

Pi

have

+ F q

~

d~dt

,

z

in § I (or in t h e v e c t o r b u n d l e case ~(~,t) c V W V = e ~ in § I w i t h I N , and ~a . V ~as e ..... e denoting the 6 E i l standard orthonormal basis for ~N) . On t h e r i g h t of 2.9, and subse-

quently,

~ i = Vi~

~

we

repeated

as

Greek

indices

are

summed

from

I to N a n d

repeated

f

213

Latin

indices

are

Similarly

summed

from

I to P.

we g e t

2.10

(Mzu,v)

= -I JZ

(F a6i p.

a z

1

Using for

integration

M, M E

2.9,

2.10

yield

explicit

expression

In p a r t i c u l a r

2.11

M u = F (~,u,Vu,6)u~.e e + R Z ~ 8 13 PiPj

where

u~3

where

R

•uB

= V.

•

we

by parts,

,

V.

1,3

denotes

the

"Hessian

operator"

1,j

terms

of o r d e r

at m o s t

one.

V V. - (e T .D T e 3 )V k i 3 ei

In the v e c t o r

bundle

,

case

get V

2.12

Mzu

= F a B(~,u,V

u,u)uBije

+R

PiPj V B = e B.V= .u where e~a is the o r t h o g o n a l p r o j e c t i o n of e ~ onto V , u.. 13 1,3 V V V V V= = ViV j - (ek.D Te3) Vk , and R d e n o t e s terms of o r d e r at m o s t i,j e. one. 1 _

Similarly

2.13

M(u)

=

IF

B

(HB_~B) + [Fp

q q

+F .~qB

q~p8 ui l'

+ terms

with

a similar Thus

are

M

expression

, MZ

are

(inhomogeneous)

ficients We

depending shall

Specifically

assume we

second

linear on

adopt

in the v e c t o r order

in the

~,u,Vu appropriate the

bundle

quasilinear

second

(and on

order u

ellipticity

following

+F

~ Buij] PiPj

of o r d e r

at m o s t

I ,

case. operators; derivatives,

in the case conditions

assumptions

ea

i.e. with

they coef-

of M ) for

henceforth:

M , ME .

214

2.14

q'F

(w,z,p,q)

q

> 0,

z ~ V

,

w

q E V ~{0}, ~

p (: T E @ V , ~ w

w e E ,

and -2

lim

inflql+0,qe v

lql

q.Fq(W,z,p,q)

> 0,

p < Tw ~

@Vw,

w e E,

and

W

2.15

S n ~i~j > 0'

F a B (w'z'p'0)q

D eVw~{O},

g e TwE~{0},

w e E,

Z eVw,

PiPj p c T E

@V

W

for

strong

tial we

(which

is e x a c t l y

the

usual

"Legendre-Hadamard"

condition

W

ellipticity

Notice

that

reason

for

note

that

of

condition assuming

it

u ~ C 2 (~;V) .)

Mzu , 2.14

is of

it w i l l

implies

(for

a rather

become

clear

(w,z,p) e ~ x V

special later,

2.16

F a B(w'z'P'0) o D.

: 0

,

but

x (T E @ V W

form.

Our

for

essen-

the moment

))

W

W

F a 8(w,z,p,0) q z

= 0

,

and

2.17

F

so t h a t 161 the

is

M(u)

(as in

sufficiently

linearization

M~ = 0

must

§ 3. K E Y

be

is

small.

of

Mu

strongly

section

give

Our

aim

the

general

2.14,

2.13)

strongly

q e V

elliptic

In p a r t i c u l a r ,

at

a

on

~ {0}

if

2.14,

t-independent

elliptic

w

,

u e C 2 ( Q a , b ;~)

2.15

guarantee

solution

and

that

¢ ~ ~(w)

of

Q.

EXAMPLES

In t h i s of which

(w,z,p,0)~en 8 > 0 ,

~B qq

here

2.15

the is t o

we

harmonic show

functionals hold.

discuss maps

that

the

energy

and

minimal

both

discussed

these in

and

area

functionals,

submanifolds

functionals

§ 2 , and

that

extrema

respectively.

take

the

form

both

conditions

of

215 3.1 in

Example N

~

The energy

I:

, equipped

n [ i,j:1

g =

i,j,k

with

= 1,...,n.

presented

funotional.

a smooth

i qi.dx ]

way

by

N

~ 1,

BI(0)

, the

unit

ball

metric

j dx

(0) '

(Of c o u r s e

in this

Consider

gij

any

Riemannian

introducing

~gij (0) k

= 6. zj'

- 0

,

3x

manifold

appropriately

can

be

scaled

locally

normal

re-

coor-

dinates). Also

for

any

consider

Euclidean

space

~

N

equipped

with

N

some

other

smooth

metric

h =

~

h ~dy~dy B

a,B=1 For

u ~ C 1 (BI(0) ; ~ N )

a given

E(U)

I

r

= ~

I

we

~u a ~u 6

' '

g 13 (x)h

:

(gij)-1

sically

,

(u(x)

~g dx

a6

JB 1 (0) (gij)

define

8x

Notice

g = det(gij)

i

that

8x 3

this

can

be more

intrin-

written

1

r

~:Iu) = ~

2

I Idul

,

JM

where

M

linear

is

map

the

Riemannian

induced

by

u,

N

manifold and

(B1(0),g),

Idu I

2

(the

dUx : TxM÷

"energy

~

is

density")

the

is g i v e n

by n

)dul2(x)

=

[ h(du

T I ..... T n

any

orthonormal

basis

(T.), X

i=1

for

du

l

M

at

(T.))

,

X

1

x

(i.e.

g(T

,T.) 1

NOW as

consider

the

r + 0) , ~ = x/Ix I .

variable Noting

change that

t = - l o g r , r = Ixl

g

. (x) 13

can

then

write

=

6

. + O ( r 2) 13

= 6

]

(thus as

r ÷ 0

.) . 13

t + we

216

:

~(u)

1i

[

~- Jo

J sn-1

e-(n-2)th

-(n-2)t

+ e

where

u(~,t)

sn-1

dient on

like 2.5 w i t h Thus m = n-2

E

= u(r~) u.a

,

(employing

= ~ua(w,t)/~t

8(u(~,t)) (6e68 + ? U a . V U 6)

E(~,t,u,Vu)}d~dt

,

a s l i g h t a b u s e of n o t a t i o n ) , ,

and where

E

satisfies

V = gra-

a condition

c O = 2. has the f o r m of the f u n c t i o n a l

F

0, ~

(u)

in 2.4 w i t h

and w i t h 1 = ~

F(~,z,p,q)

Evidently

the c o n d i t i o n

2.12,

r [h 6(z)

n q~q6 a 6 _ + i i_ _ iP ~iP ]

2.13 hold,

because

(h 6)

is p o s i t i v e

definite. Notice

that

in this c a s e the E u l e r - L a g r a n g e

operator

M

of

§2

is g i v e n by

(*)

M(u)

= h B(u)[~e-

(n-2)~ ~ + Aue]e B

+ r6(u)(~fl

e I ,...,e N

where

orthonormal

basis

= ~I (hax,6 (y) + h y B , a (y) - h aS,y (y)) '

F ~y(y)

and w h e r e (V.,

is the s t a n d a r d

B + Vue. Vu~)e Y ,

A

is the L a p l a c e - B e l t r a m i

relative

to

S n-1

in

IRN,

h ~6,y n(y) = ~h 8 ( y ) / ~ y Y

operator

are as in

]RN ,

for

(

[ i=I

ViV i)

on

,

S n-1

§ I).

l

3.2 E x a m p l e nal m i n i m a l

2: The a r e a f u n c t i o n a l on a cone. P c o n e in ~ of the form

C = {l~:

where

Z

is a s m o o t h c o m p a c t

sional m i n i m a l

submanifold

~{

Z, l > O }

Let

be an n - d i m e n s i o -

,

(not n e c e s s a r i l y

of the s p h e r e

C

orientable)

S P-I .

Thus

Z

(n-1 - d i m e n has

zero m e a n

217 curvature

sP-I

in

For

and

0 ~ Pl < P2

C

has

zero

mean

= C n

(B

Pl "P2

=

P

{x~

P

in

we write

C

(B

curvature

~B

)

P2

Pl

: Ixl < p } )

P We

are

here

"graphically"

interested

over

C

in submanifolds

M

in t h e

sense.

following

which

can Let

Pl ' P2 normal for

bundle

each

tangent

M

C

are

Z V

at

relative is t h e

e at

S P-I

cone

which

the

e e Z , to

to t h e

over

e. e

to

P-n

That for

expressible

S

P-I

dimensional

is,

V

in t e r m s

of

V =

subspace

Then

a

{V

=

consists

e e e Z .

any

; thus

we

of

e

}

expressed

V =

be

e~

all

the where

•

normal

to

Z

vectors

consider

and

normal

submanifolds

h e C 2 (C

function

be

;V)

in

Pl 'P2 =

form

(*)

M

where

we

use

the

=

G

Pl ' P2

(h)

,

notation

Gpl,P2(h ) =

re + h ( r e )

: pl < r < p2 ,

e c Z f.

/1+lh(re)/rl 2 Notice

that

for

fixed

r

the map

rw + h ( r e ) .....

re ~

/ 1+lh(r~)/rl into

~Br,

The

hence

area

M

as

functional

in

on

(*) h a s

boundary

C

is d e f i n e d

A(h)

Pl,P2 = Hn(G

c ~BpIV

takes

C n 9B r

2 9Bp2

by

(h)) Pl 'P2

We want 2.2 that

to

(after by

the

show

that

a change area

this

can

of variable

formula

be

expressed

as a functional

t = l o g r ).

(see e . g .

[SL2]

or

To [FH])

of

see

this

note

we

have

for

the

form

first any

con-

218 tinuous

C

on

M = G

(h)

Pl'P2 t

t

f

~dH n =

I

JM

where

(~oH)ax

f

JCpl,p 2

JH

P2

f

I

=

dH n

I

JPl

is the J a c o b i a n

of the m a p

~(H(p~))JH(p~)pn-ldwdp

,

JZ

pc0 + h (OCt)

pco ÷

On the

¢/7+ I h ( ~ ) / p 12 o t h e r hand,

with

M = G

(h),

the c o - a r e a

formula

(see e.g.

[SL2]

Pl 'P2 or

X

[FH])

~: ]R P ,

and l

where

x

of

at

M

r ~ dHn : [Q2 i f I ~iVrl dHn-1 dp JM J Pl J 9B nM

gives

t a k e n onP

V = gradient

is the o r t h o g o n a l x,

x ~ M.

M.

Thus

projection

of

On the o t h e r hand,

,

where

Igrl =

x

~l-

r =

Ix±/r12

o n t o the n o r m a l

again using

ix I ,

the a r e a

,

space formula,

we h a v e f

~lvrl dHn-1

I

J ~B

r

=

I

nM P

and by s c a l i n g

J SB nM P where

(E

H

P

nC)dH n-1

this g i v e s

~IVrIdH n-1 = P n-1

[

~oHIVroHIJ(HI~B

) ~B r/C P

(p) : E ÷ S P-I 2

(1 + l h ( p ~ ) / p l

)

-1/2

I 6(H(P~) 7E

IVr(H(P~)) IJEH(p) (w)dHn-1(~)

is g i v e n by H(p)(co) = p (co + h ( p ~ ) / p ) ) ,

and

-I

,

H(pw)

JZ H(p)

is the J a c o b i a n

of

r H(p)

(Thus

r

JZI Jz H(p)

is the a r e a

functional

for

H(p) ;

i.e.

IdH(p)#V(~) I ,

v(~)

n-1

JE] JF'H(P)

= H

(H(p) (Z))

unit o r i e n t i n g Comparing

and

(n-1)-vector the above

JZ H(p) (co) =

for

E .)

expressions,

and u s i n g

the a r b i t r a r i n e s s

we see that

JH(p~)

and h e n c e

:

-IjE IVr(H(p~) ) I H(p) (co) , Pl < p < P2' ~ ~ ~]

t h a t the a r e a f u n c t i o n a l

a

A(h)

is g i v e n by

'

of ~ ,

219 P2 A(h)

Next x(p) ~ M

t' [

JPl

JZ

the

space-curve

=

consider we

r I

have

-I (oj d ~ p n - l d p I J}]H(p)

IVr(H{p~))

~(p) { T x ( p ) M

;

x(p) on

= H(p~)

the

other

x(p)

+Q

.

(e ~ Z

hand

by

fixed). direct

Since

computa-

tion

£(p)

:

-1

p

,

where

[

]

0)+h (p~)/p

I = (1+Izl 2)

r

-I/2

z = h(p~)/p

Notice

that

p

I~

(x ^ v) / Ix ^ vl , an o r i e n t i n g

q = p ~ op

,

for

n-vector

v

a unit

for

ip I x ( p ) ± 1 2

because

x(p)-Q

= 0 , where

T

x(p

2 I 1 Ix(p) I : ~ p

vector )M

,

in

:

l~vl-21Q^~^vl

=

I ( p-1

:

(I+ IQ~v12) -I IQ

x(p)+Q)

Thus

we

have

finally

I -I

] I,

Izl 2 J

2

= I , hence

n Tx(p) ZBp)

, is

means

2 A V 1

2

A vl 2

orthogonal

Then

IVr(x(p))

p

-2 -I ^ v I IQ A p x(p)

± (T x(p) M)

1+

hence

IQ'I2

l

d ~

/n_l(Tx(p)M

=

( )

......

<

(h(p~)/p)

I I d = ~ p- ~

(p)-x(p)

q-z(z+~)

I q

2 J

:

(1 - I p - l x ( p ) ± 1 2 ) - I / 2

:

/I + i~ A vl 2

projection

onto

220

P2 A(h)

=

P n-1

I JPl

where

v(~) Next

claim

~dH(p)##

orienting

see

this

let

(= H(p~)

map

GO : y ÷

order Tp

(n-1)-vector

IQ ±

x(p)

at

C

.

where

for

x(p)

here. -1 p q-z/

(I =

because Now

a local

, where

x = x(p)

V = Q

be

unless

inverse

lyl~(y) / Iz(y) I ,

Hence

dG

G

> 0

= p(~+z) (I+Iz12) - 1 / 2

x(p)

~

(v)

=

Ixl

if

is

notice

that

the

u(~,t)

= h(r~)/r

T

Q = 0

.

change (r =

A(h)

near

z = h(p~)/p) ,

agrees

orthogonal

~(v_____~)

Q± = 0

injective. our

H

, and

note

G

u p to

with

projection

~(x)

that

the

first P ~ onto

of

~(x)

l~(x) l + l~(x) l <x,v>-Ixl

2) (1+Iz I ) , w e

making

for

,

Since

dGx(p)

for

,

y eM

is t h e

v e Tx(p)M

. Now

(after

12 d ~ d p

that

(**) To

v ( ~ ) 1 2 + IQ ^ d H ( ) # # v ( ~ ) P

Jg

is a u n i t

we

[

(i.e.

= 0

and

see

that

Thus

final

Q e T ~(Q)

expression

then

we

can

set

= l~(x(p))

d G x ( # ) (Q)

(**)

of v a r i a b l e

x(p)M)

<~cx),~(v)>,

IW(X) I 3

= 0

, and

hence

Q=

0

is p r o v e d . for

A(h)

t = -loglx I ,

above

can

be w r i t t e n

~ = x/Ixl,

Ixl))

- F(u)

=

r I JQ

-nt e

F(~,u,Vu,~)dtd~

,

a,b where

Q = Z × (a,b)

,

a = l o g ( I / p I)

,

b = l o g ( I / p 2)

, where

F(~,z,p,q)

=

V(~)

(i+i z 2 ) - n / 2

/(1+izl2ll(i+A(~,z,p))#v(~)12+lqA(w,Z)A(i+A(w,z,p))~V(~)12 ,

an orienting

unit

(n-1)-veetor

for

T Z

,

A(e,z,p)

=pA(~,z)-z.B

221

with

A(~,z)

= I-(1+Iz12)

-I

zt(z+~),

B

the s e c o n d

fundamental

form

0~

of

C

at

of

N × N

~

(see the a p p e n d i x

matrices.

q'F

(by

(**) above),

course

standard

non-parametric

Notice

> 0 ,

q

and h e n c e that

area

pA(w,z)

is m u l t i p l i c a t i o n

z eV

,

q~ V

this

functional

0J

~ {0},

peT

~

satisfies

8 V

2.14.

,

It is of

2.15 is s a t i s f i e d , functional

because FZ is just the u s u a l P-I Z a S In any c a s e it is e a s i l y

on

for

F

above.

if ~P is e q u i p p e d w i t h a g e n e r a l s m o o t h P Riemannian metric g = [ gi d x l d x 3 , w i t h gij(0) = 6ij , ~gij(0) i,j=1 3 = 0 , t h e n the c o r r e s p o n d i n g a r e a f u n c t i o n a l has the f o r m of ~x h G

in

§ 4.

§ 2, w i t h

JACOBI Let

sider

Notice

and

in p a r t i c u l a r

checked using our expression 3.3 Remark:

below,

the

solution

(and F)

as in 3.2 a b o v e a n d

e0 = 2 .

FIELD OPERATORS

M

be the E u l e r - L a g r a n g e

linearization of ~(~,t)

~ #(~)

M

of

operator

for

at a s t a t i o n a r y M~ = 0

This

d L v = ~ s M(~+sv) Is=0

4.1

and

F

that

,

F

(i.e.

as in

§ 2.

time i n d e p e n d e n t )

is g i v e n by

v e C2(Qa,b;V)

,

L~v e C0(Qa,b;~) In v i e w of the d e f i n i t i o n

v c C 2 (Qa,b;~)

and

w e C ~ ( Q a , b ; ~)

M

, we t h e n have,

,

2

-mt (e

2.6 for

~s1~s2 E ( % + S l V + S 2 W ) Is1=s2= 0

L~v,w) =

(e-mtL~w,v)

,

We c o n -

for

222

( , )

where

The

is

satisfies

4.3

now We

Indeed

( , ) can

by

4.4

is

also

2.10

:

at

E

the

L 2(Qa,b;]RN)

~

is g i v e n

inner

,

'

by

v { C

(Qa,b;V)

condition

v,w { C

product

easily

.

2

ME(%+sv) Is=0

(L w,v)

rather

we

(L#u,v)

;4

for

a self-adjointness

(L v,w)

where

product

of

d = d--s

LCv

also

inner

linearization

4.2

and

the

2

(Qa,b;V)

for

compute

the

L2(E; N)

.

form

L

of

L

'

explicitly.

have

=

I JS

(F(}) @ ~ . (~) 8 ~ (~) @ ~ (¢) @ u @ v a) <~ @ u j v I' + (F @ c U V i + F ~ @ u i v ) + F z pi z Z piPj Z Pl•

,

(~)

where

F ~ @ = F e 8(w,~,?%,0)

P'l p'] ning right

terms, we

and with

thus

u. 3

as

=

and where

in

ij

L u = Vi(a

e

with

similar

notation

for

the

2.10.

After

integration

by

parts

(= e e ( ~ ) )

is t h e

@ ~

@uj@

i

) + b

8~

@uie

orthogonal

ij

a 8 = F ~ @(w'~'V~'0)

Now

+

c [3u

iJ(w)~i<jq~B a~B

By a similar

4.7

on

projection by

B~

e

of

e

onto

2.15

PiPj 4.6

remai-

get

V 4.5

where

,

PiPj

> 0

computation,

L u = a

'

~ c T Z ~ {0} w

beginning

({iB-m~@)e ~

with

+

,

n cV

2.9,

L~U

,

we

~ {0}

have

V

w

the

223

where

a 6(w)

: F

(~,¢,V¢,0)

so that,

aS qq

4.8

a ~(~)q~n B> 0 ,

T h e n we can w r i t e

L%u : a~¢

4.9

n <~v ~ - { 0 }

with

~¢u : d - m u

-I

where

L~u = a

linear

transformation

by 4.8; V

,

(~)L%u

,

and

varies

smoothly

a(w)

g i v e n by

it is also c l e a r l y in

by 2.17,

+ Lcu

: Vw ÷ V~

a(~)~

smooth

in

is the s m o o t h

= a BqBe~ w

.

because

(a(w) F,%

invertible

is i n v e r t i b l e

are smooth,

and

w .)

W

This m o t i v a t e s on

Qa,b

with

us to look at s t r o n g l y

linear operators

the f o r m

4.10

for

elliptic

Lu = ~(t) - m u ( t ) + L(u(t))

u e C 2 ( Q a b ; ~ N)

self-adjoint

on

with

u(t) e C2(Z;V)

~ relative

,

t { (a,b)

to the inner p r o d u c t

, where

< , >

for

L

is

L2(Z;y)

g i v e n by

4.11

=

for a n y

f,g c L2(E;~)

linear transformation

where V

study

For the m o m e n t need them)

to d e r i v e

the p r e v i o u s functional

section. E ,

a(w)

,

is a s m o o t h l y v a r y i n g

invertible

÷ V W

We shall

r 1 f(w) .a(w) (g(w))dw JS

W

such o p e r a t o r s

in the n e x t

it is i n s t r u c t i v e expressions First,

for

section.

(although we shall not e x p l i c i t l y L%, L%

for the k e y e x a m p l e s

by the e x p r e s s i o n

3.1(*)

we have,

of

for the

224 LgU = hc~B (9) [[iC~-(n-2) ~Ic~+AuCL]eS

4.12

+ F By(9) (Vge'VuS+FgS-Fua)e ~/ +

where

F By(y)

h y,~(~)Ag~+F

is as in 3.1 (*), so that

Lgu = heB(9)Auae B + F 8 Y(9) (?9~'?uB+V9 B'?u s) e Y

4.13

+ (h y , 6 ( 9 ) A ~ +

For the area functional with

By,~(~)Vg~'V9 B u6e Y ,

F Sy,~(~)Vg~'V~S)u 6eY

of 3.2 we have

(see Appendix

I below),

9 H 0 ,

{ L0u = H - n 6 + A l u + q ( ~ ) u

t

}

4.14

L1LO u = A i u + q ( ~ ) u

A±

where with

V=

is the "normal the normal

Laplacian"

P

V

[

?~

i:I

here

(with notation

as in

§I and

bundle as in 3.2) by

A±u =

Also,

given

q(~) : V

+ V

u .

111

is given by

W

n

q(~) (V) =

(v-B(Ti,Ti))B(Ti,Tj)

+ (n-1)v

i,j=1 where

B

is the second fundamental

TI,...,T n

is any orthonormal

basis

form of for

C

T C . W

codimension

I case

(i.e. P = n+1) we thus have

q(~)V

-= (IBI2+ (n-1))v

at

~ eC nS

Notice

P-I

and

that in the

225

where

IBI

§ 5.

DECAY

n

2

= squared

length

ESTIMATES

FOR

Motivated linear

of

second

LINEAR

of the

of

§ 4, we h e r e

Lu = H - m f i + L u

u e C2(Q+;V),

v = {V

=

on

with

self-adjoint

L

Z

as in 4.11.

}

relative

to some

< , >

5.3

of

LU = a

= ,

-1

§ I v & Vi

-I a is the i n v e r s e where al3(~), b i(~) , c(~) : V

Lu

u , v e C2(Z;V)

5.4

assume

the

< ,>

for L2(Z;~)

,

form

V

V

of

"

+ V

+ bl(60) V i u + c ( m ) u

in 4.11,

a

are

and

linear,

,

for e a c h

i,j

= 1,...,P

and

> 0 ,

~ ~ T~ Z ~ {0},

Q e V ~ ~ {0}

.

that

locally

have

smooth

we c a n c h o o s e

dependence

orthonormal

on

bases

0~ .

TI(~),...,TNI(~)

NI

for

V

which

vary

,

W

a ij (~) , b i (~) , c (w)

that

has

""

q.al3(~) (q)~i~j

also

thus

(al3(~)V-i) 3

W

Notice

product

as in 4.11.

In the n o t a t i o n

5.5

inner

Thus

We

,

~

(operating

5.2

with

by c o n s i d e r i n g

as in § 4 and L is an e l l i p t i c ~6~ on u e C2(Z;~) a n d w i t h v a l u e s in C0(Z;~))

=

operator

begin

form

5.1

where

[ IB(Ti,T j) I2 i,j=1

form =

EQUATIONS

by the d i s c u s s i o n

operators

fundamental

smoothly

in

~

;

then

u(~)

=

[ U6T 8 6=1

and

226

N1

V

V.u± = =

V

~

(ViuS) T 8 + uS~ 8i ) ,

where

n8 i = V.-[8 ~l

At the same

time we

6=I can i n t r o d u c e One then

local

readily

coordinates

checks

that

x

for

L , L

Z

and write

uB(x,t) = T s " U ( W , t ) .

can be w r i t t e n

5.1'

(repeated med

from

Greek

indices

summed

from

I to NI,

repeated

Latin

indices

sum-

I to n-l)

5.3' ( 7B ?x j )

where

~7

a

definite,

ij

, ays,

i

b78,

are

b iy8 = -bBT"i

smooth,

(aay)

positive

and

5.4'

a_< 8 [ i <~j i J ~ b

Thus

cyB

78 ~x i

in p a r t i c u l a r

for solutions

of

> O,

we may apply

Lu : f,

Lu = f

-< c(o,p)

llull

NI

~ c~

-{0},

n-1 [ c ~

the usual

in order

elliptic

regularity

theory

to deduce

ItLull

w2'P (Qa, b )

~ {0}

+ LIUltL2 LP(Q a_ <7,b+<~ )

1 (Q a- 0, b+o ) )

5.6

<- c(~,p) ILUIcO,~(Qa_o,b+o )

u ]C2,~(Qa,b )

Ull

<- c(p)

IILull

W 2,p (~)

+ llull 2

+ ]]ull L 2( Z)

LP(z)

5.7

u IC2 '~(E)

-< c(u)

ILul

+ [lull C0'P(E)

L2(E)

L

] (Qa- o,b+g ) J

227

for

~ > O,

2,~(

to

C

in

5.6

p >- 2

E)

do

etc

not

Recall L

on

We

depend

also

~ ~

(0,1)

emphasize on

that

a,b

there

, where

that

the

we

abbreviated

constants

C2'~(~;V)

c(~,p)

,

c(o,~)

. is a d i s c r e t e

set of

eigenvalues

1

]

for

~ :

11 and

and

< 12

corresponding

<- 13

orthonormal

.

%1 ' ~2'

(I

...

.

+ ~

as

j + ~)

3

eigenfunctions

.

.

c C 2 (Z;v)

' ~3

,

=

oo

and

such

any

% ~ L2(E;V)

can

be e x p r e s s e d

in the

form.

~ =

[

aj¢j

!

j:1 where

a.

= <%,¢, >

3 all

with

J

distribution

(which

are

larity

theory)

< , >

the

inner

product

of

5.2.

solutions

automatically have

u

of

smooth

the

Lu = 0

on

on

Qa,b

by

Qa,b the

standard

elliptic

m ~t

--t ( a j c ° s u j t - b j s i n l l j t ) e 2 Cj

U =

regu-

form m

5.8

Furthermore

'

+

(a-b t)e 3 J

Jcl2

Jcl I

J

+

L' +

[ J~13

-7,t

la.e < 3

-'~jt~

3 -b.e 3

]@j

,

where

11 = {j : i < - m 2 / 4 } ,

12 =

{j : ~

3

and

where

7 ±. : - m / 2 3 course

This

is of

ble"

solutions

give

possible

tions on

then

the We

± /m2/4

choices the

completeness emphasise

+ lj

standard.

a(t)¢(~)

have

= -m2/4} J

One

, and

for form

a, 5.8

of

the

that

the

,

j ~ I,

begins finds

# . then

by

13 : '

{j : I > - m 2 / 4 } J

=Be 7 + for j ~ 11 3 3 looking for "separate varia~

that

The

¢ = ¢_ and ~- ma - i a : 0 3 rigorous check that all solu-

involves

an

elementary

argument

%. 3 elliptic

'

regularity

theory

in p a r t i c u l a r

based

228

guarantees Qa,b

that if the s e r i e s

' t h e n the series,

differentiation,

for any

c,d

converges

a < c < d < b

We n o w w a n t to c o n s i d e r

(*)

uniformly

u(t)

decaying

ry to i m p o s e

the i n h o m o g e n e o u s

at a s p e c i f i e d

2

s e n s e on

series obtained

and a b s o l u t e l y

on

by

Qc,d

on

Q+

problem

,

of w h e n we can find s o l u t i o n s

restrictions

L

.

LU = f

and the q u e s t i o n

in the

t o g e t h e r w i t h all the f o r m a l

termwise

with

in 1.4 c o n v e r g e s

rate as

t ÷~

on g r o w t h of

u

of this p r o b l e m w i t h

It is of c o u r s e

f(t)

as

t ÷~

necessa-

In fact we

take co

(**)

q > X I,

q I {Rey+j : j _> I},

fl

and

( --- [

II

i (eq tIf(t)II) 2dt]I/2 J0

where

II II is the Also

L2(Z;~)

< ~

)

norm.

for a n y n o n - n e g a t i v e

integer

J

we let

Hj : L 2 (Z;~) + L T ( z ; ~ )

J

be the p r o j e c t i o n ± let Hj = i d e n t i t y

5.9

LEMMA.

~ + ~ j=l aj~9 j=laj~j - Hj

Suppose

q, f

tive integer such that otherwise.

(*) with

~ ~ LT(2;V)

suPt~0eqtllu(t) 11 < ~

The lemma

required decay

Hj

J = 0)

, and we

there is a unique solution

u

of

and

lim H~u(t) = H~¢ (weakly in L 2 ( Z ) ) . t+0 flu(t) II ~ c(ll~ll+llfll(q))e -qt , t _ > 0

Remarks:

(I)

the

if

(**), and let J be a n o n - n e g a + q < Re XI and Re Xj ~ q < Re Xj+I

if

This solution in fact satisfies

5.10

Hj = 0

are as in

J = 0

Then for any

(and

Hj±

says that we are a b l e to find s o l u t i o n s

(for

f

as in

p a r t of the b o u n d a r y

p a r t of the data.

be m a d e e x p l i c i t

This

w i t h the

(**)

), a n d i n d e e d we c a n e x a c t l y

data;

h o w e v e r we have no c o n t r o l

is c o n t r o l l e d

in the p r o o f below.

by

f,

in a m a n n e r

specify of the

that will

229

(2) because that

The d a t a

is in fact t a k e n

the s o l u t i o n

constructed

in the s t r o n g

L2(Z;~)

sense,

in the p r o o f b e l o w has the p r o p e r t y

lim flu(t)II = II~I] t+0 0o

P r o o f of 5.9. s o l u t i o n of

Any

L2(E)

convergence

(*), p r o v i d e d

a. (t) 3

series

satisfies

(I)

W-m~-

We c a n e x p l i c i t l y + Xj = Xj (so X 3•

get the f o r m of s o l u t i o n s

is e q u i v a l e n t

is c o m p l e x

for

f.(t) 3

j c 11 )

(e

w

of

a n d let

.

(I) as f o l l o w s : # = eXjtw.

-(2x.+m) t -(y.+m) t 3 ~) " = e 3 f. (t) 3 Thus

(2) can be s o l v e d by i n t e g r a t i o n .

w = a

= 3

gives a

let

Then

(I)

to

(2)

and

l.w = f. , 3 3

[ a.(t)~. (~) ._ 3 3 the3-10.D.E.

for

,

(I) we get s o l u t i o n s

of the f o r m 3

(3)

a.(t) J

= Re

I -X.t -Xjtrt (2x.+m) s ~ -(x.+m) T [e 3 Xj - e I e 3 I e 3 f.(T)dTdsl , :Sj Js J )

where

~., 8. are c o n s t a n t s . (For the m o m e n t w e are not w o r r i e d 3 3 t h e r or not all s o l u t i o n s are e x p r e s s i b l e in this form.) S i n c e we a s s u m e

Cauchy using

inequality the C a u c h y

fact g e t t h a t tion, w i t h

I q > Re YI '

the e x p r e s s i o n

a n d the f a c t t h a t inequality

IIfll(q) < ~

(a.3

as in

flu(t) II ~ c(llfH (~) +II~ll)e -qt

=

~ ,

8j = 0 ,

The fact t h a t the s o l u t i o n representation

aj

=

0

(3))

s e n s e by the

Directly

a n d the o r t h o n o r m a l i t y

u = Z aj(t)#~

Bj

(3) m a k e s

of the

gives

whe-

computing, ~j,

w e in

the r e q u i r e d

solu-

, p r o v i d e d we s e l e c t

,

j

~j = <~,~j>

is u n i q u e

_< J

,

j a J+1

is e a s i l y

checked

5.8 for the difference of any two s o l u t i o n s .

by u s i n g

the

230

Of

course

corresponding sume

that

if w e decay

p ~ 2

5.11

are

estimates

llfll(q)

that

we

5.12

for

all

5.6

, then, (we a l s o

immediately

we

now

For

+ Ilfll(q)

by using need

deduce

Let

q,

p > n

w ~ W 2 , p (Q+;V) = with

the

case.

in p a r t i c u l a r

instance

< ~

the

decay

}

, the

if w e

as-

,

the version

that

the

of

estimate 5.6

solution

of

valid

u

J and

with the

consider

consequences We

are

continue as

in

B = B

of

5.9

up

5.9

to

in the

satisfies

,

THEOREM

of

the

to u s e

the

the

Banach

[(

Lu =

I

± ~oU

linear

g : g(~,t)

,

a 3

results

notation

introduced

space

functions

eqtllWllw2,

of

) < ~ P ( Q t - 3 rt

[I IIB

general

above

quasilinear

problem

a 1 - D 2u + a 2 . D u +

a 3-u+

± = ~j~

for above;

5.9.

be P,q [kwllB ~ sum. t~1

norm the

EXISTENCE

g

on

6.1

where

and

l f(((c~) +11%II ]e - q t _< c [II p w2,P(E))

QUASILINEAR

examine

quasilinear

We

improve.

f

t a I.

A GENERAL

equipped

u

about

LP(Qt_I, t

Ilull 2 , p w (Qt_1,t)

§ 5.

the

~

with

boundary),

for

more

H s u o t > I eqtlLf[i

% ~ w2'P(z;~)

combination

to a s s u m e

and

p and

willing

on

: a.(~,t,u,Vu,u) 3

2 :

~Q+

, with

, llullB <

Q+

'

B

being

231

r

la. (u~,t,0,0,0) I -< ~ a - S t 3 q0 t

6.2

{

laj(~,t,z,p,q)-aj(~,t,z,p,q)

I L

Hence

{I q i l

c, e, s > 0 Because

THEOREM

There

that

0 S a ~ a0

o f 6.1, a IIU - U a l t B ~ c a 2 • tion

u

are

with

U

=

2,p

if

<- a

+ae

.

theorem

5.9 and the e s t i m a t e s

of the c o n t r a c t i o n for small

enough

c = c(L,q0,s)

constants and

(z;v)

existence

6.1 has a s o l u t i o n

6.3

-st (iz-zI+Ip-pI+lq-ql)

are c o n s t a n t s .

application

such

if

ll~ll

q0 < q

of the g e n e r a l

that

,

w

and

it is then a s t a n d a r d to p r o v e

-< a

B

I -< ce

6.2 h o l d s ,

then

lluall B ~ c a . A l s o ~ -X~t Re [ < ~ , ¢ >e 3 Cj

mapping

principle

a, p r o v i d e d

,

6.2 holds:

s 0 = a0(L,q0,s)

there

is

a unique

, provided

5.12,

tlgll B

> 0

solu-

~ a

2

j=J+1 6.4

Remark:

rem, u

Notice

g = 1-n/p. is

C

(Q+;V)

P r o o f of 6.3.

that

(w)

a. 3

standard

elliptic

provided

a

C

For any

the

w c B

=

a~W)

D2 (w) (w) ° w+a 2 .Dw+a 3 w+g

2 < c(IIWlIB+C~)

IIF (w)iIp(q)

(2)

IIF(w I) -F(w2)ll;q)

c = c(q0,s)

of

this e x i s t s

u

(and

are p r o v e d

of

(and is unique)

,

I[F(w)[Ip(q) _< c(IlwII;+~ 2)

llw21IB+a) IlWl-W211B

by d i r e c t

if

ilgIIB
,

computation

b a s e d on

IIwllB .

We n o w set up a n o n - l i n e a r to be the s o l u t i o n

we d e d u c e

and we n o t e the e s t i m a t e s

~ C(IIwIIIB +

, which

6.2 and the d e f i n i t i o n

are

estimates

theo-

we let

= a.(~,t,w,Vw,Q), 3

(I)

where

by the S o b o l e v e m b e d d i n g

T h e n by u s i n g

F(w)

where

B c C I '~(Q+ ;~)

operator

Lu = F(w) by 5.9.

with

T :B ÷ B

by d e f i n i n g

llullB < ~

T h e n by 5.9,

and

5.12 and

T(w)

Z]u = ~ (I),

(2)

232

above,

we h a v e

(3)

lIT(w) IIB -< c(llwII~+a)

and

(4)

IIT(Wl)-T(w 2) IIB -< c ( N W I IIB+IIw2]IB +e) llwl-w211B ,

c = c(q0,e).

Hence

c = c(L,q0,c)

for

sufficiently

is a c o n t r a c t i o n

of

has a f i x e d p o i n t

large,

(a0

in this set, w h i c h

u e - Us,

because

from

(3) a n d

into itself.

is the r e q u i r e d

I]u -U IIB )

(for

small enough)

we c o n c l u d e

{u e B : llullB ~ ca}

The last e s t i m a t e te 5.12 to

~ ~ e 0 ( L ' q 0 'E)

u e - Ua

and (4) t h a t

T

In p a r t i c u l a r

T

solution

f o l l o w s by a p p l y i n g

vanishes

on

~Q+

u the e s t i m a -

and

L(u a - U s) = F ( u )

§ 7.

APPLICATIONS Let

M, N

functionals of

M~ = 0

be the E u l e r - L a g r a n g e

F , G

as in

(so t h a t

c h a n g e of v a r i a b l e we c a n w r i t e

§ 2,

a n d let

t ÷ t + T ,

T = T ( ~ , ~ , G , e , e 0)

N(u + ~) = 0

Thus

g = 0

the g e n e r a l

is the m i n i m u m

in 6.1

theorem

eigenvalue

§ 5 with

e < e0

L = L~

THEOREM

and

~ c w2'P(~;~)

§ 2).

solution

Then,

after a

sufficiently

large,

arbitrary

and w i t h

of 6.2 if

E ~ 0

q0 = 0, Further-

the f o l l o w i n g ,

(= m i n i m u m

: w c L2(~;~),

in w h i c h

eigenvalue

w ~ 0})

,

11

of

and

X~

are as in

as in 4.9.

Suppose

7.1

L~

for the

E H 0 .

6.3 i m p l i e s

of

L% = i n f { < w , w > -1 • (-)

if

mt

in the f o r m of the e q u a t i o n of

in the f i r s t two c o n d i t i o n s

m o r e we m a y take

e

be a t - i n d e p e n d e n t

in the n o t a t i o n of

the e q u a t i o n

e = 0

~

times

MZ# = 0

6.1, a n d the c o n d i t i o n s 6.2 hold, w i t h and w i t h

operators

,

q > max p > n .

{O,Re(-m/2 + / m 2 / 4 + 1 1 ) }

Then there is

,

q ~ {Rex~ : jal},

s 0 = a0(q,G,~) > 0

such that

233

lal

if

a

T = T(q,~,G,a)

and if

< SO ,

w2,P (Q+;~)

solution

I

u

is s u f f i c i e n t l y

there

is

of the p r o b l e m

(u + ~) = 0

[

large,

Hjua(~o,T)

on

QT, ~

= ~ nj,(oJ)

,

oJ~ Z ,

with

suPt>T+leqtllu

I~2

~ cI~ l . 'P(Qt_l,t )

Furthermore tional

if

E ~ 0

then we may

suPta I eqtIIu

U

is the solution

- eUI~ a w2,P(Qt_1,1)

L#U = 0

i 7.2

by

U(~,t)

Remark:

Notice

about our principal

7.3 a nal for

the addi-

~ ca

2

,

Q+

=

particularly family

that

~j I . ) (*) implies

{ua}lai<~0

It is perhaps worthwhile

functional

on

= Re I [ <~,~j>e k j=J+1

is, the l-parameter

energy

and we have

of

)

given

T = 0

estimate

(*)

where

take

examples

~

has velocity

to explicitly (as discussed

-1

ua ÷ U .

U

at

That

a= 0 .

mention what 7.1 tells us in

§ 3).

Firstly,

for the

of 3.1 we have the following:

THEOREM Let ~ : S n'1 + (AN,h) be a C 2 harmonic map (i.e. ~ is C 2 (A n ~{0}) homogeneous degree zero extremal of the energy functioE(u) n ~

of 3.1 with as in 3.1.

gij = 6ij)

Then for any

'

and

q > 0

let

g

there

be any smooth is an integer

metric

J = J(q)

+ (= the

least n o n - n e g a t i v e

integer

with

q < Re yj+ I)

such

that,

if

234

c C 2 ( S n - 1 ; ~ N) and

and

a harmonic

if

lal

H u

lim r+0

for

(p~)

a

H]~

= a

[r-qlu L

g

is

the

sufficiently

Note:

small,

there

is

p = p (c~) > 0

: (B

(~)

p

~ {0},g)

, ~ < S

÷

n-1

(]RN,h)

,

and

+rl-qlau (rv)/arlc0,

(r~)[ CI'~(S n-l)

] = 0

~(sn-1) )

~ c (0,1)

each If

for

sufficiently

map

+ u

satisfying

is

Here

standard

small

Xj± , H ±J

a

Euclidean

metric,

then

we

may

take

p

~

1

.

are as in

§ 5

with

L = h(~)-IL t

L¢

as

in

4.13.

7.3

THEOREM

smooth

(V

Let

compact

the

normal

C

be

manifold bundle

a minimal (as

as

in

cone

3.2).

in

For

3.2) , there

in

~

P

any

with q

> 0

J = J(q)

is

E : C n S and

P-I

a

~ { C2(E;V)

(= l e a s t

non-nega-

+ tire

integer

Is[ < a 0

with

then

q

there

< Re is

yj+1 )

a solution

Mh [jh

with

lim r+0

each

# ~ (0,1) M

is the m i n i m a l

for the area

tive to the J a c o b i

7.4

0

h

on

C n BI

(e) = ~ Hjp(c0)

field o p e r a t o r

such

that

,

(r~)/~r

A(h)

of 3.2)

Ic0 ,~(Z;~) j] = 0 for

(i.e. and

the E u l e r - L a g r a n g e ± ± yj, Hj are r e l a -

of 4.14.

Remarks: (I)

Riemannian

Notice case,

that

6.3 also g i v e s

if

(0)

surface operator

functional

> 0

of

[r -q-1 lh (r~) I +r-~l~h t C 1 ,v(Z;V_)

Here operator

=

a 0 : ao(q,Z, ~)

and

such e x a m p l e s

w h e n we take the area

functional

in the g e n e r a l A (g)"

relative

to

235

the m e t r i c with

g

as

M(g) ( h )

with

in 3.3.

=

,

0

In this

~

H~ha(p~)

lim [r-1-qlh.(r~)Ic I r+0 (2)

LOt

Theorem H~@(~)

a whole

6.3

also

I having

component

Another

C

the v a r i a b l e

and

(Cf.

at

4.14)

of

point

t = + log r,

the o p e r a t o r

form L

2.2,

has

0

the

6.3 can be a p p l i e d

7.4

THEOREM

(= l e a s t such

~. ~ ( 0 , 1 )

any

non-negative

that

with

For

lim r+~

if

lal < a 0

as in 7.3,

~ c C2(E;~) integer then

to

C

velocity

also

~ c C2(E;Z)

at r a t e vector

In this

this

case

we

in 3.2,

time

faster

at r a d i u s

get m i n i m a l

we have

submanifolds just m a k e and we m = -n,

form

£±u+q(w)u

thus

where

given

= h(r~)/r

but

,

submanifolds

6.3.

u(e,t)

L0u = N + n i +

Then

can

by a p p l y i n g

- @(r~)

for e v e r y

decaying

is that we

and

] = 0

h (re) - -

initial ± HjT .

to

of the

(p : p(~))

C 0'~ (Z))

of m i n i m a l 3.2)

ha { c 2 ( c n Bp;V)

(r~)/trl

lim a+0 Thus

~ { E .

equal

a functional

w ~ E

and with

infinity

change

,

a

family

Ixl + 0

important

to

,

in the n o t a t i o n

[j±

get

a~(~)

r-qlah

+

l-parameter

as

again

a

guarantees

= H~@(~)

Ma( = G O 1(ha) ' l+q than Ixl

decaying

6.3 g i v e s a f a m i l y

~I

'~ (E)

= 0 ,

we h a v e

case

=

giving

.

the

following

theorem:

q > Re X 1- , t h e r e is J = J(q) + q < Re Xj+I) and a 0 = ~0(q,~,E)

and

with there

is

0

a solution

on

of

C - BI

Mh

=

~jh

(~) = HjT(~)

+ r q l ~h [r -1+q IC I L lha (r~) '~(E)

ha

,

~ c E ,

] I = 0

(r~)/~rl C0,~(E))

for

each

236

APPENDIX

THE

AREA

FUNCTIONAL

E c S P-I

be

(e + h ( ~ ) )

, where

h(~)

over in

~

as

3.2.

venience

in

3.2);

(Notice

that

we

We

view

first

3.2,

and

consider

is

C2

and

normal

h ~ C2(~;~)

the as

compute

(y)

image a map

the

=

I

in

thus

H

dH

as

PART

FOR A SUBMANIFOLD P-I P S c IR

SPHERE

Let

TO

H c S P-I

Z ÷

~P)

map

By

A

is

definition

as

in

of

3.2 and D h V V= , we have

)T

definition

of

for

by

(T + D

denotes

V V=h T

h = T

(

normal

H : ~ ÷ (I+Ih12) ~

(C =

bundle

denotes

orthogonal

second

fundamental

(D h) T I

an

H

:

h A(~,h))

+

directional

(D h) T T

projection form

T

orthonormal

=

we

are

viewing

B

± (T

E)

~9 T

7. ÷ T

~

.

Thus

n-1 [ j=l

-

basis

(D h) T

where

for

cone E

computational

for

as con-

T e T

w

,

for

as

a

h.B

for

T

=

T ~ ~

linear

(T,Tj)Tj

T~E

- h.B

W

in

~P.

,

onto

B

derivative

at

Notice w

w

TI,...,Tn_

the

-I/2

%

D

where

at

THE

.

w

where

C

, but

induced

(I + lh12) - I / 2

map

the

of

linear

the to

, ~

OF

,

Thus

(T)

,

transformation

from

, we

that, have

by

237 V

dH

(T) =

(I+ lh12) -1/2 (T + ( V = h ) A ( ~ , h ) - h ' B T

and hence

the area

functional

is g i v e n

(with

(T) W

B(~,h)

= h-B

)

by

W

Hn-I(H(Z)

2

=

(1+lhI)

-

V /2 I (I+V=hA(~,h)

(n-l)

- B (~,h))

JZ where

H

n-1

v(~)

= T I ^ ... ^ Tn_ 1 , or,

(H(Z)

(1+lhl 2 )

=

equivalently,

- (n-1)/2

where

(b..) l]

V (T.+(V = h)A(~,h) - h ' B

n-1

IA

JZ =

1

i=I

T. 1

(n-l) x (n-l)

matrix

( [ ) I d H n-1 ~

Ir (1+lhl 2) - (n-1)/2.a e t ( b . (~)) dHn-1 JE 13 is the

V (~) IdH n-1

#

1

,

g i v e n by

P bij

=

~ ai~aj£ £=I

,

with

aii = d H ~ ( T i ) - e

(I + lhl2) I/2

V

v

= T i + ( v T h ) - e £ - ( h . VT h) (e .h+~z)• - (h.B~(Ti)) .e l

1

1+lhl 2 Hence V b

= 6

V

+ (V

h)-(V 1

h) + (h.B

(T)).(h.B

- ((h.B (T.)).T + (h.B ( T ) ) " T

Since

det

(I+P)

= I + trace

(T))

1

) +O(lhl 3)

P + ~ I (trace P) 2 - ~I t r a c e ( P 2) + O(IPI 3)

,

we

get det(bij)

where

H = trace

B

V = I + IV=hl 2 - lh-B

= mean curvature

12- 2 h - H + 2 1 h . H I 2 + O ( l h l

vector

H 0

in this case,

3)

,

because

238

is minimal. Thus since

(1+lh12)-(n-1)/2

/1+x : I + ½ x + O ( x =

I

-

2)

(n-1)lh12/2 + o(lhl ~) , we get I

V

2

(lh'B

Thus we conclude

and

in particular

I + (n-l) lhI2)) + O ( l h l 3 ) } d H n

the expression

for

L0

in 4.14.

239

PART

ASYMPTOTIC

Our main existence of the

a i m in this

of asymptotic

f o r m of

In

§I

somewhat

G

we

lengthy,

the p r o o f

and

in t h e i r

theses

in

discussion However

of

§§ 5,6)

the method

§ 6, w h e n

of

§ I and

§ 5.

STATEMENT

OF MAIN

for

to m i n i m a l and

be the

more

surfaces

the

of f u n c t i o n a l s

in

t

general

a much

§ 5

not

has

result, maps.

The hypofurther applicable.

the a d v a n t a g e

AND A FIRST

APPLICATION

in

that

and Allard

so w i d e l y

is n o t t r u e

as

out

simplified

(see t h e

, which

functionals

is

and harmonic

cone problem.

those of

the r e s u l t

The proof

by A l m g r e n

it is a p p l i c a b l e ,

THEOREM,

G

about

§ 5 we point

is b a s i c a l l y

than

therefore

ral

F,

a somewhat

tangent

decay

let

In

of t h e u n i q u e

exponential

Here we

gives

restrictive

of

§§ 2-4.

suggested

establishing

§ I.

in

theorem.

of a method

and

is k n o w n

of extremals

of the m a i n

§§ 3-5,

version

discussion

in

t + ~)

what

I, § 2.

actually

§ 6 are more

theorems

(as

its a p p l i c a b i l i t y

(and g e n e r a l i z e d )

THEOREMS

is to p r e s e n t

is d e s c r i b e d

§ 6 is i n d e p e n d e n t

[AA]

limits

one version

as p r e s e n t e d

and we discuss

LIMIT

section

of P a r t

state

II

of

for the gene-

§ 2 of P a r t

I.

Thus

a < b

Fa,b(U)

r 1

=

e

_mt F

(~,u,Vu,~)

JQa,b G

b (u) = F a b (u) + a,

,

I

e - m t E ( w , t , u , V u , 6)

jQ

a,b where

m @ 0

is c o n s t a n t .

I 2.7 a n d

M,

as

of P a r t

in

§ 2

N, M

FZ

continue I.

continues to d e n o t e

to d e n o t e

the

functional

the E u l e r - L a g r a n g e

of

operators

240

It w i l l be e s s e n t i a l addition

to

I 2.14,

real-analytic there

in

is a D o w e r

to i m p o s e

2.15).

z,p

restriction

on

we m u s t a s s u m e

: w e in fact a s s u m e

that

F

(in

F(~,z,p,0)

for e a c h

is

( z 0 , P 0 ) c ~P N x~ ~-

series expansion

~(~,z,p,0)

1.1

a further

Specifically

~ %S(~)(z-z0)~(p-p0

=

)B

BeZ~ N (Z+ = the n o n - n e g a t i v e bourhood

of

integers)

(Zo,Po)

ding series

uniformly

in

w ,

for the d e r i v a t i v e s

of

F

(~ • ~) a l s o c o n v e r g e 1.2

Then there is a lim t+~

satisfy

for

G

+

and h a v e

finite

C2

although

the e x t e n t

with

l~(t)

in some n e i g h -

and that the c o r r e s p o n -

of

to the

(Zo,p O)

2.14,

on

2.15,

F

= 0

variables in m.

in addition

and

with

Q+)

m

uniformly

of 1.1 above,

= 0

MZ(@)

(z,p)

respect

I 2.5,

N(u)

with

u

is a

lUlc2

and

< ~. (Q+;~)

] = 0 ICI (~))

T h e a-priori

Remark:

m e ~ ,

assumption

(i.e.

~ ~ C2(Z;~)

[lu(t)-~] L C2(~)

1.3

F, E

the real-analyticity extremal

for

in some n e i g h b o u r h o o d

Suppose

THEOREM

satisfies 2 C (Q+;~)

converginq

necessity

that

u

be d e f i n e d

n o r m can in some c a s e s be r e l a x e d to w h i c h

for all

t > 0

(see § 5 b e l o w ) ,

this can be done g e n e r a l l y

is n o t yet w e l l

understood. As a first a p p l i c a t i o n G

corresponds

u(m,t)

that

to the e n e r g y

= h(re)

( N, 7) • 0

of the a b o v e t h e o r e m , functional

(t = - l o g r)

L e t us s u p p o s e

h

where

space

is real analytic.

to e n s u r e

t h a t the a n a l y t i c i t y

According

to r e s u l t s

an isolated

IDh(r~) I -< c/r

is a h a r m o n i c

of

the case w h e n

n = 3, map

so t h a t

(BI (0),g)+

locall~ minimizing

near

h , and that the m e t r i c

(This last h y p o t h e s i s

hypothesis

of [SU] a n d

discontinuity

as in 3.1 w i t h

is a c t u a l l y

is a p o i n t of d i s c o n t i n u i t y

the t a r g e t

h

consider

1.1 a b o v e

[GG] we k n o w t h a t

for

h

and that for some

and

ID2h(r~) I -< c/r 2 ,

0 , ~

for

is n e c e s s a r y

is s a t i s f i e d ) . (in case n = 3) p > 0

0 < r < p .

0

is

241

Then Thus

(since (after

order

1.4

t = -log change

to c o n c l u d e

we h a v e

of v a r i a b l e the

If

THEOREM

r)

lUlc 2

,N

<~

(QT, ~ " t+T) , we

t ~

) can

n = 3

and

h

harmonic map

C2

: (BI (0),g)

: sn-1

t

lira IIh(rw) - % ( w ) IC2(Sn_1 ) + r+0 ~ Notice

that

discontinuity to h i g h e r

§ 2.

has

that

us

+

1.2

I/p

above

. in

FORMULAE

real analytic,

then

with

c2(sn_ I

precise

I = 0 ))

description

extension

AND ASYMPTOTIC

in the p r o o f

(i.e.

for e a c h

X

Ir - - I r

a partial

is a locally mi-

of this

of the application

§ 5.

sequence

LIMITS

of T h e o r e m

t-independent)

1.2

involves

asymptotic

t. + ~ 3

there

limit

proving

that

points:

in fact

is a s u b s e q u e n c e

it.,} 3

that

2.1

u(tj ,) +

in the

CI(z)

(Of c o u r s e Q

apply

(JRN, ~)

(~N, X)

÷

a rather

We g i v e

in

step

stationary

we p r o v e such

dimension

first

gives

0 .

MONOTONICITY

The u

this

at

T = log

following:

nimizing harmonic map as d e s c r i b e d above, with there is a

'

which

In the that

M(~)

with

sequences of

discussion following

~ = #(~)

= 0

first

of the p r o o f

preliminary

where

that Theorem

one has

particular part

also

is c o n s t a n t

Notice because

norm,

if we

is a s m o o t h

think

respect 1.2 does

of

not

immediately

that

{tj},

we

1.2, of

and

as a f u n c t i o n

of

MZ(~) = 0.

defined

on

to the t - v a r i a b l e ) .

to e s t a b l i s h {tj,}

~

solution

~

does

choose

is f i n a l l y

follow not

to use.

carried

out

from

depend This in

this,

on w h i c h is the

§ 4 after

hard the

§§2-3.

calculations

K > 0

will

denote

a constant

such

242

22

lulc2 ~ ~ TO establish

which the

we

now

full

continue To subset

proceed

force

In a l l

2.1

of

that

of

need

assumpt:~on

follows

(0,T)

Ga, b

I 2.14,

let

the

to d e r i v e .

the

to assume begin,

we

~ for

be

it is h e r e

I 2.14

q'F

and

that the

q

> 0

that

for

functional

we

q { V

use w

§ 2,

- {0}.

of

I

and we

~

a compact

2.5.

C I (~)

a

some

Incidentally

denotes

2.'5

monoton{c{ty formuZac,

appropriate

function

T > 0 , and,

with

spt

ii

small,

for

let

ul 't) = u ( t + I~ (t)

(uA se

is t h e n u

a

C

2

function

(Q+;V)

is e x t r e m a l

for

O

we

(*)

On

for

I~ I

sufficiently

small) . Becau-

have

d = 0 d--~ GO ,T (ul) IA:0

the

ul(t)

other

hand

direct

computation

= u ( t + l ~ ( t ) ) (I + l < ( t ) )

s = t + l~(t)

and

G 0 , T ( U l) =

noting

[

, and.

that

shows

that

after

introducing

t = s - 16(s) + O ( I 2 ) ,

e-m(s-l%(s))

the we

new

see

variable

that

{ F ( w , u ( s ) ,Vu(s) ,G(s) ( 1 + l ~ ( s ) )

]Q0,T + E(m,s-}~%(s),u(s),Vu(s),~(s)

as

II I ÷ 0

deduce

2.3

.

Hence

taking

derivative

the

general

I

e-mt{mF(~:u,Vu,6)%

I = 0

and

using

+ O(I 2 )

(*)

identity

+

J Q+

=

where

at

(1+l~(s)))]

(~.F. u

I

JQ+

(w,u,Vu,~)

e-rot(El ~ + E 2 < )

(by I 2.5) -s

IEII + IE21 ~ c e

0

t

- F(w,u,Vu,6))~]

,

we

243

with

E 0 > 0,

and with

Replacing the

%

by

characteristic

f

f

11~ I

2.4

L

e e

mt

depending %

and

function

~.r.-

JQ

of

r

taking Q

We

0 0 in

2.5

(and in

u

on

we

2.3

that

all

can

of

let

in o r d e r

Im e m P F

fact

%

m

tion

< 0 of

we Q0,p

can

(u)

Ime

-

instead

+

2.5

can

l~p

r

case

2.6

2.6' Integrating

2.7

can

zero

u

is

the

in c a s e C2

l U I c 2 ( Q a , 7 ) -
E - 0).

extremal

on

.

characteristic

(F-~-F.) I < c e

function

% 2.3

-eO p

,

p ~ 0

.

u

approximate

the

gives,

p > 0

for

-£1

1 (F-~.F.) [ s c e ;Zx{p} u

characteristic

func-

,

p

'

£I = r a i n { s 0 ' Ira l}

be written

emPE

k

while

approximating

of

is

if

f

(u)

P

2.5'

is v a l i d with

°

] -E0~ (F-~.F.) I _< c e u )

side

approximate

let

, in w h i c h

E0,

Evidently

right

JZx{p}

mp 2.6

m

to d e d u c e

P'~

If

Q+)

and

get

;Zx{r}

the

this

K

l I

+

p,T for

on

a seauence

, we

Q,T

(F-~.F.

';Zx{p}

u

only

(u)

-

h -gO p ~ ' F ~ ] I _< c e ,

[

P,~'

JQ

m>0,

p,oa

be written

emPFn

d lqZj <

2.5',

2.6'

lemTF

~,o

we

T, oo

(u)

(u)

get,

-

[ u'F~I JQo,o

for

- emPE

p,~

] I -< c e -s I p

T > p > 0

(u)

+

,

m

< 0

,

f

I jQ

u-F. I -< C e u P,T

-~0 p

n

m >

O,

244

and

2.8

Je

mT

F0,

(u) - e m P F 0 T

In p a r t i c u l a r

2.9

we

'P

(u) -

t

J G-F~J JQp,T

~ c e

-~1 p

,

m < 0 .

conclude

0(u)

Z lim emPF

(u)

exists

(and is real),

m > 0 ,

(u)

exists

(and is real),

m < 0

pt ~

2.10

@(u)

Usinq nary

ce

the a b o v e

asymptotic

arbitrary

~ lim emPF

facts we

limit Points

and note

t. , s u c h ]

that

¢

u.

is

in 2.1. lul

that

÷ ¢

by

of

T o see

(¢)

E @(u)

in

CI

we can

on

'

Q

this

note

that,

be

find a subsequen-

,

only

for

statio-

t. + ~ ]

-t.], < t < ~

Thus we have

and

pt ~

for every quired.

-~ 0,

that

we have

bv

2.9 t h a t

emPF

{ 0

the e x i s t e n c e

SDecifically

= u(t+t.3,)

~ K

t.

2

locally

uj,(t)

i ¢Ic1 ,I(Q)

with

is i n d e p e n d e n t 2.7 a n d

as

C

is d e f i n e d

C 174

can n o w d i s c u s s

(since

uj, where

0,p

in e a c h

Then by

identical

m < 0 case

I 2.14 w e h a v e

argument

is l e f t

{ ~-F~(~,{,?{,~) JO"'p,%

to the

(based o n reader.

~ H 0 2.8,

= 0

on

2.10)

Notice

Q

as r e -

to d e d u c e

that we

also

con-

that

FZ(¢) = ImJe(u)

2.11

Notice

also

elliptic

that

¢

regularity

Because

of

2.11

is a u t o m a t i c a l l y theory

(see e.q.

we can conclude,

smooth by [MCB])

I 2.15

because

by using

2.7,

and

standard

MZ¢ = 0 . 2.8 a n d a q a i n

using

245

2.5,

2.6,

2.12

that

m

t

I JQ

f

2.12'

f

~.F.u ~

(F-~.F6)

+ c e

-eoP

,

m > 0

,

m < 0 .

t~x{p}

fi'F6 ~ F2(~)

{m{ ;e

- FZ(¢)

f

- ; ~ x { o } (F-fi.F.) U + ce

Ow ~

One now easilv = -(Mz(~),¢)

213

= 0

checks,

(by d e f i n i t i o n

lml[

r

jQ

e = m i n { E 0 , e 1}

M(d#) = 0)

2.14

~¢

2.15

2.12'

lu-¢ 2)

imply

+ c e -ep

,

u

Q

satisfies

fl(u) = 0

for the d i f f e r e n c e

{RI

-<

c e-st

u- {

and since

an e q u a t i o n

¢

of the

+ a 3 " (u-d#) + R

{aj(t) l -< c{lt lu(t)-¢lcl (z)

Hence,, p r o v i d e d

we c a n

f

,

use

the

W2 ' 2

+ t~(t)

lu-~IC1(Qp_1,p+1)

estimates

of

I

5.6

(16{2+]v(u_d#){2+lu_d#12)

satisfies form

,

l

1

c°(z) j

is s u f f i c i e n t l y

(with

L = ~¢)

2

to

give

+ c e -ep

_< c Ilu-d#ilL2(Qo_l,o+l)

:~:x{p} c = c(K,E,F)

c = c(K,E,F)

2.16

2.12,

~ s FE(~+s~) Is=0

is as in 4.9 and

C = c(K,E,F)

with

that

~¢(u-d#) = a 1 " D 2 ( u - ¢ ) + a 2 " D ( u - # )

where

small

(since

we get

I 2.18);

(1612 + IV(u-d#)t 2 +

p >0

for all

that

d

I 2.14 a n d the fact

J ~:x{p}

u

P,6

Notice

usinq

r jQ

> 0 ,

A l s o by I 2.14 we h a v e hence

2.13 y i e l d s

lul 2 -< C IIu- d#ii~

p,o~

6.F.u ~ clul 2 ,

+ c e _ep 2 (Qp-1

,p+l )

,

c = c(K,G)

,

246 p z 1

for any

such t h a t

lu-¢l

1 C

( d e p e n d i n g o n l y on K,G) 2.17 if

Remark: u

is

is s u f f i c i e n t l y (Op_1,p+i)

In v i e w of later g e n e r a l i z a t i o n s

merely

then e s s e n t i a l l v

C2(Q0,T;V) =

small

for some

the same a r g u m e n t

(in § 7)

T > 2

as a b o v e

with

we note t h a t

lu 1 C2(Q0

g i v e s us in p l a c e

s K

I

)

pT of 2.16

that

(*)

(

[C~l 2 _<

2

o ilu- ~IT 2

L

;Qp,~ for any

1 < p < ~-1,

is s u f f i c i e n t l y FE(u(t)) if

~ FE(¢)

provided

2.12 w i t h

N e x t we w a n t about

Q

then 2.4 a l o n e in p l a c e

p,~

to s h o w that

-U(tl)ll

satisfies

)

ME¢ : 0 ,

l¢-ul

of

Q

2.16 gives

g r o w t h of

2=

is e n o u g h

u-¢

.

to give

I

C (Qp-1 ,0+1 )

and p r o v i d e d we k n o w a-priori that -~t - ~ e if m > 0 a n d fE(u(t)) ~ FE(¢)

the p o s s i b l e

Ilu(t 2)

(Qp - _ I ,p+l

-~p

small,

m < 0 , because

2.12,

¢

+ c ~e

-st + o e

inequalities

like

p,~ us some r e a s o n a b l e Specifically

I1[ t2

u ( t ) d t l [ 2-< [ [ t2 Jtl L Jt 1

information

we n o t i c e

that

IIG
r

t

(t 2 - t I)

I~l 2

JQtl,t 2 for any II

0

< tI < t2 ,

II L2(E;~)

lu-¢l

cI(Q

where,

unless otherwise

)

h e r e and s u b s e q u e n t l y , specified.

is s u f f i c i e n t l y

small

Hence

(depending

if

II p > 0

means

II

and if

on K,E,F),

we d e d u c e

p,p+2

from 2.16 that

flu(t2) - u(tl) II 2

for any

t 2 > t I z p+I

-~ c ( t 2 - t I)

Hence,

C llu-~ ]I2 [ + e -ep ]) L 2 (Qp, p+2 )

integrating

with

respect

to

tI

over

247

(p+1,p+2) =

and

( u ( t 2 ) - ~)

notinq

that

- u ( t l ) - ~)

,

we we

can write

u ( t 2)

2 _<

ct

I llu-¢ll

for

t > 2.

In p a r t i c u l a r

IIL2(Qa,b)

II

2.18

,

we

for

< n

}u-¢IC1(Qp,p+2)

However

since

(Qp,p+2) R a 2 , writing

( < cr < ] flu- ¢II ( p , p + 2 )

,

2 < K

lul

+e

(a,b)

have

l[u- ~II (p,p+R)

provided

any

-~p

2

k all

=

deduce

Ilu(p+t)-¢ll

for

- u ( t I)

where

we

have

-£p] + e

c = c(K,E,F),

by

j ,

n = q(K,E,F)

interpolation

C

lu-¢l

2.19

~ ~ + e<~)~

-n

C1(Qa,b )

for

any

0 < a < b-1

,

hence

2.18

actually

Ilu-¢II (a,b)

gives -~p

Ilu-¢ll(p,p+R)

2.20

flu-eli(p,p+2)

whenever Next

we

introduce

_< cR(llu-¢ll(p,p+2

< nl the

'

~I = Tll (K,E,F)

v :

at

(w,t)

in

> 0

e

)

.

function

2.21

(with value

) +

V

(u-C,e -ct)

x IR

c ]RN+I)

.

~ cT

IIv II (p,p+2)

co

and we

note

that

all

T a 2

implies

2.22

Ilvll

whenever

tl tI 2 L

II V 11(p,p+2) ~+I

(Qa,b; ]R

) )

(p,p+T)

-< n 1

(where

now

II [I (a,b)

for

means

2.20

> 0.

248

We shall another 2.12,

return

important

to this

in

consequence

§ 4,

but f i r s t we n e e d to r e c o r d

of the m o n o t o n i c i t y

2.12' we c l a i m that for any

formulae:

for

m > 0 :

Fz(u(t))...-> FZ(¢)_. - c e

2.23'

for

m < 0 :

Fz(u(t))_

that

supzl~(t) I

In fact we k n o w that t h e r e

is

2.24

Izl +

Hence any

it f o l l o w s t

from assumption

§ 3.

GROWTH ESTIMATES

2.23'

Lw = W - m W + L w

We continue

(as in 2.14

where

to a s s u m e

assume

condition

I 2.17)

in

q

for

lql S 61 -

for 2.12,

as in

§ 5 of P a r t

I 5.4; we h a v e section

condition

I, w i t h I 5.2 and

in m i n d the c a s e

and in I 4.9).

of the f o r m

'2 h ~ W 2loc

(for the moment) small

is c o n v e x

the s e l f - a d j o i n t n e s s

+ a 2 - D h + a3.h

(Qa,b)

s U P Q a , b laj}

is

(and the c o n s e q u e n c e

as c l a i m e d .

of the p r e v i o u s

V (V = , ~ t ) ,

3.2

Here

F(~,z,p,q)

L h = a1.D2h

D =

I 2.14

is a n y o p e r a t o r

we look at e q u a t i o n s

3.1

small.

such that

respectively,

the s t r o n g e l l i p t i c i t y L = ~¢

-et

F ( ~ , u , V u , ~ ) - 6.F.u (~,u,Vu0~) ~ F ( ~ , u , q u , 0 ) supzl6(t) j ~ 61 , and for such t then have

~

m ~ 0.

-ct

that

2.12'

Here

> 0

Ipl ~ K

such that 2.23,

_< EZ(¢)__ + c e

is s u f f i c i e n t l v

61 = 61(K,F)

from

t > 0

2.23

provided

Viz.

n L2(Qa,b )

~ 6 ,

an a r b i t r a r y

in the r e s u l t s b e l o w ) .

and

j = 1,2,3

parameter.

(We are q o i n g to

249 The key In this

result

lemma

of this

~,a

0 < a < min{IRe

are

section

arbitrary

y] : Be X

is g i v e n

in the

parameters

~ O,

with

following ~ > 0

T7

and

for

±/m-/4+7.

7 =-m/2

lemma.

some

j ~1},

3 where

11,12,...

also

use

in this

3.3.

the e i g e n v a l u e s

the n o t a t i o n

I. = 3

of

L

as

(T + (j-I)R,7 + jR)

in

§ 5 of Part

and

3

There

i f 3.1,

that

I.

R 0 = R0(~,a,L)

are

3.2

hold with

(a,b)

~ 2 =

and

L

(ZxIj)

60 = 60(~,a,L,R)

(Y,T+3R)

We

[lhlli. = IIhll 2

lemma.

LE~94A

such

are

for

> 0

R ~ R0

some

and

~ 6 0 , then:

aR (i)

0 ~ IIhlll2 _> e

(ii)

0 ~ Ilhlli2 -> e

aR IIhIlll ~

]lhlIl3 > e

IlhlIi3 ~

llhIIi1 > e

aR

(iii)

if neither

(then of c o u r s e

hold

tlh(t2) In case then

at

Proof:

one

first

= 0 in 3.2. there

tions

o f the

conclusions

neither

hypothesis

are of

Indeed

j ~ I

that (i),

we

can

that

the

suDpose

find

SUPo 0

there

after

normalizing

{n

If

(ii)

(ii)

n •a

must

at

fails

least for

R a 2

of

h. 3

e ~R lIhj ili1 } ~ ]1 hjlli2

so that

IIhjlIi2 = 1

and

( 12. L~

=

0

case w h e n such

Then

3.1

one

then

hold.

lemma.

h.]

(ii)

tl,t2,t of

to the

and

in the

solution

so t h a t

¥

t c C2(Z;~)

(i),

exist

(i)

(i),

can hold),

tt

can be r e d u c e d

W l2,2 oc but

(i),

ll ~lth(t)

as c l a i m e d

a

lajl < I/j ,3R (ii), (iii) fails.

in

solutions

lemma

60

llhIII2

in the i m p l i c a t i o n s

tl£(t)

conclusions

max {e-aR flhjl ti 3 , and hence,

,

no n o n - z e r o

the

show

ilh(tl)ll

is no c o r r e s p o n d i n g

integer such

aR

-<el+n)

there

least

We

ll

Iih1112

on

of the

that

for each Q0,3R implica-

then we h a v e

# 0 , using

the W 2'2

250

estimates

of

solutions 2,2 WIo c and

I 5.6,

h

of

for

strongly

that of

this

Similarly

if

(respectively)

(ii) fail

and

2,2 Wloc(Q0,3R)

a

(weakly

in

Q0,3R ) ,

and

In p a r t i c u l a r

or

(iii)

for

h

fail

with

(i)

for

any

fails

for

h, we deduce 3 n' < q in p l a c e

n. we

have

llhlli2 ~ 0

. Now

sentation

for

h

have

for

any

that

IIh (t)II

2

spect

t

for

in

solution

J~ll

]

to

of

I 5.8,

(a.cos

on

reduced

solutions

as

real

only to

actually

[

=

suitable

pending

of

the

Lh = 0

Uj, I9~ =

on

means

~

(a 2 + b 2 ) e m ~ R

11 ul 2

where

c - I R - 2 a -< b-< c R 2 a The

right

where

c. 3 P-I < -~ ' required

side

depends

<

3

Pl > ~

"

of

'

+

(a _ b . t ) 2 e m t

[

]

,

b2e 3

27-ZR J

=

,

c = c(L)

R

Using

implications:

can

{ j c 13

3.2

and

repreI 5.8 w e

]

0 < t < 3R

de-

this

not

on

+ R a2e 2 Y j Z

~ i3

3

2 2Y3(Z-I)R b.e 3 13

+ [

: ± y

be w r i t t e n

but

+

3

13

here on

notation

J~12 ~2 I )

explicit

Q0,3R

]

13

1,2,3,

the

in

a, b . Evidently if R >-R 0 (R 0 a 2 3 3 j c 1 I) we t h e n h a v e b y i n t e g r a t i n g with

~

+

i =

have

6 = 0

[ (£-I)R,ZR]

i

for

when

in the

~ , t - b •sin Z. t ) 2 e m t

constants

over

we

hence,

] + ( -~" e Y] + [ Ia . e 3 -b~ J~13 k 3 3

the

case

Lh = 0

and

]

IIh,12i

the

hj,

with h = l i m h. 1,2 2 3 WIo c a n d in L on

in

(iii)

Thus

for

a subsequence

e ~ R !lhl~1 } < llhlIi2 = I.

~ , R.

(ii),

have

Lh = 0

m a x { e - O R llhll , 13 h

we

> 0}

in the Z '

it is q u i t e

,

form

P~J < P~+Ij

and where

a ~ b

2 2pjIR c. e

j=_~ ~

straightforward

3

j, P0 = 0 , to c h e c k

re-

251 Specifically

if

llhll

~ e

2eR

2

I]hlIi2

,

then

(in the n o t a t i o n

above)

3

(1)

c j=_~

and if

2

4pj e

R

-> eR

{lhil~2 _> e2~RI{hllll 2

(2)

~ c2 e j=-~ J

Then writing

c

-4

e

-2~R

[ j:_~

3

4pjR

= c e ] 3

c2

e

6p .R 3

3

, then

cR-4

k

20 R 3

e

+2eR

~ 2 2pjR ~ e. e j=_~ ]

and adding

0o

(i),

(2) we see

co

N2

[ c a cR j=_~ 3

-4

[ c o s h (2 (pj-~) R) ~ 2 j=_~ ] co

cR

Thus

for

R > R0(~,L)

fact t h a t

-4

e

2~R

~2 ~ c , j=_~ 3

this g i v e s

= min{e,pl-e}(<0 )

Z c7 ~ 0 , thus c o n t r a d i c t i n g 3

IIhll ~ 0. 12

By s i m i l a r

r e a s o n i n g we e s t a b l i s h

see that if b o t h c o n c l u s i o n s

(3)

Z c2 e 3

4p R ]

in

(i), -4

>- cR

(ii) and at the same t i m e we (ii) fail then

-2~R

e

2

~ c

6p,R

e

3

3

and 2 (4)

E

c.

4pjR e

-4 ~

cR

-2~R e

2 E

3 Adding

(3),

(4) and a g a i n w r i t i n g

c.

2pjR e

] c

: e2pjRc 3

,

we get

]

~2 a cR-4 e - 2 e R ~ c. ~2 cosh( 2pj R) ] 3 and hence

(5)

the

this time

for s u f f i c i e n t l y ~2 -4 ZR c 0 -> cR e

large

R

we d e d u c e

~2 (IPj I- ~ ) R [ c. e j~0 3

252

with

a = min In v i e w

> 0

{p1-a, Ip_ I I-~} of the

definition

and a f u n c t i o n

I2 :

•

(~+R-~,T+2R+~)

of the

~ = ~(w)

with

-sR

(iii)

follows.

this

last

vial

solutions Next

following

3.4

=

(T,T+qR)

for

R z R0(£,L)

of the

impossible

lemma when

also

there

follows

because

are no n o n - t r i -

L} : 0. repeated

growth

some

> 0 ,

application

picture

Theorem)

for

Jl

integers

part

is e v i d e n t l y

(Growth

60 = 6 0 ( L , R , a , Q ) are

remaining

that by

general

THEOREM

(a,b)

of

2 IIh!Ii2

II 2 -< e

The

inequality

notice

is

F

Ilh-~

and

C. it f o l l o w s that there 3 L~ : 0 such that, with

for s o l u t i o n s

Suppose

integer

q

0

h

> 0

~ 3 ,

R 0 = R0(L,~,q)

' J2 w i t h

of L e m m a

> 0

as

in

Lemma

such

the

3.2:

3.2 h o l d

6 ~ 60

where

I ~ ii ~ J2 ~ a

in 3.1,

3.1,

and

as

3.3 we get

,

with

R z R0 ,

3.3.

Then

there

that

-~R (i)

(ii)

11hl[

Ij+ 1

< e

][hll

I ~ j ~ JI-I

1.3 '

IIh(t2) II s (1+q) I[h(tl) If,

II~(t) I[ s qllh(t) II

Itl-t21 s m

(iii)

Ilhll z e ~ R IlhllI. " IJ +I 3

Furthermore

3.5

Of c o u r s e

(i) is v a c u o u s ,

J2 = q" (2) has

if

L@

tl,t2,t

~

( T + J I R , T + j 2 R)

J2 + I s j s a-1

=

0

has

no

non-zero

< c2(z;g).

solutions

Remarks: (I)

case

Jl = J2

,

,

q-1,

period,

allow

case

speaking,

of e x p o n e n t i a l

followed

the p o s s i b i l i t i e s

Jl = J2

in w h i c h

Roughly

a period

we

by a p e r i o d

in w h i c h (iii)

(ii)

Jl = 1 , in w h i c h

is v a c u o u s ,

or

is v a c u o u s .

the a b o v e decay,

case

that

theorem

followed

of e x p o n e n t i a l

says

by an growth.

that

the

"almost

solution

stationary"

h

253

§ 4.

PROOF OF M A I N T H E O R E M

Throughout ding only on With

L

Lw =

w N+I { C 2 (~;~)

c( ~ I)

, K

V= = {V }

an o p e r a t o r

4.1

the proof

K,E,F

on

Z

(as usual) , let

C2(Qa,b;V')

( ~ ( w ' ) , g w N+I)

,

w =

is u s e d

to denote

a constant

depen-

as in 2.2.

, with

,

V'= = {Vw ×19 }~eZ A = Laplacian

on

and define Q , by

w' = (w 1 , . . . , w N) e C2(Qa,b;V)

(w' ,wN+1 )

(~¢

,

as in I 4.9).

L

is then an o p e r a t o r of the same type as c o n s i d e r e d in the p r e v i o u s N+I N in place of V= . section, w i t h in place of and { V x ~ }~cZ Now 2.14

let

v =

(u-%,e -st/2)

and note

that the e q u a t i o n

implies

4.2

Lv = a 1 " D 2 v + a 2 " D v + a3"v

for s u i t a b l e

a. ]

satisfying + e -st/2 )

4.3

laj(t) I s ci (lu-¢IC1

Notice E(u)

as in 2.20,

also that by d i f f e r e n t i a t i n g

= 0( <=~ N(~+(u-%))

= 0

with

the e q u a t i o n

respect

to

t , we get an e q u a t i o n

the form

L#~ = b I D 2 u

+ b2D~ + b 3 6 + R ,

with

Ibjl

In view of the d e f i n i t i o n

4.4

s clu-¢]C1

of

and

v , it follows

IRI

s c e -st

that

L~ = b I D 2 ~ + b 2 . D ~ + b 3 - ~

of

254

for some

b

satisfying

J -ct/2 4.5

Ibj I < c2(lu-*l

I + e

) ,

c 2 = e2(K,E,F)

C For l a t e r use we a l s o n o t e h e r e t h a t if pendent w =

solution

of

(u-~,e -st/2)

M9 = 0

with

(Cf. the a r g u m e n t

4.6

Lw = e 1 - D 2 w

~

is some o t h e r

I~I s K+I

, then,

defining

leading

to 2.14,

4.2,

t-indew=

4.3)

we h a v e

+ e2-Dw + e3°w

with -st/2) 4.7

lejl

<-e3(lu-~l

I + e

,

C where

c 3 = c3(K,E,F) N o w we

set up n o t a t i o n

, 8 e (0,~) R 0 = R0(~,~,L)

4.8

with

d e p e n d on

not

for use

are a r b i t r a r y

as in 3.3,

6 > 0

(ci,c2,c 3

does

as in 4.3,

4.5,

j' = j ' ( ~ , q , L , R , @ , B )

4.9

e

4.7),

in the r e m a i n d e r

parameters

R ~ R0,

is such that

+

of the proof:

to be c h o s e n

60 = 60(~,q,L,R)

max{cl,c2,c3}66

s 60

T = t., ] to e n s u r e

Ivl

s

later

as in 3.3

and we s e l e c t

large e n o u g h

-cT/2

~ .

~

.

as f o l l o w s

:

(as in 2.1)

C I ( Q T , ~ + R) We a l s o d e f i n e

4.10

1( )

if

IV~

<1

otherwise

q = q({,B,R)

I(Q

)

6B

< Z

then

q ~ 4

is a r b i t r a r y ,

T, ~

Notice

q

that by 4.8,

is the l a r g e s t i n t e g e r s u c h t h a t

4.10 we can a p p l y

IVlcI(QT, T+qR)

3.4 to the f u n c t i o n s

~66

v,v,w

255

of 4.2,

4.4,

4.11

4.6;

let

(jl,J2),

(kl,k2) , (Z1,Z2)

as in 3.4 with

denote

h = v,%,w

the integers

(Jl 'J2 )

respectively.

and f max{kl

I 4.11'

kI

! kI

if

Jl }

if

Jl'kl < k2

kI = k2

|

L k2 - I

if

F r o m now on w r i t e

Jl >- k2 > kl

III3

II

=

II IIL2 (ExI,)

Then by 2.22 we have

3

4.12

Ilvll

provided

6B ~ ~I

'

Now we select (c as in 4.12);

Ij+ 1

cRIIvll

~

~1 = 61(K'E'F) R > R0

then

4.13

large

,

enough

llvll

s

that we can then take

1 sj

to ensure

R = R(K,E,F,a,L,D)

,

that

cR < e

~R/2

and

eaR/2jlvlt

,

1 < j

s q

.

I. 3

(in v i e w of 4.13

4.14

4.15

I. 3

> 0 .

Ij+ 1

Notice

for 1.3 = (T+(j-I)R,~+jR).

and 3.4)

J2 = q-1

Remark:

then by 3.4,

Notice

that

if

4.13 and 4.14 we have

e -~R IIvll I

(*)

q+l (by 4.9).

L9 = 0

Thus

has no n o n - t r i v i a l

Jl = J2 = q-l,

_< e-aqRIivlI I1

(by the i n t e r p o l a t i o n

2.19), < -1 6S + c

I V I c I ( Q T + q R , ~ + ( q + I ) R)

-

2

solutions,

and

s 6e -aqR

for

8 < (n+1)

61-nB

< 68

-I

,

256 for

~

small

q ~ I.

Then

> 0

with

enough,

(again

thus

showing

(see 4.10)

by i n t e r p o l a t i o n )

that

it f o l l o w s

-< c e - ~ q

(*) h o l d s

from

for all

(*) t h a t

there

is

%' q -> I

I V l c I ( Q T + q R , T + ( q + I ) R)

Hence

(by the

estimates

a-priori

I 5.6)

we have

-u't

#' > 0

V t>-0

such

that

.

Vlc2(Qt,t+l ) ~ c e

Thus we

the p r o o f

is c o m p l e t e

did not n e e d

nential

decay

the

of

Of c o u r s e

analyticity

u

to

the p r o b l e m

(for e x a m p l e

non-trivial

solutions to c h e c k

We now vial

turn

solutions

3.4 a p p l i e d

to

4.17

for

With

t e

kl,

remarks

also

is that

functional

= 0 ; in any

In this

case

case,

case

I s Jl s q-1

Tj+ 1

-< (1+n)

(JIR, (q-1)R)

k2

4.18

-< e

of

in this we

as in 4.11,

-aR

when

such

IIv[I

with

expo-

in m a n y

cases

I 3.2)

there

ll

there

are

is o f t e n

m a y be n o n - t r i -

in m i n d

that

r. 3

Hv(t) [I < qllv(t)II

Itl-t21

_< R

.

we h a v e

< e - a R II~lli

11~111

II~(t)

case

get

the c o n d i t i o n

(keeping

llv(t I) II ,

j+1 4.19

and

that

, and

IIv(t2)II

t I, t2,

area

difficult

, we h a v e

llVll

I ~ j ~ JI-I

1.1,

also

or the other.

= 0

4.16

for

L%~

to the m o r e

v

hypothesis

for the

of

L%~

Notice

in the a b o v e

one w a y

of

case.

¢ .

of i n t e r e s t

difficult

in this

<- q I 1 @ ( t ) l l

,

1 _< j < k 1 - 1

3

,

II~(t2)[[

_< ( l + q ) I I @ ( t l ) l l

4.14),

by

257 for

t I , t 2, t ~ (T+kIR,T+k2R)

ligli I

4.20

~ e

, Itl-t21 s R , and ~R

iigli

j+1 Notice

that the first

4.21

liB(t)

Notice

inequality

I1 - < n ( l l 6 ( t ) l l

also that by the

of the s e c o n d

4.22

inequality

k2+1

13

tI < t2

+ e -Et/2 )

W 2'p

of 4.19

II,

and u s i n g

of I 5.6 we have (in view 6 for 6 small enough

and 4.4)

t ~ (T+klR+I,T+k2R-1)

III t2 v(t)at!I 2 -< [t2 Ilv(t) fiat I Jtl t Jt I )

inequality,

4.18,

t ~ (~+klR,~+k2R)

estimates

tt2 t t l % ( t ) II 2 d t Jt I

(t2-t 1 )

the t r i a n g l e

,

we have

IIv(t 2) - v ( t l ) II2 =

Using

< j s q-1

-

in 4.19 can be w r i t t e n

I I D $ ( t ) ll s c I l $ ( t ) ] 1 ,

Now for any

'

integrating

we then e a s i l y

with

see that

IIvll

4.16,

IIvll

Now let and note

that

<- c 6 ,

(k~

s c 6,

tI

over

hence,

by

as in 4.11')

-st/2 :flu(t) II se V t c (T+k R,s)], -£t/2 flu(t) II >e at t = T+kIR By 4.9

S : sup{s e [ T + k I R , T + k 2 R ] S = T+k~R I

if tS I

IT+k~R

IIvIldt

~

c

Also,

6

.

taking c q <E/4 (here and s u b s e q u e n t l y ) , -st/2 flu(t) I}-e is s t r i c t l y i n c r e a s i n g w h e n e v e r 4.24'

to

Ikl

4.23

4.24

respect

flu(t) II -> e -st/2

for all

by 4.21 we see that it vanishes,

t ( [S,T+k2R-I]

hence

258

Next

we

is s o m e w h a t fically

examine more

we n o t e

4.25

for

another

precise

way

than

of w r i t i n g

the

the e q u a t i o n

"linearized"

version

Nu = 0

of

which

2.14.

Speci-

that we m a y w r i t e

~(t)

t c [T,T+qR]

- mu(t)

,

+ Mz(u(t))

= a1.m6(t)

+a2"u(t)

+a 3

where

-st

!ajl < clu-¢Icl

,

j = 1,2,

la3!

s e

(Q<,T+qR) (Justification vative

parts

for

of

In v i e w

this

the

of 4 . 2 1 - 4 . 2 4

4.26

with

simply

operator

involves

"splitting

off"

the

time

deri-

M ).

we

-mG(t)

then

have

that

+ Mz(u(t))

liE(t) II s c(65+q) II6(t)]I

,

for

= E(t)

so that,

t c (S,T+k2R-I)

,

assuming

c(66+q)

~ Iml/3

, we

have

4.26'

liE(t) II < (Iml/3)

To c a p i t a l i z e u(t)

follows

on this

a "curve

(which,

a general

inequality: Viz. I 9 = 9(F,%) s (0,~) such t h a t

4.27

function

quality

depends

shall

not

We do note tion

of the

is a

with

a = u(F,%)

> 0

E ,

for

says

EZ)

that

, we n e e d

and

2 < ~ " The p r o o f of this C real-analitycity h y p o t h e s i s 1.1.

crucially

on the

give

the p r o o f

here,

here

that

is s i m p l y

result

term

or a s c e n t

IPE(¢+h) - Fz(¢) II-8

h ~ C2(~;~)

4.27

the e r r o r

descent"

there

IIMz (¢+h)II

for any

modulo

of s t e e p e s t

flu(t)II -

but

lhl

instead

refer

an i n f i n i t e

(due to Z o j a s i e w i c z

[Z])

the r e a d e r

dimensional

that

to

ineWe

[SLI,§2].

generaliza-

259 IDf(x) i ~ If(x) - f(0) 1I-8, in some n e i g h b o u r h o o d Assume

of

Ixl < ~ , n 0 c

for the m o m e n t

that

if

f

is a real a n a l y t i c

k2 > kI .

function

We are g o i n g to use 4.27

with

¢ + h = u(t) in o r d e r to k e e p c o n t r o l of fly(t) I! in the i n t e r v a l . (T+kIR,T+k2 R) We give the a r g u m e n t in the case m > 0 ; the case

m < 0 t

is a l m o s t

on

identical.

(S,T+k2R-1)

F i r s t note that

In fact,

Fz(u(t))

by d e f i n i t i o n

is d e c r e a s i n g

I 2.8 and 4.26,

4.26'

in

we

have

4.28

_ ddt FE(u(t))

Hence

taking

=

S < T I < S+2

(Mz(u(t)),6(t))

with

T I s T 2 s T 3 s T 4 = T+k2R-1 Fz(u(t)) on

f Fz(@)

(T3,T 4)

and

u(t)

with

Fz(u(t)) > FZ(@)

~ 0

on

(T2,T 3)

(T1,T 2)

- FZ(¢))@ = ( 9 ( E E ( u ( t ) ) 1

>- ~ e ( M z ( u ( t ) ) 1

~ 8 ll~(t)II

Thus by i n t e g r a t i n g

over

llMz(u(t))II lift(t)TI •

T I ( [T+rR+I,[+(r+I)R-I],

On the i n t e r v a l

d (Fz(u(t) - d--~

I

>-~

(T1,T 2)

,

and

we h a v e

F~.](¢)) 8-1 e-1

-Fz(¢))

on

r ( Z , (TI,T2),

Fz(u(t)) < FZ(@

(Cf. 4.28)

(M~.](u(t)) , L l ( t ) )

liMz(u(t))

I1

E1G(t) II

by 4.27

we have

T2 flu(t)II dt _ < c ( E z ( u ( T I ) ) - EZ(@)) 8

4.29

JT I By a s i m i l a r that

argument,

starting with

Fz(u(t)) ~ FZ(@) - c6

~t

(EE(@) - Ez(u(t)))

by 2.23 and 4.9, we d e d u c e

T

4.29'

4

flu(t)II dt < c

JT 3 Since d d-~ F ~ ( ¢ + s ( u ( T 1 ) - ¢ ) ) l s = O

=

(-MZ(¢)'u(TI)-¢)

= 0 ,

and n o t i n g

260

we h a v e

4.30

IFE(u(TI))-FE(~)

By the

W 2,p

estimates

I < c IIu(TI)- ¢II 2 1,2 W (Z)

I 5.6 for

v

(which can be a p p l i e d by 4.2,

4.3) we h a v e (s k2-I)

IIu(T1)-~III,2 ~ c IIvll , where r is the i n t e g e r W (~) Ir+1 such that T I ~ [~+rR+I,T+(r+I)R-I]. By 4.29, 4.29', 4.30 we

then have

T4 llv(t)Ildt

4.31

< c(68 + IIvllI _

JT I while

) , r+1

from 4.24 S

I JT+k~R

4.31'

llv(t) lldt

Now for any

T+k~R < t I < t 2 < T 4

4.32

llv(t 2) - v ( t I) II =

< c 6 .

we h a v e

II

It2

v(t)dtll

<

Jt I so that by i n t e g r a t i n g 4.23,

4.12,we have

with respect that

to

Ilvll Ik2-1

with

respect

f i r s t that

tI

over

< cR3@ 8

--

l}v(t) flat

Jt I to

IIvll Ir+ I

It+ I

It2

tI

over

s cR26

Ik~+1

.

and u s i n g

T h e n by i n t e g r a t i n g

in 4.32 we o b t a i n

from 4.31,

4.31'

and h e n c e by 4.12

t

IIvll

~ cR46 e

Ik 2 This was derived case o b v i o u s

subject

from 4.23 in case

Let us now a g r e e 6 = 8/(2n+2)

to the a s s u m p t i o n

(8

k 2 > k *I , b u t

k2 = kI

to f i n a l l y

select

as in 4.27 d e p e n d s

on

8 : we in fact c h o o s e F,¢)

lity says

4.33

II v[l

-< cR46 (2n+2) B Ik 2

it is in any

T h e n the a b o v e

inequa-

261 If

k 2 a q-4

we

then

have,

by

4.12

-< cR

II v III

8 2 (n+2) ~

q But

by the

interpolation

4.34

ivi

and h e n c e

by

cl(ExI

4.33

2.19 we

< e ~

q

have

-2n6

l]vli

)

Ivl

< 68

I

for

28

+ q

sufficiently

(depen-

small

C1(~XIq) d i n g on R,c),

thus

proving

(see 4.10,

4.31)

that

q

is a r b i t r a r y

and

~oo 4.35

i if+If < ~

Of c o u r s e

the

first

,

c

inequality

exists

relative

to the L2(Z)

gether

with

estimates

respect

the

to the

C2(Z;~)

finition

as

in case

k 2 ~ q-4

in 2.1,

Finally

Ivl

this

~ 6

I

(Q

)

(in v i e w

norm,

I 5.6,

and

of

4.32)

then

the

guarantee

norm.

On

limit

must

zero,

that

lim v(t) t+~ i n e q u a l i t y , to-

second

that: the

the o t h e r be

proves

hand, and

limit

is w i t h

since

~

the p r o o f

is by deis c o m p l e t e

.

we w a n t

for c o n t r a d i c t i o n ,

to s h o w

assume

that

automat~oa~Z9

k 2 ~ q-4

,

define

k 2 ~ q-4.

Therefore, I (k 2 + ~ ) R , and

T = • +

select 2 ~ C

, (Z;~)

such

that

M~

= 0

6B l~-¢l

2 ~ c

and

4.36

flu(T)-9

= inf {IIu(T)-
: I~-¢I

2 s ~8,

MZ~

= 0}

C We n o w n e e d which ticity

again

a second

makes

I 2.15.

inequality

for

use of the a n a l y t i c i t y

Specifically

we n o t e

that

FE

and

the o p e r a t o r

hypothesis for g i v e n

1.1

and

ME,

the e l l i p -

~ ~(0,I)

there

is

262

~1 = ~1 (F'$'~)

> 0

and

IlM]: (h) I[

4.37

L h { C2,~ (Z;~)

for any This

is an i n f i n i t e

that

IDf(x) I k

4.37 we

to

in v i e w

lu(T)-%l

2,U C c ~@ ~ ~I

for

linf {Ilh-%ll

that

: ]~-%1

2,
2 < oi , M ~ : 0} Z J

lh-@I

: 0}}) T

neighbourhood § 2]

of 4.36

< c 68

such

C

with

[SLI],

~ 2

(E)

(dist{x,{y:f(y)

in some

Then

2

dimensional

real-analytic refer

y = y(F,$,D)

of the w e l l - k n o w n

for

of

0

ixl < °I

in

n

.

'

fact

where

For

f

is

the p r o o f

of

.

and

by v i r t u e

4.37 of

(together

with

the e s t i m a t e s

the

I 5.6)

fact

that

we have

that

]IME(u(T) 11 > IIu(T)-~II Y By 4.25

and

the e s t i m a t e s

I 5.6 we then

have

IIu(T)-~IIY s cl]vIIIk +I 2 and

so

4.38

IIw(T) }[Y < cil+ll

with

w =

(u-~,e

Notice

-et/2

that

by

)

Ik2+1

(as in 4.6).

4.33

and

4.12 we h a v e

II~-61[ s cR 4 6 (2n+2)8

and

cR 5 6(2n+2) 8

llwll Ik2+1 Since in p l a c e of

4.39

and

v,

EE(~) of

~, h e n c e

so that

Ilwtl

= F~(~)

(for

(p,p+S)

in p a r t i c u l a r

6@

S

(by 4.27)

the a r g u m e n t s

in p a r t i c u l a r sufficiently

csllwll

(p,p+2)

we h a v e

of

§ 2

2.22 w i t h

apply w

in p l a c e

small)

,

s ~ 2 ,

with

7 < p S T+qR-2

263 4.39'

~ cRllwli

IIwll Ij+ I

,

I < j
I.]

Now we c l a i m I

4.40

IIw(T) li a ~ suPt~ I

IIw(t) II , k2+I

since

otherwise

we w o u l d

I

have

liw(t)

from

II

4.32

that

(using

f I II%(t) lIdt Jlk2

s

suPt~Ik2+1

C RIIv[[ I

] : cR[[Q]I I Ik 2 +1 ) ~

k2+1 and by 4.20,

I 5.6,

and 4.39'

v ~ Q)

this

_< c R e

-mR I IQTI Ik2+2

c R e

-mR

ilwll

(T+k2R, T+ (k2+3) R)

< c R 3 e-~RI[wll

and for

R

large e n o u g h

statement

that

[lw[I Ik2+1

Thus

4.40

is proved,

4.41

to ensure

c R

4

e

and h e n c e

4.39')

(since

implies

the false

Ik2+1

(q-k2-3)y

SO that

this

< cIT%ll

and 4.20 we see that

IlwllY Iq-2

I

<~

4.38 gives

k2+I 4.39

-mR

= 0

Ilwll~

Now by a p p l y i n g

Ik2+l

_< (cR)

~ : w) by I 5.6

4.41

gives

-m(q-k2-2)R e

llvll I q-1

(applied

to w) we deduce

(again u s i n g

264

-a(q-k2-2)R

(q-k2)Y

,y

Ilwll I

< (cR)

I1wll

e

(~+ (q-2) R-1 ,T+ (q-1) R+I )

q-2

(q-k2)Y (cR)

-e(q-k2-2)R e

R2 c

IIwll

, Im_ 2

so that -~' ( q - k 2 - 2 ) R IIwll

s c e

,

~' = a / ( 2 ( y - 1 ) )

Iq- 2 and h e n c e

(yet a g a i n by 4.39')

]lwll I

4.42

_<

c

e -~' ( q - k 2 - 2 ) R R 2

q Since

(as we a l r e a d y

remarked

I~-~]C1

above)

< R46 (2n+2)B

r

we d e d u c e

from this that

4.43

IIv I~

-< c e

-a' (q-k2-2) R 2 5 (2n+2) 6 m + c m 6

q

Let us s u p p o s e

for the m o m e n t

the l a t t e r a l t e r n a t i v e (with

q+l

in p l a c e

that

in 4.10 holds).

of

q )

o

is not a r b i t r a r y

T h e n by the i n t e r p o l a t i o n

we d e d u c e (2n+1)6

11v[I I

>

c

c

R

q+l and h e n c e

(by 4.12)

Ilvll I

4.44

_>

-1

(2n+1)

q On the o t h e r h a n d by 4.12 and 4.33 we h a v e

llv]l and h e n c e

I

~

q-k2+4 (cm)

6 "2n+2"~(~

q

f r o m 4.44 - (q-k2+5) (cR)

(so that

_< ~

4.34

265 Using

this

in 4.43 we

then

get

llvll I and by a g a i n

using

the

c R56 (2n+2)B q

interpolation

4.34

Ivl

c R 6 ~26

cl(3xI if

@

is s u f f i c i e n t l y

latter

alternative

Thus

we

small,

in 4.10

conclude

We

k 2 ~ q-4

easily

then

the g r o w t h

rule

from

theorem

the

thus

q

iv I

can n o w

q+l

out

cl(Qz, ~)

of

to

~

q

in p a r t i c u l a r

÷ ~ I

bound

the

and h e n c e

that

q ~ k2+4

Indeed

, and

from

we

4.45,

if

see that

~ j a k2

as

' j ÷ ~

, thus

contradicting

the C I

,

]

4.45. This

§ 5.

completes

the p r o o f

MODIFICATIONS

A careful

Remark

2.17)

with

parameters

lution of indeed we

material

will

show

lul

2 < ~ C ~,6,£ c

ME(~) shall

of T h e o r e m

1.2.

AND APPLICATIONS

examination

the p r e l i m i n a r y

Q+

in 4.10,

1.3

llvll

that

, gives

edRII~ll

so that

fact

~ ~B

the p o s s i b i l i t y

IlvllIj+ I

the

to hold.

is a r b i t r a r y

applied

< ~B

contradicting

arbitrariness

3.4,

4.12 we h a v e

)

is s u p p o s e d

that

4.45

and

= 0

of the p r o o f

on m o n o t o n i c i t y

that

(~

be w o r k i n g

,

in

the h y p o t h e s i s

can be c o n s i d e r a b l y (0,1)

in the p r e v i o u s

TO ~ 1

is no

and

longer

in a s e t t i n g

§ 2

C

Specifically,

any given

an a s y m p t o t i c where

2

be a-priori

u

relaxed. ~

and of

(see in p a r t i c u l a r

that

for

section,

limit

for

so-

C2(E;~) of

we do not k n o w

on

u

;

a-priori

266

that

u

is e v e n

defined

over

P : P(6,8,s,m,T0,%,E,F) Nu : 0

on

5.1

b

Q0,b

is

'

to b e

TO ~b s ~

C2

a

of

the

O+ ) , we

class

of

(b = b(u)

for

maximaZ

to be

all

u

solution

;

of

C

define 2

is,

u = 0

on

class

solutions

such

that

tha

u

of

that:

u

cannot

Q0,b'

be

with

extended b' > b

;

5.2 lu-~IC2

-< ~ ; (00 T

)

' 0

l I J ] 1 <

5.3

if

m > 0

,

7z(u(t))

~ E~(~) - ~e

if

m < 0

,

F~(u(t))

~ F~(~)+

5.4

] u _ % I C 2 ( Q 0 , 7 ) £ 6B

and

and

lu-~[

6e

-st -st

• a TO ~

,

0 < t
,

0 < t
;

b _R T + T 0

~ 1 C2(9

) T, T +T 0

By will

carefully

see

ly y i e l d assume

5.5

that, the

checking

with

only

following

I 2.5,

2.14,

THEOREM

6 , 8 , 8 { (0,1)

exist

l i m u(t) : ~ , t+~ in the C2(Z;V) turns

It

niently

We manifold smooth

points first N

TO ~ I such

also

and that

§ 2 and

~ ~ C

2

,

1.2;

analyticity

(Z;~)

u { P

~ ~ C2(E;V)_

same

of Theorem

the

if

the

with

: 0

the

,

reader

arguments we

actual-

continue

to

assumption

1.1.

M~

there

= 0 ,

u { C 2 (Q+;~)

then

MET

~ 4,

and

where

,

and

the

limit

is

norm.

that

this

version

information

about

of minimal

discuss of

and

of

modifications,

generalization

where

out

to g i v e

singular

arguments

minor

2.15,

Given

the

the

area

minimal

P

and

case. and

the

theorem

asymptotic

surfaces

dimension

n-dimensional

of

an open

submanifolds

behaviour

harmonic

Given

applies

M

of

near

conve-

isolated

maps.

a smooth

subset

very

U

complete of

dimension

N

Riemannian

, we n

consider

such

that

267

(M-M) nU = {q} , q

a point By

of

U

virtue

.

of

(Thus the

is

always

true

that

to

assume

that

C

cing

normal

we

assune

theory M

has

T N G

that

a smooth there

an

of

varifolds

is

some

for

(see

point

e.g.

C

at

[AWl] q

.

regular.

and

That

origin

corresponding

have

(as

3.2

minimal

sequence

of

Part

submanifold

I. + 0 ]

such

or

We

with

in

of

[SL2])

here

it

have

is,

introdu-

to

q

I)

of

M).

and

that P-I S

, and

that

-I l. M ÷ C ]

the

measure

each We

for

M

continuous

Suppose

at

q ;

is a

ve a n d

sense

that

I

f dHn

j {l-ly:y~M}_ ] f with compact

function

then

p > 0

using

such

C

is a m u l t i p l i c i t y

C

is

the

that

the n o t a t i o n

M n Bp(a)

unique

(using

h c C2(C

= GO

5.5

n Bp;~)

the

above.

w = x/Ixl priate

p support

in

]R

(~

form

as

above

To in

(as

regular

normal

cone

tangent

for

coordinates

M

cone

at

q

and

as d e s c r i b e d

abo-

(h)

,

vP

the n o r m a l

Ir-lh(r~) I +

prove

I

tangent

r / x+h(___x) [-- ~ I + lh(x)/ixl

To

f dHn C

o f I 3.2)

-

where

÷

claim:

THEOREM

there

theoretic

fixed

5.7

rem

singular

cone

one

compact

5.6

in

isolated

tangent

a

for N P ]R , we

with

, ~

is

multiplicity

is

coordinates

identifying P-I C n S : ~

q

see

this

we

I 3.2;

then

described

in

we

0

have

introduce the

area

I 3.2,

: x

bundle

IDh(r~) I +

theorem,

12

as

to

~ C

of

I 3.2)

r + 0

reduce

variables functional

3.3).

Newt

] ] n B p f )1

with

.

to

the

t = - l o g [xl does

we

general

note

take that

the

theo, appro-

(since

C

268

is m u l t i p l i c i t y theorem

([AWl]

in the

C2

large

I and or

sense

j , using

smooth

away

from

0 ) by the A l l a r d

[SL2,

Chapter

5])

on

;

for each

C

thus

notation

the c o n v e r g e n c e

as in I 3.2,

in 5.6

0 < Pl < P2

we

can

find

regularity is l o c a l l y

and s u f f i c i e n t l y h. E C 2 ( C ;V) ] Pl,P2 =

with

n (B

(I-IM) 3

5.8

,

and w i t h

lhj I 2 ÷ 0 C R e c a l l i n g I 3.2,

as

) = G

~B

Pl

P2

3.3,

as in 6 > 0

(~,t)

: r

-I

,

u. is an e x t r e m a l of a f u n c t i o n a l ] § 2, s a t i s f y i n g I 2.5, 2.14, 2.15. Furthermore

Q0,T

5.3 a b o v e

minimal

= ~ x (0,T)

(for the

one

submanifolds.

m > 0)

Ixl

I

JMnB

(q)

for s u f f i c i e n t l y

claim

holds

that

for

the

u

f o r m of

we have,

Specifically of

M

large

G

for any

see

formula

satisfies

j ,

"energy" To

3 of the m o n o t o n i c i t y

the d i s c u s s i o n

r

5.9

< 6

We a l s o

case

of the v e r s i o n s

for e x a m p l e

of the

T > 0

IUjlc2(Q0,T)

recall

h. (re) 3

then

a n d any

where

g

and d e f i n i n g

3 that

,

j +

u

we note

(h) 3

Pl,P2

the

inequality

this we

of

just

parametr¢o

for

inequality

(see

[SL1; § 73)

±2 -< (pl-nHn-1 (M n ~B

+ cp

(q)) - H n-1 (E))

P

Ixl n+2

P ±

for all

p

such

that

B

(q)

c U

where

x

is the o r t h o g o n a l

projec-

P tion of

x

putations

of

I 3.2 [Tj

c

JO

where

FE

(TxM)±

onto

is the

we n o t e ( I 16jlmd~

3E

area

Scaling that

(x ~

this

t-lx), and recalling 3 gives for any n > 0

the

com-

-t dt

-< (Ez(U.(t)) 3

functional

of

E

-EE(0))

+ cl.e 3

(see the A p p e n d i x

to P a r t

I)

269 and where

T. = l o g I/oj , o ~ (0,1) 3 j r e p r e s e n t a t i o n 5.8 h o l d s w i t h Pl : oj (Notice of course

that we can

find

h. as in 5.8 w i t h a n y f i x e d 3 with u = u. , 6 = cl. , s = I. ] 3

a rather formula

only

straightforward 5.9 a n d

3] so w e

[SLI, § 7] is g i v e n

the r e a d e r w i l l We have theorem

thus

and

repeat

it r e a d i l y

implies

that Theorem

5.7

lhjIc2 ÷ 0

Thus we have

property

5.4.

than

5.4 w i t h

follows

5.3

This

is

is g i v e n

(Actually

form

for

monotonicity

The proof

it here.

different

lhjlc2 + 0

because

theorem.

the

the

in lemma

5.4 a b o v e ,

a suitable

but

T O ).

from the general

5.5.

It is p e r h a p s give

not

such that

of the p a r a m e t r i c

regularity

in a s l i g h t l y

shown

~'+ 3 0

the p r o l o n g a t i o n

shall

see t h a t

P2 = I ,

0 < PI < P2 < ~ )"

consequence

the A l l a r d

[SLI; § 7, L e m m a of

to c h e c k

,

such

any

It r e m a i n s

any real number

information

worth

about

mentioning

asymptotic

that

5.5 c a n a l s o b e a p p l i e d

behaviour

of minimal

to

submanifolds

of

P

on approach submanifold growth

of

to i n f i n i t y . ~P

bounded

with

Specifically

(M~M)

according

~ BI(0)

suppose

= ~

, and

M

is a m i n i m a l

suppose

M

has volume

to

En(M n B

(0))

< cp n

P where gent

c

is a f i x e d c o n s t a n t .

cone

C

o n e of t h e s e

at

~

cones

(prior to 5.7).

;

we are

It is t h e n interested

is m u l t i p l i c i t y

Thus

I 3.2 a n d a s e q u e n c e

we assume I. + 0 ]

one

that

standard

here

regular

there

that

M

has

in t h e c a s e w h e n as d e s c r i b e d

is a m i n i m a l

cone

a tan-

at l e a s t

above C

as in

such that

I M ÷C ] locally same tly

in

P ]R

in t h e m e a s u r e

reasoning large

j

as b e f o r e , we

find

given

M)

n

(B

Pl

any

h. ~ C 2 ( C

] (t3

theoretic

~B

sense

(as for

0 < Pl < P2 < ~ ;V)

' for all

so t h a t

Pl,P2 = P2

) =

G

Pl 'P2

(h)

J

, lhjl

C

2 ÷

5.6).

0

.

By the sufficien-

270

We

can

goes

very

the

change

now

similarly of

functional leave

again

with

details

Finally

we

harmonic

analytic

target

5.5,

1.4.

except

"tangent map to

of

S

[SLI,

§ 6.

the

manifold.

now

at

the

into

AN

ALTERNATIVE

we

(see

4.15)

when

any

use

of

case gets

The

the

theorem

basic

to k n o c k

of

idea

5.5;

a

good

as

make

obtain

a

before).

information manifold

was

already

similar

We

at

by

mi-

a realin

applying

least

represented

detailed

about

into

discussed

result

that

be

more

out

EAA];

of

of

one

of

the

a smooth

as

discussion,

coefficients

the

of

the

actual

submanifolds here

is

0

goes

-1

is 0

we

refer

"easy

case"

actually the

to

De

exponential a trick

is

a generalization

proved

by

Allard key

and

of

in

here

the

sense

formulae

result

but

works

generated

Giorgi,

I 5.8.

simpler,

which

by

a

that

(~s - })

back

Such

the

a method

in

monotonicity

positive

for

form

first

somewhat

of

:

s

up")

present

MZ% s =

lim s+0

up

the

we

L%~

of

blowing way

("blowing

indeed

solutions

special

expansions

proof

3

can

method

~ c C2(E;y)

family

a rather

minimal

we

we

METHOD

} =

and

some

n=

For

(,)

tion

reasoning

time

and

m = +n

Riemannian

point

target.

i ,

with

a-priori a s s u m e

a different

solution

l-parameter

in

~:x/Ix

this

the

§ 8].

Here

use

,

gives

The

must

5.5;

check.

smooth

one

theorem although

than

to

5.5

singular

the

Ixi

a

n > 3 we

case,

log

reader

from

case

general

(rather

that

maps

that

maps" n-1

the

the

previous

m=-n

for

In

to

t = +

mention

nimizing

Theorem

to

variable

F

the

reduce

here

§ 2

in o r d e r

certain

was

also of

Almgren

ideas

but

are

we

eigenfunc-

used

in

[HS]

a result [AA].

for

The

essentially

the

same. The nential

advantage decay

of

of the

the

method

solution

here u

to

is its

that

it a c t u a l l y

asymptotic

limit.

yields The

expodis-

271

advantage

is that

it is g e n e r a l l y

(*) ; the h y p o t h e s i s of the a r e a The the

and e n e r g y

notation

functionals

continue

and

Also tional

6.1

F ,O

that

such

will are

that

assumption

the

be as in

such

that

~

We a s s u m e

that

is a l - p a r a m e t e r

satisfied

§ 2 of P a r t I 2.5, m

2.14,

the

special

above)

{~s}0<s<1

2.15 hold.

we

1.1

Also of

E

is

holds. the

C2(E;~)

following

solution

solution of

section. that

we n e e d

for e v e r y

cases

I, and we a s s u m e

in the d e f i n i t i o n

is a f i x e d

family

for

the h y p o t h e s i s

in the p r e v i o u s

hypothesis

mentioned

in w h i c h

to c h e c k

considered

analyticity

~

of

solutions

addi-

of ME% = 0 :

L~}

of

= 0

ME# s = 0

that

lim s+0

6.2

Remark:

suitable lhi

always

the c o n s t a n t

(as we a l r e a d y

Assumption:

there

is no~

functionals

here

to a s s u m e

non-zero,

(*)

not p o s s i b l e

there

into

man~fold

M

2 C

It is s t a n d a r d

6 > 0

2 < 6}

Is-1 (~s - ~ ) - 41

(see e.g.

: 0

[AA]

is a r e a l - a n a l y t i c

L2(E)

with

containing

all

image

of

.

(Z;~)

or

[SLI, § 23)

embedding T

that

~ : ker

for

L% n {h

an N - d i m e n s i o n a l

:

real-analytic

solutions

s 6 = {~:Mz~

= 0 ,

I+-~I

2 <~12} C

as an a n a l y t i c tion {n

6.1 M

for there

.

we

subvariety

see

that

the

In p a r t i c u l a r

~ c S6 is

in p l a c e 60

0

such

if

~

of

M

n <W: I~-TI 2 < ~/2}

solutions

it f o l l o w s

of

% ,

S6 that

C contain

a

Then

~e~g;~bourhood

the c o n d i t i o n

6 sufficiently

under

small.

6.1

holds

More

assumpof

uniformly

precisely,

that

S60

and

if

L~

: 0 ,

i~IC 2 < 60

,

then

(*) there

is

$ ~ $26

with

$-~ : q~+6

i%I '

0

where

c

is a c o n s t a n t

depending

C

only

on

6

and

F

.

2 <- cl~i 2 2 C

Notice

that

'

272

then we a l s o

of c o u r s e

have,

(**)

for

FE(~)

60

sufficiently

- FZ(~)

small,

that

~ c S6

for all

0

(because smooth

M ~ ~ 0

paths

in

S60

in

guarantees

that

FZ(~)

is c o n s t a n t

along

of MZ~ = 0

and

).

$6 o From s > 0

n o w on

is also

a class

Q

~

is a f i x e d

fixed.

o,6,b

For

C2(Z;~)

parameters

of

solutions

u

C2

at least

on

solution

~ c (0,1)

of

,

Nu = 0

6 ~ 0 ,

such

b ~ I

we d e f i n e

that _<

(i) (ii)

u

is

for as l o n g

as

of

we h a v e

Nu = 0 l

(*)

(iii)

u

Q0,3b

and

lu-}IC2(Q0,3b )

can be e x t e n d e d

to give

a

C2

solution

-st

for

m > 0 : FE(u(t))

k FE(%) - 6e

for

m < 0 : FE(u(t))

< FZ(~) + 6e -st

]

if

T ~ 3b

and

lu-~]

~ o

,

then

u

extends

to be a

C2(Q0, T) C2

solution

lu-~(

as

have

C(6,~)

6.3

÷

clear

that

and m i n i m a l

as

C(6,a) 6 + ~

t = -log

for any

cases

(i),

given

(subject

(6+0) -I/2

in the p r e v i o u s (iii)

are

above

satisfied

which

have

cones

respectively

Thus

to A s s u m p t i o n

>- I

of

in

and

(iii),

regular

the m a i n

section

(after

so long

tangent at

maps

0 ; this

o > 0

theorem

it s h o u l d

a suitable

by m i n i m i z i n g

6 > 0 , for any g i v e n

b z I.

It is c o n v e n i e n t b 0 = b0(¢,F,E)

(ii),

Ixl/p)

tangent

s = I , for any g i v e n and

Q0,T+(6+o)-I/2

+0).

submanifolds one

in p l a c e

the d i s c u s s i o n

properties

multiplicity

on

~1

From

of v a r i a b l e

at l e a s t

(Q0,T+(6+~)-1/2)

any

~

Remark:

Nu = 0

2 C

(We c o u l d

of

applies

6.1).

to f i r s t

prove

a lemma;

in this

lemma

maps

regular

is true

sufficiently below

change

harmonic and

be

with small,

to t h e s e

273

6.4

LEMMA

b a b0

Under

the a s s u m p t i o n s

there is a c o n s t a n t

of ~ )

such

u E Qo,6,b '

that if then

above

~ > 0

(depending

MZ~ 1 = 0 ,

there

I~I-~I

exists

(including

~2

on

~,e,b

MZ~ 2 = 0

I

, but i n d e p e n d e n t

6 e [0,1)

2 < q ' if

C with

6.1), for g i v e n

,

and if

and

, c~ I / 2 }

,

lu-~21C2(Qb,3b ) -< ~ max{ lu-<~1 IC2(Qo,2b) c = C(~,F)

6.5

Remarks: (I)

c1

Automatically

I%1-~21c2 s Cll (%1-u) - (u-%2) IC2(Qb,2b )

max{lu-~11C2(Q0,2b ) (2)

, c61/2}

By induction on

,

c 1 = el(~)

k , provided

~1/2 + lu_%iC 2

< qo

for

small enough

(Q0,2b) q , repeated application of the lemma (making a change

of variables

t ÷ t-b ,

{~k } c C2(Z;~)

with

6 ÷ e-Eb6

~0 = ~ '

max {c6 I/2 -c(k+1)b e , ~ max

at each stage) gives a sequence

MZ~ k = 0 ,

lu-~k+11 C2(Q(k+1)b, (k+3)b)/

c61/2e-Ckb

t

lU-~kl 2

f

C (Qkb,(k+2)b) and -k l~k-~k+1 I 2 C provided we take C2(~;~)

b z 2/~ .

and (ii)

THEOREM

are c o n s t a n t s that if

b ~b 0

Hence we deduce that (i)

lu-~I2

~ cqq2 -k

~k ÷ ~

in

Thus the lemma actual-

C (Qkb, (k+2)b)

ly implies

6.6

<- cqq2

If the h y p o t h e s e s

b0 z 1 , a n d if

q,a

0

are as above (depending

u c Qq,~,b

with

on

(including

~,e,m,F,E

6.1), then there only)

~I/2 + lu_~iC 2

< (Q0,2b)

such qq

,

274

then

u

extends

to be

a

C

2

solution

2 < cqoe -~t

1~l~u,t,-~ r ~

Nu= 0

of

,

on

all

of

Q+

and

p : ~(b) > 0 ,

C for

t ~ C2(Z;~)

some

Proof

of Lemma

find sequences

(I)

inf

where Q

Rk ÷ ~

as

Mz~ = o

If the lemma

{~k } ~ [0,13

,

is false {Uk}

iUk-¢ i : 74Z%= 0 C2(Qb,3b )

MZ% k : 0

we know

6.4

with

and

that

uk

k ÷ ~

for some

c Q -I k ,@k,b > ~ max

b , then we can such

lUk-%kl <

that

2 C

' k/6k

By the e x t e n s i o n p r o p e r t y 2 to be a C solution of Nu= 0

(iii)

I¢k-¢ I < k -I extends (note that

otherwise

6k + 0

/

(Q0,2b)

on

we c o n t r a d i c t

of

Q0,Rk ,

(I)),

with

(2)

iuk-¢kT

÷ 0

as

k +

C2(Q0,Rk ) Using = %k

the m o n o t o n i c i t y

by p r o p e r t y

(ii) of

Rk l II2 < c J0 16k -

(3)

inequality Q

2.17(*)

and by Remark

(which we may do with 6.2(**))

we have

2 llUk-%k11L2(Q0,2b ) + @k I 2 lUk-¢k}c2(Q0,2b ) + @kl)

Then

(see the argument

tly large

following

2.17)

it follows

that

(for s u f f i c i e n -

k )

llUk-%kllL2(Q0,R)

cR

lUk-% klC2

+ V6 k J1 (Q0,2b)

and hence by

that

(2)) we have

(by the a - p r i o r i

estimates

1 5.6, w h i c h

can be a p p l i e d

275

(4)

]Uk-~klc2(Q0,R ) < CR[lUk-~klc2(Q0,2b)

In p a r t i c u l a r

we m a y as w e l l a s s u m e

(5)

+ /@k ]

> k/6 k lUk ~ k I c 2 ( Q 0 , 2 b )

otherwise using

(4)

(with R = 3b)

already

gives

to

(I) by

~ = @k Then

~k =

set

(2),

(with

, % = ~k ) ,

u=u k

C2(Q+;~)

(3),

function

-I Wk : @k (Uk-%k)

lUk-~klc2(Q0,2b ) '

that then by

w

(4),

we h a v e a s u b s e q u e n c e S w : 0

and note

<

lemma

in

c {k} and a

(for e a c h

C

R

R > 0)

,

(3))

]~

(7)

{k'}

2.14

F

IWIc2(Q0,R ) (by u s i n g F a t o u ' s

'

(5), in v i e w of the l i n e a r i z a t i o n

with

(6) and

a contradiction

I]~]1

2

<

oo

JO

However Then

since

L w = 0

we h a v e a r e p r e s e n t a t i o n

(7) t e l l s us that all c o e f f i c i e n t s

I 5.8 are zero.

Thus

for

of the f o r m I 5.8.

of the p o s i t i v e

exponentials

in

m > 0 + -y .t

(8)

w(~,t)

= ~(~)

+

[ a•e] {j:l >0}

]

%j (~) , (~,t) ~ Q+

,

]

for s u i t a b l e

constants

for

w

m < 0 ,

a. , w h e r e ] has the f o r m

} c C2(Z;V) =

with

L } : 0 ; while

-I< . t

(8) '

W(~,t)

= 4(00) +

[

j->l where or

each

<j h a s p o s i t i v e

a(1)+a(2)t •

J

.

]

with

a (I) ,

3

t

Re(a e

] ) 4 (~)

]

J

real p a r t and each a (2) •

J

constant.

a. 3

is e i t h e r

a constant

276

Now notice

(9)

Uk,

where

S

we h a v e for some the

t h a t this t e l l s us that

- Ck , = Bk , (~+S) + o (Bk' )

is the s e r i e s on the r i g h t ~ = lim ~k' ~k'

C2(Z;~)

with

with

ME(~k,)

norm).

Thus

L~k,~k,

in

(8),

= 0

and

= 0 .

(I) for

the fact t h a t for any

b

(because of the s t r i c t n e g a t i v i t y S ) .

there

is a

~ elSlc 2 C2(Qb,3b )

of

(8)'

T h e n by R e m a r k

B k,~k,

(All l i m i t

sufficiently

8 E (0,1)

Isl

k' ÷ ~ ,

6.2

= ~k' - ~k' + ° ( B k ' )

statements

h e r e are in

(9) c a n be w r i t t e n

Ukt - ~k' = Bk'S + °(Bk')

thus c o n t r a d i c t i n g

as

as

k' ÷

large,

t

by v i r t u e of

b 0 = b0(8,S)

for

(5) a n d

such that

b~b 0

(Q0,2b) of the e x p o n e n t i a l s

in the d e f i n i t i o n

277

REFERENCES

[AA]

F°J. Almgren,

surfaces Math. [AW]

On the radial behaviour

and the uniqueness

113

(1981),

95

of minimal

of their tangent cones, Annals of

215-265.

On the first variation

W.K. Allard, Math.

W.K. Allard,

of a varifold,

Annals of

(1972), 417-491.

[CHS3 L. Caffarelli,

R. Hardt,

singularities, [FH]

H. Federer,

[GG~

M. Giaquinta,

quadratic

L. Simon, Minimal

surfaces with isolated

TO appear in M a n u s c r i p t a Math.

Geometric m e a s u r e theory, E. Giusti,

functionals,

The singular

Springer Verlag

1969.

set of the minima of certain

Ann. Scuola Norm.

Sup. Pisa 11

(1984),

45-55. [GT]

D. Gilbarg,

N.S. Trudinger,

tions of second order, [HS]

R. Hardt, L. Simon,

Elliptic partial d i f f e r e n t i a l equa-

2nd Edition,

Boundary

Springer Verlag 1983.

regularity

for the oriented Plateau problem,

and embedded solutions

Annals of Math.

110

(1979),

439-486. [McB] C.B. Morrey, M u l t i p l e Springer V e r l a g

integrals in the calculus of variations,

1966.

[SLI~ L. Simon, Asymptotics

for a class of non-linear

tions, with applications 118 [SL21

to geometric problems,

evolution

equa-

Annals of Math.

(1983), 525-571. , L e c t u r e s on g e o m e t r i c m e a s u r e theory, P r o c e e d i n g s of

the Centre for M a t h e m a t i c a l Analysis, A u s t r a l i a n National Univer-

[SU]

sity, Vol.

3, 1983.

R. Schoen,

K. Uhlenbeck,

J. Diff. G e o m e t r y 17

A regularity

(1982),

307-336.

theory for harmonic maps,

Harmonic mappings in the plane

Read more

Harmonic Mappings in the Plane

Read more

Immersion

Read more

Immersion

Read more

Mappings

Read more

Universal Spaces and Mappings

Read more

Instant Immersion French Deluxe

Read more

Multivalued mappings

Read more

Polynomial Mappings

Read more

Polynomial Mappings

Read more

Polynomial Mappings

Read more

Quadratic mappings and Clifford algebras

Read more

Stable mappings and their singularities

Read more

Conformal Geometry and Quasiregular Mappings

Read more

Minimal surfaces

Read more

Mappings in Thought and Language

Read more

Stratified Mappings - Structure and Triangulability

Read more

Minimal Semantics

Read more

Minimal Semantics

Read more

Minimal surfaces

Read more

Minimal Semantics

Read more

Mind Design and Minimal Syntax

Read more

Lectures on Quasiconformal Mappings

Read more

Mind Design and Minimal Syntax

Read more

Harmonic synthesis and superposition

Read more

Harmonic Analysis and Applications

Read more

Minimal submanifolds and related topics

Read more

Mind Design and Minimal Syntax

Read more

Lectures on quasiconformal mappings

Read more

Classification of Lipschitz Mappings

Read more

Recommend Documents

Harmonic mappings in the plane

CB638-FM CB638/DUREN January 22, 2004 13:13 Char Count= 0 This page intentionally left blank i CB638-FM CB638/...

Harmonic Mappings in the Plane

CB638-FM CB638/DUREN January 22, 2004 13:13 Char Count= 0 This page intentionally left blank i CB638-FM CB638/...

Immersion

IMMERSION Gregory Benford Gregory Benford is one of the modern giants of the field. His 1980 novel Timescape won the Ne...

Immersion

Mappings

Mappings Mappings FEMINISM AND THE C U LT U R A L G E O G R A P H I E S OF ENCOUNTER • SUSAN STANFORD FRIEDMAN P R ...

Universal Spaces and Mappings

Instant Immersion French Deluxe

Multivalued mappings

288. A. Manitius, "Optimal control of h e r e d i t a r y s y s t e m s , " in: Contr. T h e o r y and Top. Funct. Anal....

Polynomial Mappings

Polynomial Mappings