Harmonic Maps Between Surfaces: (With a Special Chapter on Conformal Mappings)

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: Mathematisches lnstitut der Universitiit und Max-Planck-lnstitutfur Mathematik, Bonn Adviser: E Hirzebruch

Jurgen Jost

Harmonic Maps Between surfaces (with a Special Chapter on Conformal Mappings)

Springer-Verlag Berlin Heidelberg New York Tokyo I984

Author

J~irgen Jost Mathematisches Institut der Universit~t Wegelerstr. 10, 5 3 0 0 Bonn, Federal Republic of Germany

A M S Subject Classification (1980): 58 E20; 30 C 70, 32 G 15, 35J 60 ISBN 3 - 5 4 0 4 3 3 3 9 - 9 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0 - 3 8 7 4 3 3 3 9 - 9 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright.All rights are reserved,whetherthe whole or part of the material is concerned,specificallythose of translation,reprinting, re-use of illustrations,broadcasting, reproduction by photocopyingmachineor similar means,and storage in data banks. Under § 54 of the GermanCopyright Law where copies are madefor other than private use, a fee is payableto "VerwertungsgesellschaftWort", Munich. © by Springer-VerlagBerlin Heidelberg1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 214613140-543210

Dedicated

to the

memory

of

Dieter

Kieven

P R E F A C E

The p u r p o s e

of these L e c t u r e

give a fairly harmonic

complete

maps b e t w e e n

Notes

surfaces.

also serve as an i n t r o d u c t i o n ral;

therefore,

dimensional

whenever

results

and I try to give

should consult

On the other hand,

account

I want

of the results these notes

to on

should

in gene-

I p o i n t out w h i c h of the two -

to h i g h e r

some references

the several

On one hand,

account

to the theory of h a r m o n i c maps

appropriate,

pertain

For a more c o m p l e t e

is twofold.

and self h c o n t a i n e d

dimensions

and w h i c h

do not,

and an idea of the r e s p e c t i v e

in this direction, excellent

however,

survey a r t i c l e s

proof.

the r e a d e r

of Eells

and Le-

maire. An e s s e n t i a l

a i m of this book

the i n t e r p l a y particular

of d i f f e r e n t

the calculus

differential

geometry,

the c o n c e p t

of this book

In particular,

Nevertheless, cations

I believe

tion of u n r e l a t e d

proved.

until

thorough way

differential

conformal

Thus,

unified

treat-

contains

several

simplifi-

available

the s e q u e n c e

of the three are

in

as a m e r e e n u m e r a -

and m a n y d i f f e r e n t

mappings

than in the e x i s t i n g

equations,

n o r desirable.

On the contrary,

the results

in

to the view of a mere

is n o t i n t e n d e d

order,

and

maps,

analysis.

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

This b o o k

a logical

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

opposed

treatment

compared

results.

ters also r e f l e c t s

partial

that a c o m p l e t e l y

possible

that this

literature.

be constructed,

I think

is n e i t h e r

of m e t h o d s

theory of h a r m o n i c

topology, and c o m p l e x

is s t r o n g l y

and u n i f i c a t i o n s

the e x i s t i n g

in the

of variations, algebraic

specialist.

m e n t of the topic

is to show the v a r i e t y

fields

final

of the chap-

tools have

chapters

to

can be

used in a much m o r e

literature.

An outline

of the con-

tents n o w follows. After

giving

of h a r m o n i c in c h a p t e r discs

an a c c o u n t of the h i s t o r y maps

some

on surfaces,

are no c o n j u g a t e Moreover, controlled Chapter

points

geometric

points

Christoffel

"Multiple

contains

is r o u g h l y

then there are

w e show the e x i s t e n c e symbols,

following

c o n f o r m a l mappings.

a mistake.

These

I, we start concern c o n v e x

that if on a disc there

also no cut points.

of local

Integrals...",

the d e f i n i t i o n

of view in c h a p t e r

considerations.

a n d the result

3 deals w i t h

in Morrey, proof

from several

2 with

and p r e s e n t i n g

coordinates

Jost-Karcher

[JK1]

We first p r o v e T h e o r e m

Springer,

The d i f f i c u l t y

with curvature

which

1966, leads

9.3

since M o r r e y ' s to this e r r o r

is

VI

overcome

by minimizing

H 2I w h i c h

space

vertheless

conclude

sired

properties

11).

Furthermore,

morphism ever,

and

will

that

we

4, w e

the boundary

first

general which

a - priori We

maximum

estimates

then attack

compact hard

to s e e

homotopy group

of the

obtain [LI],

[L2].

image

In c h a p t e r

maps

and prove

Kaul

[J~KI]

problem

problem

values,

results

detail

lent harmonic

unit 6 and

mappings

boundary

disc

between

maps,

of

disc.

results

results

and we

thus

of Lemaire we

can al-

classes

for

to a 2 of harmonic

[Ht]

and J~ger-

case

of m a p s

be-

surfaces.

and since

f o r the c a s e w h e r e

latter

These

estimates map

the existence problem,

apply

in p a r with

a

c h a p t e r 3 to p a s s 9. T h e

results

and employ

several

of harmonic

7.

is t a k e n

diffeomorphisms

if the b o u n d a r y

homeomorphically

result

of chapter

we

for u n i v a -

composed

in c h a p t e r [JKI]

can

of

Heinz.

the D i r i c h l e t the d o m a i n

domain

7 where

from below

a conformal

on Jost-Karcher

assumption

of c h a p t e r

determinant

to a n a r b i t r a r y

o f E.

This

This

harmonic, w e c a n use the r e s u l t o f

8, w e p r o v e of

of t h e

functional

7 are based

ideas

in the p l a n e .

the help the

one is a g a i n

In c h a p t e r

convex

for

to c o n f o r m a l

solutions

the

unit

removed with

estimates

important

out of a

homotopy

surfaces.

disc

chapter

fall

argument,

the

map.

between

i t is n o t

of uniqueness

C 1'e - a - p r i o r i - e s t i m a t e s

from the

can

second

homotopy

is the

harmonic

again,

of H a r t m a n n

in s o m e m o r e

gives

maps

is h o m e o m o r p h i c

the q u e s t i o n

of a

the h a r m o n i c

however,

replacement

the domain

ticular

that

and Le-

also

theorems

6, w e p r o v e

then be

of

If t h e

In c h a p t e r

prove

proof

minimizing

in two d i f f e r e n t

the c o r r e s p o n d i n g

how-

to H i l d e -

to C o u r a n t

Lemma

off.

if the i m a g e

5, w e d e a l w i t h

due

for h a r m o n i c

existence

by a careful

for the c a s e

The

cannot happen,

fundamental

and then examine

closed

due

of continuity

splits

this

estimates,

a result

dimensions.

existence

vanishes, o f the

4 and

diffeo-

consists o f a c o m b i n a t i o n

limit of an energy

boundary

problem

and a lemma

two

if a s p h e r e

the D i r i c h l e t

sphere.

tween

only

(A-priori

the Courant-Lebesgue

Furthermore,

nonconstant

in

is a g l o b a l

the d e -

chapters).

convex ball,

the m o d u l u s

Using

a new proof

so s o l v e

for

t h a t the

class

in c h a p t e r s

idea

this m a p

the D i r i c h l e t

principle

the general

surfaces.

a similar

[ H K W 3]. O u r p r o o f

is o n l y v a l i d

Sobolev

map with

could expect

in s o m e

of the

so that we can ne-

is a c o n f o r m a l

that

in later

solve

lie

brandt - Kaul - Widman rather

as o n e

obtained

values

besgue

encounter prove

subclass

to the p r o b l e m ,

the m i n i m u m

shall

as r e g u l a r

in a r e s t r i c t e d

adapted

(we s h a l l

only be

In c h a p t e r

energy

is s u i t a b l y

from

onto [J3]

values

a convex and uses

map

as the

curve inside in p a r t i c u l a r

a

VIi

We can also use the proofs

of Thms.

a - priori - e s t i m a t e s

4.1 and

8.1 in chapter

to p r o v i d e

9, using

non - v a r i a t i o n a l

Leray -Schauder

degree

theory. We then apply T h e o r e m monic

coordinates

8.1 in c h a p t e r

on a r b i t r a r y

These

Karcher

[JK1].

perties

and can be u s e d to prove

coordinates

nic maps b e t w e e n

surfaces

jectivity

once

Theorem

radii,

The

final

chapter

surfaces.

First,

m o n i c maps

gives we give

the a n a l y t i c

ature

is h a r m o n i c , a s

Gauss

curvature.

A mong

the o m i s s i o n s

well

insights

into

or c o n v e r s a t i o n s Schoen

of this

Yau.

that w e

Finally,

comments

on m y m a n u s c r i p t

that the

constant

mean

curv-

on the e x p l i c i t

between manifolds [EW 2],

can c o n t r i b u t e

To H e r m a n n Karcher,

[EW 3],

anything

I owe many

of the field w h i c h he g e n e r o u s l y

I benefitted

much

Bob Gulliver,

advice

my research

from c o l l a b o r a t i o n

Luc Lemaire,

Rick

joint w o r k w i t h him),

But most of all,

for his continous

of Bonn.

stating

to [L I],

11 represents

and for s u p p o r t i n g

the means

of a w e l l

in 3 - space of c o n s t a n t

maps

the reader

aspects

Chapter

through

maps b e t w e e n

area.

Furthermore,

John W o o d and S h i n g - T u n g

with

to show that the

and T r o m b a of har-

book are results

of h a r m o n i c

w i t h J i m Eells,

Stefan H i l d e b r a n d t

due to

c l o s e d surfaces,

space w i t h

surfaces

to several persons.

(in particular,

many years,

as i m m e r s e d

the g e o m e t r i c

c o m m u n i c a t e d to me.

between

of Earle - Eells

since w e do not feel

is i n d e b t e d

the

surfaces,

p r o o f of Eells - W o o d

of E u c l i d e a n

We refer

new to the p r e s e n t a t i o n My w o r k

and in-

theory.

of the p r e s e n t

metrics.

[EL 3] instead,

bounds

of d i f f e o m o r p h i s m s

argument

the T h e o r e m of R u h - V i l m s

and c l a s s i f i c a t i o n

canonical

closed

of h a r m o n i c

mappings

some a p p l i c a t i o n s

we discuss

construction

pro-

for h a r m o -

is known.

in the class

replacement

concerning

Gauss m a p of a s u b m a n i f o l d

with

regularity

estimates

11 w h e r e we prove

between

some a p p l i c a t i o n s

to T e i c h m O l l e r

Furthermore,

of har-

to Jost -

diffeomorphism.

result of Kneser

a n d then we give

in c h a p t e r

energy

delicate

according

only on c u r v a t u r e

of c o n t i n u i t y

diffeomorphisms

a rather

is a h a r m o n i c

k nown

depending

[JS]. We m i n i m i z e

a n d then apply

the e x i s t e n c e

best p o s s i b l e

C 2'~ - a - priori

the m o d u l u s

of h a r m o n i c

Jost - S c h o e n

limit

possess

8.1 w i l l again be a p p l i e d

existence

10 to prove

discs on a surface

I am indebted

and e n c o u r a g e m e n t

to over

in every p o s s i b l e w a y

of the S o n d e r f o r s c h u n g s b e r e i c h

72 at the U n i v e r s i t y

I am grateful

for some useful

to A l f r e d

and to M o n i k a

great care and patience.

Baldes

Zimmermann

for typing it

Table

of contents I

I. I n t r o d u c t i o n 1.1.

A short history

1.2.

The

1.3.

Definition

1.4.

concept

2.

Physical

1.6.

Some

~ometric

2.2.

Convexity

2.3.

Uniqueness

2.4.

Remark:

Curvature

2.6.

Local

of

of g e o d e s i c s

the s q u a r e d

of geodesic

arcs

distance

arcs

. . . . . . . .

and conjugate

discs

10 10

14 15

curves . . . . . . . . . . . . . . . .

15

curvature

o f 2.3.

13

. .....

with

analogue

. . . . . . . .

controlled

Christoffel

mappings

16

18 3.1

surfaces

concerning

Proof

Lemma

of Theorem

3.7.

Uniqueness

3.8.

Applications

3.9.

The Hartman-Wintner

3.1

of conformal

~eorems

3.1

A maximum

The Dirichlet

principle

to p l a n e

domains

.....

18 19

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

21

. . . . . . . . . .

33

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

34

Lemma . . . . . . . . . . . . . . . . .

35

for h a r m o n i c

4.2.

representations

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

representations

of T h e o r e m

4.1.

conformal

homeomorphic

The Courant-Lebesgue - 3.6.

9

points12

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

compact

3

of harmonic

function . . . . . . . . .

in c o n v e x

dimensional

of p a r a l l e l

S t a t e m e n t o f Thm.

4. E x i s t e n c e

and terminology

1 2

12

existence

coordinates

symbols

3.3.

concept

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

about notation

The higher

2.5.

of

from the

considerations

Convexity,

3.2.

. . . . . . . . .

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

arising

significance

remarks

2.1.

3. C o n f o r m a l

maps

problems

principles

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

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

I .5.

3.1.

of harmonic

Mathematical maps

of variational

of geodesics

maps between

for energy

problem,

if the

minimizing image

38

surfaces maps . . . . . . .

is c o n t a i n e d

38

in a c o n v e x

disc . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

4.3.

Remarks

42

4.4.

The

Theorem

4.5.

The

Dirichlet

problem,

Two

different

solutions,

stant

about

the h i g h e r

of L e m a i r e

-dimensional

situation

and Sacks-Uhlenbeck if the i m a g e if

......

. . . . . . . . .

is h o m e o m o r p h i c

the b o u n d a r y

values

are n o n c o n -

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

4.6.

Nonexistence

4.7.

Existence

for c o n s t a n t

results

boundary

in a r b i t r a r y

43

to S 2

46

values

. . . . . . . . .

50

dimensions

. . . . . . . . .

52

IX

5. U n i q u e n e s s 5.1.

Composition

5.2.

The

5.3.

of harmonic

uniqueness

Uniqueness positive

5.4.

for the

Uniqueness

6. A - p r i o r i

problem

for closed

....

54

. . . . . . . . .

54

if t h e

image has

non-

solutions,

if the i m a g e

62 has

curvature . . . . . . . . . . . . . . . . . . . and nonuniqueness

for harmonic

maps

63

between 65

estimates

65

Cc~position

6.2.

A maximum

6.3.

Interior

modulus

6.4.

Interior

estimates

6.5.

Boundary

continuity

of h a r m o n i c

principle

of continuity

conformal

maps

.....

65 66

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

68

f o r the e n e r g y . . . . . . . . . . . . .

68

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

Interior

C I -estimates

6.7.

Interior

C 1'e - e s t i m a t e s

6.8.

C I - and C 1'~-estimates

estimates

maps with

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

6.6.

7.1.

functions

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

6.1.

harmonic

convex

and Kaul

surfaces . . . . . . . . . . . . . . . . . . . . . .

C 1'e

7. A - p r i o r i

maps with of J ~ g e r

Dirichlet

results

Uniqueness closed

theorem

curvature

nonpositive 5.5.

54

theorems

69

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

72

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

75

at t h e b o u n d a r y . . . . . . . . . .

75

from below

for t h e

functional

determinant

of

diffeomorphisms

A Harnack

77

inequality

o f E. H e i n z

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

77

7.2.

Interior

estimates

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

78

7.3.

Boundary

estimates

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

82

7.4.

Discussion

8. T h e e x i s t e n c e

of t h e

of h a r m o n i c

situation

in h i g h e r

diffeomorphisms

dimensions

which

solve

.....

84

a Dirichlet 86

problem 8.1.

Proof

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

tained

in a convex

8.2.

Approximation

8.3.

Remarks: Theorem

9.

Plane

in c a s e

and bounded

the image

is c o n curve.

. .86

arguments . . . . . . . . . . . . . . . . . .

88

domains,

necessity

by a convex

of the hypotheses

of

8.1 . . . . . . . . . . . . . . . . . . . . . . . .

C I'~ - a - p r i o r i existence

theorem

ball

estimates

for

arbitrary

domains.

Non - variational

proofs

90

9.1.

C I'~ - e s t i m a t e s

9.2.

Estimates

9.3.

A non-variational

surfaces

90

for

on arbitrary the f u n c t i o n a l

surfaces determinant

. . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . proof

of T h e o r e m

90

on arbitrary

4.1 . . . . . . . . . .

92 93

9.4.

10.

11.

A non -variational

Harmonic

Existence

10.2.

C 2"~ -estimates

10.3.

Bounds

10.4.

Higher

10.5.

C 2'~-

10.6.

Higher

The

diffeomorphisms 11.1)

11.3.

Extension

11.4.

Remarks

Applications

12.3.

the

of Kneser's

Proof

of T h e o r e m

Contractibility

. . . . . . . . . . .

between closed

surfaces

. . . . . . . . . . . . . . . . in h i g h e r

between

dimensions

....

Tromba's

proof

12.5.

The

12.6.

Harmonic

Gauss

12.7.

Surfaces

of

of

103 104 104 105

106

106 106 111 111

112

surfaces

certain

harmonic

Theorem

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

112

12.1 . . . . . . . . . . . . . . . . . .

113

of

Teichm~ller

maps

98 100

surfaces

space

and

and

an a n a l y t i c

the

diffeomor-

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

12.4.

References

8.1

maps

of

proof

approach

maps

between

situation

of harmonic

group

. . . . . . . .

11.1 . . . . . . . . . . . . . . . . . .

of Theorem

Holomorphicity

phism

coordinates

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

of T h e o r e m

about

coordinates

diffeomorphisms

(Theorem

....

. . . . . . . .

Conformal

95

m a p s 98

maps . . . . . . . . . . . .

of harmonic

Harmonic

for harmonic

C 1'e - e s t i m a t e s

symbols.

of harmonic

of harmonic

Proof

estimates

coordinates

for harmonic

regularity

8.1 . . . . . . . . . .

coordinates.

harmonic

Christoffel

regularity

11.2.

12.2.

the

for

estimates

existence

12.1.

of h a r m o n i c

on

of T h e o r e m

C 2'~ - a-priori

10.1.

11.1.

12.

coordinates.

proof

that

TeichmOller

space

Gerstenhaber-Rauch maps

constant

and

Bernstein

Gauss

is a

cell . . . . .

. . . . . . . . . . . theorems

curvature

in

. . . . . . .

3- space .....

116 119 123 124 124

127

GROUP

EPIMORPHISMS

AND

PRESERVING

THE

If

one

uses

various

natural

topological

for

certain

class

i.

Definition

its

commutator

To

define

commutator

class

under

is

of

class,

element, subgroup.

if t An

recall

that

subgroups of

if

each the

the

derived

to to

a group

Because

It

subgroup H

perfect

The

G.

a

'~

the

have

being

obtain

this

an

plus-

algebraic

viewpoint

P

is

perfect

homomorphic

radical

series

here of

a given

is

also

is

if

image

group

PG is

a

equal of

is G.

of

to

by

Thus

the

G~

¢(PG)

view

is

to

a

which

PG

its

must

~ P(tG) the

under commutators,

class

admits

be

because,

as

therefore

closed

generated

functorial

then

G

evidently

generate.

construction

approach

of

H's

all

: G ÷ H is a homomorphism~

transfinite

sequence

back,

Central

K-theory,

excision)

so

perfect

alternative

a fibre

pendulum

algebraic to

image of a perfect group is again perfect.

subgroup

invariant

as

matters.

[P,P].

because the

the

to

obstruction

homomorphisms.

automorphisms

union, so

swing these

a commutator,

a maximum

the

we

subgroup

is

Tbs closed

i.2

(such

the homomorphic

i.i

then

formulations

group

the

approach

example,

discussing of

Berrick

(for

Here

setting

J.

plus-construction

problems

constructive).

group

the

key

RADICALS,

PLUS-CONSTRUCTION

A.

then

PERFECT

a

fully

from

I.i,

~ PH. intersection

of

it

is to

in

We

enquire

the

image

be

an

other

sion)

words,

an

P.

have

is

F,

= P.

be

On

theory

Ep2R

to

equality

holds

restrict

consideration,

hypotheses -

P

an

to

in

ensure

Epimorphism

of

free,

other

shall

an

(1.2).

Since suppose

that

(or,

epimorphism

correspond

to

has

perfect.

the

: we

seek

example

being

can

wish

~(PG)

in

context,

is

not

= P(~G), Exten-

Radicals.

R ~+ F -~

group

PP

group

%

extreme

Thus

free

we

We

Perfect

extension

conditions

which

that

Preserving

group no

what

to

epimorphism.

For the

under %G

only

So hand,

see

a

free

any

are

Ep2R,

presentation

of

non-trivial

although there

that

which

free

PF

=

i,

no

examples

surjection

let

a perfect

subgroups,

making

tPF from

with

and

=

i,

we

finite

finite

kernel

is

Ep2R.

Note

that

hypoabelian) In

of

among

G

n-th

forces

triviality

derived

groups

of

we

PG

must

(that

have

(PG)

is,

G

(n)

~ G (n)

is

fact,

The f o l l o w i n g

1.3

(i)

(iii)

The effect

2.

PG

=

1 and for

some

i

G (i)

PG

=

1 and for some

j

G(J)/Z(G

proof

that

is

an

G(i)/Z(G

Algebraic

a

three conditions are equivalent

:-

G is soluble;

(ii)

on

solubility

since

easy

(i))

exercise, finite

is finite;

save

implies

is finite.

(j))

for

a

G (i+l)

lemma

of

Schur

to

the

finite.

results

Our

main

group

G

epimorphism

purpose and

%

as

Proposition

of

these

to is

here

normal

the

is

We

maps

what

it

can

easy

to

that

if

}

then

~ must

also

be

Ep2R;

LEMMA

2.1

: G

be ÷

H,

so,

(in

: Ker

~

Several

(for

t:

about H ÷

under

Q

what

below) to

conditions Lemma

Since

said

(2.3)

sufficient

such

useful

question.

Ep2R

see

establish N

Ep2R.

find

following

Ep2R,

to

subgroup

: G --~ Q 2.3.

is

have

to the

relate

Again,

composite

that

an

col]ated one

composition

converse?

conditions

ensure are

3.7)

evidently the

conditions

~o9 will

it

: G ÷ Q ~ be

Ep2R

of is Ep2R as

well?

: Let ?

: G

+

H,t

: H

÷ Q

be epimorphisms

some finite n, (Ker

~)(n)

~

tPG.

such that~

for

Then

(a)

~ is

Ep2R; a n d

(b)

~ is

Ep2R i f

Proof.

Let

(2.2)

below,

lemma (Ker

~)(n)

j(n+l),

J ~ H

Thus

of

which

and

Ep2R

WPG

~PG

= PH

I am which

led

one

me

got

is

Ep2R,

By

PH

=

assumption

to

the

for

all

For

Prof.

K~

B.

immediate is

PROPOSITION

(ii)

the

other

map)

(iii)

- and

of

PH

t)(n)

(PQ). it

From is

is

(b),

the

known So

equal

= FQ.

when

that

j(n)

therefore

FQ (n)

and

~PG

9.

Then

=

last

Hartley

~o@

to

This is

the

both

~PG.(Ker

subgroup

have

image

PQ,

¢)(n)

lies

in

@PG,

leaving

be an

=

fact

that

P

whenever

application

2.3

: An

extension

argue

[j(m)K(n-l)

of

an

argument

where

P is a p e r f e c t

= j(m)K(n)

m,

(iii)

suggesting

m ~ n ~ O,

fixed

hand,

for

lemma.

J-~P

finite

each

from

the

(k)

-I

(1.2)).

namely

(@PG).Ker

this

following

: Let

j(n)

An

then

= ((tPG).Ker

to

= 0 results

identity

in

j(n),

implication

= P H (n)

2.2

Then

On

as

that

grateful

Proof.

j(m)

(after

lies

image

~

However

~ j(n+l)

perfect,

j(n)

n

~)(n).

(PH) (n+l)

thence

image

required.

LEMMA

group.

inverse

j(n+l)(Ker

same

Ep2R.

~o~ i s

the

deduces

~ H.

as

if

maps.

if

one

PH

because

=

the

Conversely, from

=

being

has

(a),

composite

PH

j(n),

PH

establishes

~

only

denote

j(n)

~ ~PG

whence

- PH.

and

induction

= p(m)

with

j(n-1)

Lemma

following

extension

on the

n.

The

latter

= j(m)K(n-l)

' j(m)K(n-l)

of

the

by

it

case

the

image

follows

of

that

] = j(m)K(n)

2.1(a)

(or

(b)

with

~

as

the

result.

N~

G --~ Q is Ep2R

it is split;

G (m) ~ N.PG for some f i n i t e m; N (n) =< PG f o r s o m e f i n i t e n; o r

provided

either

the homomorphism G ÷ Aut(N/PN),

(iv)

has

REMARKS. ful,

in

its

to

as

Finally,

example

the

study

Proof.

also

G

(i)

is

borrow

finite

so

Thus we

shall

weaker

of

§3

than

may

PROPOSITION

homomorphism

since also We

~ is

b)

[PG,N]

In

fact,

by

(2.4)

implies shall

case...

therefore

an

(1.2),

infinite reveals.

extension

since

For

PQ

in

fact

will

if

(ii)

~ Q(m)

the

we

for

course of

=

establish

be

proved

by

homomorphism commuting

Q

÷ Out(N/PN)

all

of

whiah

(iii).

PG/P. the

following

topological condition

is

diagram

Aut(N/PN)

extension and

by

then P(G/P)

÷

if

by

group

consequence

G ~ Aut(N)

~ An

The

N~

G @-~Q

only

if both

induces

the

trivial

PN. is

~ the

in

the

÷ Out(N)

Now

more

and

Ep2RI

it

PQ.

is perfect,

from

replaced

= PQ.

discussion,

shall

or,

use-

= ¢PG.

automorphism

seen

÷ Out(N/PN)

=

~

of

¢@PQ

immediate

converse

its

be

2.5

PQ

a)

whose

That

~

~PG

further

we

be

most

length.

a perfect induced

~ t(PG.N)

following

work,

(iv),

below. (iv)

that

: If P ~ G

extra

not of

~PG

the

derived

application

then

¢(o(m).N)

the

2.4

a little

fact

demands

of

may

automorphisms

¢,

perhaps

extensions,

finite

(iii)

a two-fold

the

is

perfect has

presentation

to

~

and

(ii),

= Aut(N/PN)/Inn(N/PN).

(iii)

kernel

outer

just

(iv) use

strengthening in

[4].

FQ

only

COROLLARY

means

in

(1.2)

that

make

By

in

the

is

central

a free

of

in Out(N/PN)

hypotheses,

whose

right-inverse from

m,

to

of

further

image

above

occurring

the

pursued

: Q ÷

the

extensions

ordinals

ones,

is

Of

application

generally, finite

hypoabelian

induced by conjugation,

is

only

necessary

Ep2R

equivalence demonstrate

if

and of

to only

[PG,N] this

prove if

this

N/PN~

= PN

and

notationally

result G/PN--~ [P(G/PN), simpler,

when Q

PN

is. N/PN] special

= I, (2.4) =

I.

LEMMA

trivial

t is Ep2R;

b)

[PG,N]

Z(N)

~

CH(N)

From giving next

%

fact

2.7

of A.

that

So

embeds of

centre

induced

the

extension

the

~ H/N is

in Out(N).

in Out(N) an

N~H~PQ

homomorphism

H/N.CH(N)

PQ

of

is Ep2R

that

there

whence

4(PCH(N))

Suppose

:

a)

[PA,B] PA

=

and

extension

This therefore

NnCH(N)

N has

derived

CH(N)

--~ PQ

is

length Ep2R.

at

that

PG

~ N.PCH(N),

=

most

i,

However,

too.

= PQ

(2.6)(h)

we

is

have

an

immediate

application

of

that

= PB,

(From

where

BC

B,

C are c o m m u t i n g

normal

sub-

= PC; and

and

(2.7)(b) we

: If

then

(2.7)(a)

may

(b)

instead identify

the extension

of

are

equivalent.

(2.7)(a)

H above

N ~

as

we

obtain

the

N.PG.)

G --~ Q induces

the

trivial

below,

for

PG = PN.PCN.PG(N).

the

lemma

we

P

: PA

Q

: [PA,B]

P0

is

verify

the

assertions

Pa'

Qa

a.

Evidently

gives

applying

2.9

prove

ordinal

[PA,C]

(2.6)(a)

map P Q + O u t ( N ) ,

for

=

: Conditions by

COROLLARY

To

A

PB.PC.

2.8

following.

P~-I

induces

i

Then

b)

Note

QB

=

lemma.

REMARK

of

: G -->> Q

the

observe

and

satisfied,

= P(N.PCH(N)).

LEMMA

each

so

image

the

to

we

and

the

Since is

that

[4],

H = N.CH(N)

--~ PQ.

the

PG

groups

precisely

so

restrict

in

N.CH(N)

(2.3)iii)

~ G,

we

as

Consequently

condition CH(N)

first,

kernel

H/N.CH(N)

trivial.

the

(a)

has

PN

if both

and

Proceeding

H ÷ Out(N)

with

N ¢-~ G t-~ Q

if and only

= i.

obtain

PQ.

makes

extension

PQ + O u t ( N )

a)

To over

: An

2.6

map

all that

~ B(a).C

true.

B < a clearly of PA

Pa,

since

=

[PA,PA]

.

~ B (a)

If

a

forces

~

is

a limit

that

[B(a-l).c,

of

ordinal, Qa

then

otherwise,

B(a-I).c]

the the

~ B(a).C

truth truth

.

of

6

On

the

other

hand,

because

PA

we

have

Q~

induction

=

implying

Since that

the

lemma

2.3

(and

the

all

[PA,B]

an

thereby

Qa"

=

of

interesting

us

check

PA

Now

the

the

have

(after,

proof

e.g.,

any

identify extension

these B~C=+

the

result.

easier.

So

transfinite

latter,

[PB,B]

~

require

[PA,B]

BnC

to

whose

be

central

proof

in

uses

A,

Proposition

A = B.C for normal

subgroups B,C of A.

length or Out(B~C)

or

= RB.PC.(B~C)

(2.10)

=

[PA.(BnC)

=

[PB.PC.(BnC),

=

[PB.PC,

of

of finite

does

indeed

imply

2.7).

For,

, PA.(BqC)] PB.PC.(B~C)] = PB.PC.

PB.PC]

(2.10).

From

~ B/BnC

the

decomposition

× C/BnC

,

(2.3)i))

perfect

= P(B/BnC)

radicals

D -->>D/B~C

PA.(BnC)/BnC whence

=

series

that

P(A/BnC)

We

even

,

then

A/BnC

we

PB

(2.7)

: Suppose

its derived

hypoabelian~

first

is the

generalization

PA.(B~C)

Let

~ B(a).C

2.7).

P R O P O S I T I O N 2.10

If B~C has either

[PA,B].C

From

B(a)

~ ~

Lemma

Out(B~C/P(B~C))

~

holds

converse

Pc'

hypotheses

has

Q

[PA,B.C]

Pa~

clinches

when

is

by

× P(C/BnC)

appealing

Ep2R.

z PB.(BnC)/BnC

In

×

to

(2.3)iii)~

particular,

PC.(BnC)/BnC

we

;

iv). deduce

Thus that

3.

Connections In

KIA

is

This

the

Whitehead

=

derived

classical

length,

fact

and

that and

has

been

ning

The

time) the

KIA

lies

and

[I

general

GLA

we

main

their

is

which

definition use

of

of

close

It

Here

led

the

to

two

of

ring

course

to

the

groups

of

the

plus-

Since in

Ep2R

A.

matrices.

a discussion

of

natural

examples

of to

was

Quillen

study

the

class

for

of

elementary

the

the

are

over

by

K.A = v (BGLA+). 1 1 plus-construction,

relation

definition

well-suited

radicals.

motivation

in

seen

the

group

generated

a member

have

= GLA/PGLA the

for

linear

subgroup

perfect

further

then topology

maps

(at

questions

the

concer-

(respectively

[i

(6.8)]).

Suppose

(3.1)

the

making

plus-construction.

(5.11)]

starting-point

the

which

thence

much

geometry.

present

on

EA,

preserving

construction there

as

[GLA,GLA],

epimorphisms

the

lemma

PGLA

EA

plus-construction

K-theory,

identifies

finite

the

algebraic

Moreover,

of

with

f

: X

+

has connected fibre.

Y

Then the commuting

diagram qx

X

qY

Y

is co-Cartesian

+

y+

If the fibre sequence

(3.2)

is Ep2R. F

÷

E

~

(with F,E,B

B

(that is~ F ÷ is also the homotopy

connected)

p+),

fibre of

is

then

is Ep2R. The

proof

whose

evident

This

suspicion

First, is

>

X

if and only if %l(f)

plus-constructive ~l(p)

>

needed

spaces)

trivial.

can

it

is

in

order

which

are

(3.3)

only if

of

~l(p)

(3.2)

in

[I]

irreversibility be

validated

possible to

in

to

two

the

doubt

argument on

the

precisely

what

fibre

Thus

further

sequences

[3]

reveals

condition (of

the

F ÷ E ~ B is plus-constructive

induced

converse.

ways.

those

plus-constructive.

Ep2R a n d

a diagram-chasing

considerable

state

characterise

A fibre sequence is

involves

casts

action

of

connected

following.

if and

P~I(E) on ~,(F +)

is

In with

the

~I(F)

special

G

sequence

both ~ is Ep2R This

comes

PROPOSITION

3.5

homomorphism

BN

÷

The group of

proof

BN +.

of

+)

Since

This be

some

base

groups

greatly.

with

is plus-constructive

PN

=

Then

i.

if and only if

i. to

implying

filled

by

the

the

as-yet

following

An extension N ~

: PQ

÷

(3.5)

of

(free)

PN

= I,

argument

unproved

part

corollary

to

of

the

main

with PN = I induces

G --~> Q

if and only if the fibre

Out(N)

(given

[3])

hometopy this

sense

has

is

dual

subgroup

simply

classes just

consists of

AUT(BN)

3.6

the

sequence

in

identifying

self-homotopy

the

equivalences

= wo(Aut(BN))

sects

the

ring A.

split By

with

matrix

over

finite

matrices,

A.

take OA

as So

G

A) A

which

is

indeed

above,

the

upper

: GLUT

~ is

fibre

sequence

base corollary, whose

are hypoabelian.

examples of

in

the

concerning condition

linear

group

triangular

÷ GL(A

certainly

abelian

and

following

Then

2

×

(2.5) on

GLUT

the

(after

2 - matrices

epimorphism

in

an

space

the

plus-constructive.

general

ring

multiplication N

every

is

important

matrices

making

total

is plus-constructive.

of

split

~

that

irredundancy

for

The

the

therefore

both N and O u t ( N )

the

epimorphism

(2.3)i) GLnUT

[2] group

provides

We

a

result

is

with fibre B N

K-theory

ring

that

There

Suppose

:

[PG,N]. on

groups.

the

demonstrates

a given

require

fundamental

it

[i p.27])

induces

to

sequence

Algebraic (3.4);

not

spaces.

hypoabelian

COROLLARY

over

does

classifying

every fibre

and

of

[5 p.42].

space in

BQ

spaces

is plus-constructive.

BQ

AUT(BN

Out(N)

=

is

classifying

[3].

trivial BG

+

close

gap

are

be a group extension,

BG

[PG,N]

F,E,B

simplifies

-~>Q ÷

very

The

of

BN

and

above.

÷

this

Let N~

(3.4)

theorem

where

hypoabelian,

the fibre

(2.5)

case

N

@

A)

Ep2R. where

corresponds

subgroup.

of

general

Now M

is

N an

to However,

linear

= Ker

~

inter-

arbitrary

addition N

fails

n

of to

x

n

-

commute

with

PGLUT

= EUT.

For

example, 0

0

1

0

in

GL2UT

the

matrices

12 NnGL2UT 12 1 ~ el2 image

and the

sequence

12 of BN

allows @

E(A ÷

the

BGL(A

•

(since

sequences

[i

BN +

(12.3)],

The

[6].

algebraic

÷ EA

is

has

trivial

isomorphism.

of

exact

long

The in

second

of

have

Fp+

= F + , one

on

fundamental

homology these

the

(with two

a lemma

based

LEMMA

To exact

prove only

to

map

arise

= Fp

distinct

is

fulfilled

induced

$

map

is

difficult the

to

Fp+

B

and

an

the

relative

is

aspects in

(3.2)

in

terms

of

(3.8),

of

Recall

it

and thus

above

terms

Fp+.

that

P~I(F)) It

as

[I on

However, if

so

A-

K.A I.

F ÷

coefficients).

the

structure

A 1 the

thereby

F ÷ E ~

kernel

signi-

KA0(BG)

perfect).

a converse

F

÷

induce

an

consists the that,

an

to

epimor-

isomorphism

to

be

the

original

(3.9)

on

expected

below.

that fibre First,

(2.1).

: E ÷

(3.7), in

the

from

further

makes

and

sequences

(with

B

A ~ B ~

respect

we

is a fibre

C

(with connected

Then ~l(p)

÷ A.

K.A

fibres

of

groups

: Suppose

3.7

and that a map p : A×BE

fibre

map

A--~

which

BEA~,

+

whose

K-group

all

fibre fact

Mayer-Vietoris

natural

define

the

BGLUT

plus-constructive

concerning

expectation on

fact

linking

requires

should

This

the

the

after

only

BEA + ÷

(homotopy)

arbitrary

data

sequence.

is of

is

(3.4)

:

K-theory

algebraic to

is

(2.5))

This

Bt +

epimorphism

(EA B*

discussion

between

is

this fibre

of

one

from

contractible),

lower

the

ring

centre,

a characterisation

phism

any

not

of

Ep2R

sequence

relation

p'

12,13].

chs

and

(by

plus-constructive.

equivalence

G to

[i

for

the

not

additivity

enables

Hence

non-trivial, is

patently

latter

Again, of

is

a group

evidently

description the

the

generally, EA 1

A)

homotopy

= BN

of

K-theory

More

is

@

the

commutator.

non-extension and

ring

Out(N)

BGL(A

the

ch.3]

non-trivial

in

of

representation

since

A)

BGLUTB-~t

embraces

: EA

have

existence

A) +

ficance

~ E2UT

is Ep2R consider of

the

sequence

homotopy

fibre)

if and only if ~l(p the

first

following pairs

of

commuting horizontal

')

with F~I(C)

=

I,

induces is. diagram, maps.

which

10 ~l(f') ~2(C) + ~I(A×BE) -->>

Im vl(f')

I ~I(p')

~l(gop) -+

~ ~I(E)

I ~I(P)I

I ~I(p)

~l(f) ~2(C)

Since

~2(C)

(b)

that

(a)

occurs

is ,

wl(p

abelian, ) is

that

Ep2R

if

EP

Ker

Lemma 2

~l(g)

2.1

yields

R precisely

when

PwI(C) and

~l(g)

-~

precisely

hypothesis wl(p)l

÷ ~I(A)

the

when

if

PROPOSITION 3.8 :

~+

~I(B)

(a)

that

wl(p)

i

~l(POf

restriction

= 1 forces

only

to

Proof. is

affected

of

course

Then

B+

back

over

By

~l(p)

by P~I(B

is

[2 Lemma

2.1]

pulling-back +)

= i).

contractible,

) = ~l(f~p') is

lie

B + B + to

and

over

induce

is,

EPZR.

inside

and

which

However,

Ker

wl(g),

by the

making

sequence F ÷ E ~ B with F conn-

the the

lemma

the

a map

of

above,

fibre

neither

sequence

assume

that

F ~

~I(F) ~ ~l(Fp+)

is

Ep2R.

is

Therefore so

is Ep2R,

is.

For any fibre

if and only if

~I(C)

~l(f')

wl(p)l

FWl(B)

~+

ected, the induced f u n d a m e n t a l group homomorphism surjective

~l(C)

B

fibre

is

AB

fibre

÷ B + B+

already

sequence

an

over

(since

acyclic

÷ E+ ÷

Fp+

sequences

assertion

space.

B + pulls

B:

Fp+ : E +

\

/

E g E+xB

p\/ B As fibre only

the

maps

[~ p.35],

of it

total

follows

spaces that

and

of

Wl(Fg)

fibres

maps

share

~I(F)

a common

onto

Wl(Fp+)

homotopy if

and

if

~l(g) : ~I(E) + ~I(E+xB) : ~I(B) x ~I(E)/P~I(E) is

surjective.

so

that

~I(B)

Now ~ w/K.

write There

~ for is

Wl(E), a map

P of

for group

P~I(E)

and

extensions

K for

Ker

~l(p),

The metric (y~6)

=

( ~ 6 ) -I

manifold cules

by

We

(gij)

shall

the

use

a more

frames,

= {(x,y)

and

latin ones

to the

while

calculus,like

at other

covariant

x . occasions

we

derivatives,

or-

etc.,

w h a t e v e r is m o s t c o n v e n i e n t . o d e n o t e b y U its i n t e r i o r a n d

we

X , we

in the p l a n e ,

denote

a geodesic

ball with

: d(p,q)

d is the d i s t a n c e and

-< R}

notation

minology

p 6 X a n d ra-

function

,

o n X.

l o w e r b o u n d s for the s e c t i o n a l 2 2 denoted by K and -e , i.e.

__
center

i.e.

Upper

2

i.e.

: x 2 + y 2 ~ I}

where

-m

and

convention.

X at the p o i n t

coordinates,

unit disc

B(p,R) : = {q 6 X

This

the m a n i f o l d

of a manifold,

6 ~2

R by B ( p , R ) ,

often

(ye6),

t e n s o r of the i m a g e i by £ik~ " G r e e k m i n u s -

symbols

summation

notations,

by

the m e t r i c

to the d o m a i n ,

intrinsic

the c l o s e d

On a manifold dius

of

local

be denoted

closure.

D denotes D:

and

Einstein

complex

If U is a s u b s e t by U its

(y~),

refer

space

shall

prefer

thonormal

of a m a p w i l l

a n d its C h r i s t o f f e l

tangent we

the d o m a i n

always

shall use

is the

Sometimes

of

, y = det

as i n d i c e s

image. TxX

tensor

curvature

K of a m a n i f o l d

are

2

avoids

in s o m e o f

square

roots.

the p a p e r s

It differs,

frequently

however,

referred

from

the

ter-

to in the p r e s e n t

book. The

functional

determinant

J(u) (x) o r s i m p l y We

shall

use

the

J(x),

standard

ferentiable

maps,

dition with

exponent

C k'e

mapping

u at a point

~ . Here,

interval

spaces

spaces

hold

for all

(0,1).

of

satisfy

exponent

If n o t s t a t e d ~ E

from

ck'~(X,Y) of w h i c h

the H ~ I d e r

(O,1).

x is d e n o t e d

u is u n d e r s t o o d

the k - th d e r i v a t i v e s

from the open unit involving

of a m a p

if the m a p

k

times

dif-

a H~ider

con-

~ is a l w a y s

otherwise,

Whether

by

the c o n t e x t .

taken

results

the S o b o l e v

space

w i l l b e d e f i n e d w i t h the h e l p of a c o o r d i n a t e c h a r t o r w i t h H I(X,Y) 2 the h e l p o f a n e m b e d d i n g of Y i n t o s o m e E u c l i d e a n space, w i l l a l w a y s be

clear

We

try

from

to g i v e

results.

We

the c o n t e x t . selfcontained

assume,

however,

and complete basic

facts

proofs

of m o s t

from Riemannian

twodimensional geometry

as

12

well

as the

ferences

theory of linear e l l i p t i c

are

[BI] and

about p a r t i a l

2.

Geometric

2.1.

Convexity,

We start w i t h w i n g lermaa,

[GKM]

differential

equations

for the g e o m e t r y

and systems.

and

[GT]

Good re-

for the results

equations.

considerations existence

of geodesic

some e l e m e n t a r y

an a p p l i c a t i o n

arcs,and

considerations.

conjugate

points

The p r o o f o f the

of the T h e o r e m of Arzela - A s c o l i ,

follo-

is well

known.

L emma 2.1: Suppose the b o u n d a r y

M is a c o m p a c t

y is n o n v o i d ,

p and q in M , every h o m o t o p y shortest Here,

arc,

and this

a Lipschitz

goes

2.1:

Under

two d i s t i n c t points

, resp.,

Proof:

We denote that YI

class b e t w e e n ving on YI

p and q

length

tinct,

respect

, since

to p

of y).

conjugate

angles,

i.e.

of e q u a l

point,

arc w h i c h

are

is conjugate

is the shor-

. We can assume

in t h e i r h o m o t o p y

starting

e.g.

from p and mo-

to p in the same h o m o t o p y

and of

(At this point,

to use L e m m a

that they are s h o r t e s t

to cease

2.1 and a n d dis-

connections,

type o f the disc. from p into B

somewhere

the argument,

consisting

this

class

Since YI and Y2 are h o m o t o p i c

furthermore

In the limit,

p and ql

we have

. As

if we have not y e t of geodesic

of two h o m o t o p i c

are s h o r t e s t p o s s i b l e

construction

has

YI and Y2

in B to be s h o r t e s t

we get a nested sequence

configurations

length w h i c h

topy class.

that there

assume

connections

line e m a n a t i n g

line has

. Repeating

desired

2.1,

arcs by YI and Y2

of such a connection,

We n o w look at a g e o d e s i c

connection

every

to the i n t e r i o r

to a geodesic

otherwise,

a connection

we could assume

this

if t h r o u g h

is d i s j o i n t

class.

a set B of the t o p o l o g i c a l

are shortest,

a

find a p o i n t ql w h i c h w o u l d e i t h e r be c o n j u g a t e

the c o n v e x i t y

they b o u n d

t h e m contains

in M, and this p o i n t

as the s e g m e n t of T1 b e t w e e n

because

If

two points

joining p and q. Then each of the

point

the two g e o d e s i c

, we w o u l d

for the e x i s t e n c e therefore

with

of L e m m a

arcs

and Y2 are s h o r t e s t

to p or w o u l d have equal

geodesic

in its h o m o t o p y

w.l.o.g,

c o n v e x w.r.t.M,

arc w h i c h

the a s s u m p t i o n s

homotopic

test connection

Given

of q.

p and q has a conjugate

t_~o p o~_rq

w i t h boundary.

of arcs b e t w e e n

y is c a l l e d a geodesic

of M in a n e i g h b o r h o o d

Prop.

class

possibly

to be convex.

arc is ~eodesic.

curve

point q 6 y there

surface,

it is assumed

found the two-

geodesic

arcs

in their h o m o -

to y i e l d a g e o d e s i c

13

arc c o v e r e d

twice.

furthermore, topy class

The e n d p o i n t

the g e o d e s i c

f r o m p to q2

q2 t h e r e f o r e

arc is the s h o r t e s t

is

conjugate

connection

to p , and in its h o m o -

" q.e.d.

F o r the p r e c e d i n g

2.2.

Convexity

argument,

cf.

of the s q u a r e d

[B1], p. 231.

distance

For a function h on a Riemannian

function

manifold,

its s e c o n d

fundamental

form

at a p o i n t q is d e f i n e d v i a

D2h(q) ( Y , Z ) =


grad h(q),Z>

,

Y where

Y and Z are t a n g e n t v e c t o r s

at q a n d D v d e n o t e s

ferentiation

in the d i r e c t i o n Y .

Lemma

Suppose

2.2:

in a R i e m a n n i a n center.

-~

B(p,r)

manifold

M which

F o r the s e c t i o n a l 2


2

= {q 6 M: d(p,q) is d i s j o i n t

curvature

< r}

covariant

dif-

is a @ e o d e s i c b a l l

to the cut l o c u s of its

K o__n_nB(p,r)

we a s s u m e

,

and r < 2-~ We d e f i n e

r(x):

= d(x,p),

Then

f £ C 2 , and

a)

]grad f(x) I = r(x)

b)


Proof: If q(t)

a)

- ctg
follows

is a c u r v e

c(s,t)

f(x)

= ~Id

• IY] 2 S D 2 f ( Y , Y )

f r o m g r a d f(x) in B(p,r)

we have

Thus

= - ~--~ c(s,t)IIS

O

< ~r(x)

c o t h er(x)

IYl 2

= -eXPx1(p).

w i t h q(O)

= e X p q ( t ) (s eXpq1(t)p),

g r a d f(x)

(x,p) 2

= x and q(O)

= Y , putting

14

Dy grad

For each

t , Jt(s)

arc c(-,t)

The J a c o b i

D ~s c(s't) Is = 0 ~t

_

D ~ c(s,t) ~s ~t = ~

f(x)

= q(t)

= Dj(O)

a n d Jr(1)

(A 4.1) may

s i n c e we a s s u m e b o u n d s

Uniqueness

of g e o d e s i c

2.3:

face,

is t o p o l o g i c a l l y {v:

Suppose

Proof:

map.

Clearly,

= 0

f i e l d a l o n g the g e o d e s i c

. Therefore

= - J(O).

and

(A 5.1)

from

[K2] n o w i m p l y b).

also be a p p l i e d to t a n g e n t i a l 2 _ 2 ~ K g <

Jacobi

a r c s in c o n v e x discs

B(p,R):

= {q £ Z: d(p,q)

a d i s c for some

Ivl = r} = ~B(p,r)

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

is the J a c o b i

g r a d f(x)

field estimates

Lemma

eXpp

c(s,t)

that these estimates

fields,

2.3.

=

w i t h Jt(O)

Dy g r a d

Note

f(x)

R < ! K

_< R}

(K _< <2).

for all r _< R , w h e r e

Furthermore,

~B(p,r)

~B(p,r) c__eXpp

{v:

, where Then

eXpp

is convex,

Z is a sur-

: TpZ + Z is

if

r < 2-~ "

Ivl = r} c__B(p,r).

We a s s u m e n o w t h a t

(2.3.1)

eXpp

e X p p is a local

{v:

o Ivl = r} 0 B(p,r)

diffeomorphism

r e m of M o r s e - S c h o e n b e r g eXpp

[v:

(cf.

{v:

Iv I < --~}
[GKM], p.

Iv I = r} is compact,

= r} w i t h m i n i m a l

distance

y from p to q is o r t h o g o n a l < r . On the o t h e r hand, , t £ [O,1]

different

{v:

176),

Ivl = r} is an immersed, s m o o t h c u r v e

Since eXpp

expptW

on

~

to p

and therefore for r < ~-< .

w e can find some q 6 e X p p

. Consequently,

to e X p p

q = eXppW

{v:

the s h o r t e s t

Ivl = r} at q and has

{v:

f r o m y , s i n c e its l e n g t h

to e X p p

{v:

is p r e c i s e l y

length

r . Thus,

together

to see that e v e r y p o i n t i n s i d e

the f a c t that this o v e r the a r g u m e n t

to a g e o d e s i c

y and y'

c o r n e r at p . It is n o t d i f f i c u l t loop can be

j o i n e d to p by a s h o r t e s t

y' =

Ivl = r} at q and

h a v e an a n g l e of ~ at q a n d m a t c h

geodesic

Ivl

geodesic

, lwl -- 0 , and the g e o d e s i c

, is a l s o o r t h o g o n a l

theo-

geodesic,

loop w i t h this

in s p i t e of

loop m i g h t n o t be c o n v e x at p . Thus, we can c a r r y of Prop.

2.1 to a s s e r t the e x i s t e n c e

of a p o i n t p'

15

inside

this

loop which

Y'' • S i n c e

p'

comparison

theorem

I08f.).

mate

of L e m m a

on B ( p , r ) , a level

From

2.2b.

and

rank

proves

under

v 6 T Z with P eXpp injectively

this,

we easily

Therefore,

consequently,

s e t of a c o n v e x

see

if r S ~ ~B(p,r)

the

{v 6 TpZ

that every

is m a p p e d

, p.

has m a x i m a l

a shortest

geodesic

in c o n t r a d i c t i o n

on

we infer

V which

is

eXpp

noted

[KI]

to p w . r . t ,

r < -~ , this

This

since

hood

and

of M o r s e - S c h o e n b e r g .

Furthermore, above,

is c o n j u g a t e

6 B(p,r)

f i r s t claim•

: Ivl < ~} K

Iv I = r has onto

to the

, as

a neighbor-

its

image

(cf.

that we may

apply

the e s t i -

, then

= eXpp

{v:

f is

a convex

function

Iv[ = r} is c o n v e x

as

function. q.e.d.

Thm.

2.1:

Suppose

and

r < ~

(K S <2).

joined

by a unique

conjugate

Proof:

that

Then

B(p,r)

each

is a g e o d e s i c

pair

of points

ql

disc

' q2

geodesic

a r c in B(p,r),

a n d this

of L e m m a

2.3

apply

on a s u r f a c e ,

6 B(p,r)

arc

is

can be

free

of

points.

By v i r t u e

would

exist

would

find a point

i.e.

now,

two

an arc o f

Schoenberg

geodesic

arcs

joining

q3 c o n j u g a t e

length

Theorem

ql

a n d q2

to ql w . r . t ,

S 2r < !

(cf.

, we could

p.

2.1

, if t h e r e

" Consequently,

a shortest

. This would

[GKM],

Prop.

contradict

we

geodesic

arc,

the M o r s e -

176). q.e.d.

2.4.

Remark:

Thm.

2.1

B(p,r)

pertains

to

is d i s j o i n t

however,

since

dimensions.

2.5.

The h i g h e r

Cf

Curvature

the [J2]

arbitrary

to the

one of

can

assign

c , thus

a point to c

of p a r a l l e l

through

(2.5.1)

q which

double

to c on M sign

is the

if w e s u p p o s e

is

clearly

a priori

that

is d i f f e r e n t , restricted

to two

surface

. One

case.

points

locally

distance

tangent

function

through vector

to c I . If Y

grad h(q),Z>

= O

on one of

q which

of the

, Z £ TqM,

M

c divides

h on a n e i g h b o r h o o d

c I the c u r v e


on the

, and since

to this

unit

is o r t h o g o n a l

=

2.3.

. The proof

in the g e n e r a l

a C 2 function

D2h(q) (Y,Z)

2.1

of

curves

neighborhood,

g r a d h(q)

of p

of Prop.

without

a negative

obtaining

in this

, then

dimension,

for a p r o o f

the d i s t a n c e

analogue

cut l o c u s

argument

L e t n o w c be a C 2 c u r v e can m e a s u r e

dimensional

M

,

side

c . If q is is p a r a l l e l

geodesic

we have

16 in c a s e

Y o r Z is o r t h o g o n a l

to cI

and

(2.5.2)

D2h(q) (Y,Y)

in c a s e

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

Lemma than

2.4:

Suppose

o-~ (A2:

(2.5.3)

= max

= < g ( C T ; q ) IYI 2 ,

c is g e o d e s i c ,

For h

, we have

(2.5.4)

<~(h)

geodesic

further

I

IYI 2

= <2(h)

away

from

for Y 6 T p M

curvature

< (h) of the p a r a l l e l g differential equation

the

curvature.

c

Then

-< A l t a n ( A h ( q ) )

the g e o d e s i c

at d i s t a n c e

p £ M is n o t

IKI) o r a cut point.

]D2h(q) (Y,Y)I

Proof:


to c I , w h e r e

.

curves

of c

+ K(h)

g

a n d thus

I

I~ a r c t g

and since

(~
by a s s u m p t i o n

<

g

(0) = O

(2.5.5)

ling(h) I _< AItan

and

(2.5.3)

now

2.6.

Local

In this

follows

coordinates

section,

trolled

Christoffel

without

using

following We s h a l l

in o r d e r In c a s e of T Y q

to be d(p,q)

(2.5.2).

controlled

to i n t r o d u c e

symbols

coordinates

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

ourselves

from

to the

again

of

of the g e o m e t r y

is t a k e n

suppose

and

Christoffel with

[JKI ] a n d w o r k s

twodimensional

t h a t B(p,M)

is

able to a p p l y Thin. 2.1. I < ~M , we t a k e an a r b i t r a r y

curvature

a point

outside

symbols

q £ B(p,M),

B(p,M). in a n y

case,

con-

The

dimension.

however,

for

a disc with

M < 2-~ '

orthonormal

base

e I , e2

.

If d(p,q) with

We

,

curvature

any i n f o r m a t i o n

restrict

,

(2.5.1)

with

construction

simplicity.

Ah]

from

we w a n t

~ 1 ,

the

geodesics

> 1M

, we

geodesic eXpp

arc

(t-e i)

choose

e I and e 2 in s u c h

f r o m q to p is ~ stay

inside

B(p,M)

a way

. We n o w w a n t for t _< t O

that

their

to s h o w , where

angle

that

the

t o > O can

17

be e s t i m a t e d Indeed,

d(p,

where

from below

b y the

the

e x p q t - e i)

right

< de(T,

hand

side

in the p l a n e

of

having

an a n g l e

again

in terms

of

Rauch.- T o p o n o g o w

cosh(ed;(p,

constant

e x p ~ t - e i)

is the

of ~ w i t h

< cosh

et

et

~t

cosh if t S t

the g e o d e s i c s

can

terms

expqte i

([JKI]):

define

triangle

= d(p,q),

form ~ to ~

ei

. Consequently

-

- sinh(ed(p,q))

• sinh

of e

eM

(~,~M)

<

~M -

. sinh

eM

,

,

stay

local

inside

coordinates value

, < , M only,

I h. (s) : 2t l

and

B(p,M)

~ M

consequently

for t S t o

B(q,T)

for w h i c h

, and

N B(p,M)

the

T > O is b o u n d e d

of q 6 B ( p , M ) ,

Christoffel f r o m below,

symbols both

are

in

via

(d 2 (s, e x p q o

By L e m m a

, d(p, e x p q t e i)

In a n e i g h b o r h o o d

in a b s o l u t e

Proof:

194~

, say.

for t S t o = m i n

bounded

p.

I

ifts~M

we

[GKM],

cc~parison

• cosh(ed(p,q))

- 1 s i n h et 2

2.5

(cf.

de(~,~)

the g e o d e s i c

< cosh

Lemma

in the

_ 2 , with

I - ~ sinh

Then,

Theorem

,

distance

curvature

expqtei))

e and M.

Comparison

t

o

e. ) - d 2(s,q)) i

.

2.2,

~ M coth ~M lD2hi (s)] ~ %-

(2.6.1)

o if d(s,q)

I < ~ M,

(2.6.2)

dhlq

and

is an i s o m e t r y

where

h =

(h I , h2):

This

easily

implies

B(p,M)

,

÷ R2

a lower bound

T for the

radius

of the s e t on w h i c h

h is i n j e c t i v e . Furthermore,

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

symbols

are g i v e n

by D 2 h

, namely

,

18 2 Dx,yh

(2.6.3)

=
h, Y > = ddh.xdh. Y - d h ' D x Y

= ddh.xdh'Y

- dh-dxY

=-dh-r(X,Y)

since

dh is linear.

Thus,

Lemma

2.2 a l s o

- dh-F(X,Y)

,

implies

the b o u n d

on

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

symbols.

q.e.d.

Remark:

In c h a p t e r

regularity

3. 3.1.

mappings

Statement

of Thm.

Theorem

surfaces

3.1:

Suppose G bounded

coefficients

Then

S admits domain

bounded

(here

(x,y) are

denote

y can be normalized one

of the b o u n d a r y

ly,

to t h r e e

ken

as the u n i t

given

or

C~

(B),

with

better

representations

of

domain

boundary•

a chart

homeomorphic

to a

~: G ÷ S. S u p p o s e

the

of S can be d e f i n e d in this c h a r t 2 g11 g22 - g12 ~ I > 0 i n_n G . T 6 H I2 N C a (B,G),

representation

•

T x

respect

and ~ satisfies

Ty

by

where

almost

B i~s

everywhere

= 0

curves points

point

the o u t e r

£ M,

, then

also

if S is at l e a s t

satisfied

and pro-

everywhere,

points

on

respective-

of B w h i c h

T is as r e q u l a r

0 < ~ < 1) or C ~

are

three

c a n be

i m a g e of an i n t e r i o r

reqularity,

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

relations

and n o r m s

to c o r r e s p o n d ,

boundary

f i x i n 9 the

higher

in B,

of S.)

condition t namely

of S can be m a d e on

circle, or by

ck•~(k

of p o i n t s

to the m e t r i c

by a three

respectively.

the c o n f o r m a l i t y

even

gij w i t h

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

concerning

if S is of c l a s s

conformal

via

by k c i r c l e s

,

taken with

Furthermoref

tensor

with

relations

l~yl 2

=

ducts

k circles

functions

coordinates

coordinates.

to p l a n e

S is a s u r f a c e by

a conformal

the c o n f o r m a l i t y

l~x 12

concerning

of the m e t r i c

measurable

a plane

3.1

obtain

harmonic

homeomor~hic

domain

bounded

shall

namely

Conformal

compact

plane

10, w e

properties•

ta-

point. as S

, i.e.

Y £ ck'~(B)

C 1'~

•

then

a n d T is a d i f f e o -

morphism. Remark:

For

an e x p l i c i t

interior

C 2'~

estimate

for ~ , cf.

Cor.

10.2.

Ig

For h i g h e r

regularity,

In his m o n o g r a p h faces of class obviously

this

surfaces

case

together w i t h

case,

[MI]

theorem

Morrey

is not correct,

to p r o v e

constructed

and this the one -

[Lv]

metrics). this

(for

(cf.

result with

to the u n i f o r m i z a t i o n

the global

theorem

an energy m i n i m i z i n g to o b t a i n

theorem

directly

of the m i n i m i z i n g

claims

implicitly

by a

as ~:

368,

counterexamples

to me, why ~°T:

G ÷ S which

if w e do not know

curve

a priori

however,

that,

if the

b e l o w by e ,

of this disc is at to this assertion.

B ÷ S represents

is n e e d e d

from

limit map

sequence,

on p.

HI 2 map on a disc is b o u n d e d

to c o n s t r u c t

it is n o t c l e a r

sequence

a conformal

of the image of the b o u n d a r y

surface

m a 9.3.7,

he

of a c o n t i n u o u s

It is easy

Also,

for

in case k = I

onto G , in order

since

then the length

Fr~chet

[Li],

theorem yields

(for m e a s u r a b l e

did not appeal

His proof of the e q u i c o n t i n u i t y

least e/2.

of the P l a t e a u p r o b l e m

and one can again c o m b i n e

Morrey

domains

oscillation

is

method.

In his proof, circular

restriction

from the fact that M o r r e y w a n t e d

the u n i f o r m i z a t i o n

r e s u l t and tried

variational

(the latter

for sur-

3 - space).

and M o r r e y

details),

however,

or a local

theorem

a local r e s u l t was p r o v e d by L a v r e n t ' e v

metrics)

for m o r e

[M3],

3 - space only

investigation

in E u c l i d e a n

the u n i f o r m i z a t i o n In

in his

a corresponding

(k = I).

In the g e n e r a l continuous

and stems

10.3.

a local v e r s i o n was p r o v e d by L i c h t e n s t e i n

local v e r s i o n contour

states

C I'~ in E u c l i d e a n

theorem

In this case,

[BJS]

[M3], M o r r e y

superfluous

to apply minimal

see also Cor.

the same

in his proof

to apply Lem-

that • is a u n i f o r m

limit of dif-

feomorphisms. (Finally, C 1'e)

Morrey

culties

mality

I shall

and even

s equ e n c e

show,

simplify

importance

yet

to this q u e s t i o n

sketch

class

the m i n i m i z i n g

to show

idea shall also

these diffi-

the confor-

turn out to be of

can be taken over

directly

from

[M3], we

these.

The C o u r a n t - L e b e s g u e

In the p r o o f of T h e o r e m due to L e b e s g u e

proof by c h o o s i n g

large enough

(A similar

overcome

in later chapters).

some of the arguments

shall only

come back

how one can easily

Morrey's

in a restricted,

of the limit map.

crucial

We shall

(in case S 6

3.5 and 3.6.)

In the sequel,

3.2.

that • is a d i f f e o m o r p h i s m

or at least a homeomorphism.

in s e c t i o n s

Since

did not prove

Lemma

3.1, we shall make

and C o u r a n t

(cf.

[Co], p.

use of a well - k n o w n 101).

lemma

20

We s t a t e it h e r e since

in a m o r e g e n e r a l

f o r m than we n e e d for this chapter,

this form w i l l be n e e d e d in s u b s e q u e n t

S u p p o s e ~ is an o p e n s u b s e t of some of c l a s s

Lemma

C 3 , while

S satisfies

for the c u r v a t u r e

K of ~ , ~ < m i n

Then

some r 6

there e x i s t s

ly c o n t i n u o u s

(d,/~)

(1,i(E) 2

u(x2))

coordinates

We i n t r o d u c e

rr

G

(cf.

(3.2.1)

polar

[B1], p. 153),

G(r,B)

N o w for x I , x 2 6 ~B(xo,r)

(3.2.2)

I/I 2 )

u]DB(xo,r)

N ~ is a b s o l u t e -

_< 4 ~ - D I/2. (log I/6) -I/2

Proof:

function

r

for w h i c h

D

"G

uI~B(xo,r)

d(U(Xl),

on B(xo,r),

we

.

and a l m o s t all r ,

u(x2))

ds 2 = dr 2 +

infer

~ I/I s i n h l r

is a b s o l u t e l y

i.e.

continuous

since u is a S o b o l e v and

2~ S S ~8 (x) id8 O

< 2z

w h e r e we a s s u m e d w . l . o . g . The D i r i c h l e t

integral

E(u;B(Xo,r))

lusl 2d8

B(Xo,r)

c ~ .

of u on B(xo,r)

= I/2 ~ (fUrl2 + I B(xo,r) V

Thus, we can f i n d some r 6

(3.2.3)

2z o~ luB (r'8)I2d8

is

luSl

2)

G dr d 8

(~,/~) w i t h 2D I

< /6

2D -<

~-C~7~7, G e ) d~ sinCe

for r < /[ < I/I

The l e m m a

3.1.

~ D , x ° £ ~ , -I 2 a l o w e r b o u n d

for all x I , x 2 £ ~B(xo,r)

Since K -

manifold

as in Thm.

and

d(U(Xl),

G2(r, 8)de 2

Riemannian

the same a s s u m p t i o n s

I L e t u 6 H2(~,S) , E(u)

3.1:

chapters.

twodimensional

follows

from

, G(r,8) (3.2.2)

_< 2r by and

(3.2.1).

(3.2.3). q.e.d.

21

3.3. By

Proof

of

Theorem

assumption,

there

to G.

Using

outer

boundary

satisfies We

then

the

three

define

circle,

onto

let ~ be of

maps

t

We

by

G

the

assume

be removed

of

~

T 6 ~ By

in

show

terms

find

(3.3.1)

'

that

all

Now

~B(xo,r)

P3

from

only.

S into

two

B

B(Xo,r)

the

three

the

as

zero,

6 aoes

to

equivalent Since has

estimate choose

B n are

E ( ~ n)

implies point

that

selection ting

of

domain T

n

are

the

one

unit

is a c i r c u l a r uniform

do-

limit

of

on

Of

_< D

a map

assumption

p: B + G as

of

continuity

of

any

, say.

every

6 B

x°

and

6 < I , we

1/8) - I / 2

and

a

one

Likewise

of

at most

limit

reaion and

reeien

D B) one

of

the

Pl

' P2

'

divides the

points

diffeomorphisms.

aoes

in S

uniformly

the

, and

to

zero

is u n i f o r m l y

small we

region

obtain

T n : B n ÷ G in ~

seguence required

hence

the

type. T n are

of

the

domains

to a n o t h e r

one

in t h e

the

where

points

in B

the

de-

continuity.

the

and

circle

of

its m e t r i c

diffeomorphisms,

of

region, the

~(~B(Xo,r)

containing

small

of

D,

"large"

at most

minimizing

required B

of

This

of G.

small

domains

the

E(T)

one

of

finite

tangent

defined

B

the modulus

S is c o m p a c t

a subsequence, B of

the

1,2)

• is a u n i f o r m

the

no b o u n d a r y

or b e c o m e

bound

contains

the modulus

some

a circular is

continuous.

-< 2 ~ ' D I / 2 " (log

limit

circular

~ D for

from

I and H2

energy

3. I , f o r

small

since

energy

are

(~ =

condition.

the

into

gij

the

Let

Euclidean

of an

the

condition.

the weak

into a"small" D B

T is a u n i f o r m

now

the

all

the

that

n B.

diameter

to b e m a p p e d

sired We

to

assume

B on-

with

T(pl) , T(p2) , T(p3) , s i n c e Furthermore,

the

Lemma

point

regions,

t is

can

T(x2))

divides

domain

diffeomorphism

of which

B + G where

'

x I , x 2 6 ~B(Xo,r)

region

can

this

boundary point

p3dx

we

(~,~J~)

d(T(Xl),

for

small

r 6

t:

define

piD

of E(T)

some

we

that

diffeomorphisms

outer

that

We

the Courant - Lebesgue

can

circular

.

in 3.6.

to

and

three

and

a moment

I

now want

all

all maps

E(p) = ~ f g i j (p (x) )D

We

of

the

considered

for

circle

the

, satisfying

type

some

condition.

class

k circles,

class

of

transformations,

the unit

point

as

: B + G from

n

now

will

D

the

the

M~bius

of B is

bounded

main

is a d i f f e o m o r p h i s m

elementary

domain

We

3.]

B n then

type,

have

and we

We

can

aaain

assume

equicontinuous.

B n can limit

as

collaps n + ~.

to c o n v e r g e

can

, where

assume

to

to a After

some

w.l.o.g,

This

limi-

that

22

ThUs,

a subsequence

T 6 ~

. Because

with

respect

this point

that,

as a m a p p i n g

[Co],

we

on

the r e a l

nor

that

(Note f o r

S , the energy

integral

the

by

comparison

maps, map,

T minimizes

i.e.

pp.

of T n

of

, regarded

o f S).

Fur-

~o

that

T is w e a k l y

, where

id:

o f B1

with

secondly

after

B 1 ÷ B is a

B depending

B + B

We observe the

is t h e u n i t

a conformal

that

~i:

B 1 onto

t h a t ~i p r e s e r v e s

boundary

con-

369 - 372).

• • ~I domains

, with

that aaain

T o ~I

d ~I d--~ E(T o

(3.3.2)

I

composition

E in ~

[M3],

T with

to r e a u i r e

o f T o ~I a n d

a conformal

or

argument

from circular

the outer

ging

Since

for

integral

in ~

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

standard

compare

parameter have

can be achieved energy

the

169 - 178,

shall

this

sible

using

pp.

this,

that we don't

condition

with

G , because

of diffeomorphisms

smoothly first

from B n onto

energy

domain

to t h e D i r i c h l e t

to a l i m z t

of the Dirichlet

T minimizes

t a k i n g G as a c o o r d i n a t e equivalent

to show,

(cf.

To a c h i e v e family

I and uniformly in H 2

weakly

semicontinuity

T is c o n t i n u o u s .

We now wnat formal

lower

I HQ convergence,

to w e a k

~n is u n i f o r m l y

thermore,

converges

of t h e

three

point

circle,

since

map without

therefore

chan-

T ° ~I a r e a d m i s -

a possible

composition

6

, we

)Ii= O

infer = O

We put

E:

~T i ~ J ~x ~

= gij

(note t h a t

, F:

= gij

~T i ~T j ~x ~y , G: = gij

I , these expressions T £ H2

since

~T i ~T j ~y ~y

are d e f i n e d

(only)

everywhere),

Of:

=

-~-II=O

[

+

iq

=

v

+

ie

.

Then

E(T)

= 1 S

(E + G ) d x d y

B

and

E(To

a~l )

=~

1 Bj. {E((y2 + qy2) (~xqy - ~yq x ) - I d x d y

,

almost

23

and

(3.3.2)

(3.3.3)

thus

implies,

f(E - G ) (~ B

x

noting ~

-ey)

o

(z) = x + iy ,

+ 2F(~y + e x ) d x d y = O

P u t t i n g ~: = E - G - 2iF , (3.3.3)

means

Re f ~(9 + i e ) z-~xdy = 0 B

Replacing

v + ie by e - i9 , w e see t h a t the i m a g i n a r y

vanishes,

hence

(3.3.4)

I ~ ( 9 +ie)z-dxdy

part likewise

= 0

B

First we observe with

compact

(3.3.5)

t h a t in

support

~(

): = x + 19(z)

t h e n is for s m a l l Hence

(3.3.4)

(3.3.6)

(3.3.4)

~z = O

Yj,r:

in B

s m o o t h e and

the r e q u i r e d

properties.

i.e.

.

circles

(z,yj)

~ and e w h i c h

(j £ {I, ....,k}).

with

t h a t ~ is h o l o m o r p h i c ,

= {z 6 B: d i s t

at v a r i a t i o n s

arbitrary

+ i(y + le(z))

III a d i f f e o m o r p h i s m

implies

L e t y 1 , . . . , y k be the b o u n d a r y resp.

we can insert

in B , s i n c e

of B , with

= r}

(3.3.3)

zl,...,z k ,

(j 6 {I, .... k}). W e n o w look

are s u p p o r t e d

Integrating

centers

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

by parts

o f ¥j

(noting ~ £ L I ( B ) ) ,

we

obtain

(3.3.7) Choosing

lim ~ (E-G) (~dy+edx) r÷O Y j , r variations

which

translate

+ i~ = a 3. + ib.3 = const,

(3.3.8)

f

~(z)dz

n e a r ~j

= 0

+ 2F(edy-gdx)

= O

the c e n t e r

z. , i.e. p u t t i n g 3 (and o I aaain_ as in (3.3.5)), w e g e t

(r > 0 s m a l l enough)

Yj,r If w e a p p l y a h o m o t h e t i c (ej + i f j ) e i8

dilation

(ej + if 3. = const.)

of yj i n s t e a d , n e a r yj

, where

i.e.

c h o o s e ~ + ie =

8 parametrizes

we obtain

(3.3.9)

f Yj,r

(z - z j ) ~ ( z ) d z

= O

(r > 0 s m a l l enough)

yj

24

Thus

we

tion, This

see

the

that

implies

~"(z)

Since

since

that

exploit

fore

can

thus

put

there

periods

is an a n a l y t i c

2 ~ £ HI(B),

that

compose

simplicity

(3.3.7)

¥j w e r e

subject

to v a r i a -

to vanish.

function

then

(3.3.10)

and h e n c e

T solves

T with

~ on B w i t h

~ is c o n t i n u o u s

a free b o u n d a r y

diffeemorphisms

+ ie = A ( 8 ) i e i8

For

circles

of ~ h a v e

= ~(z)

~ 6 LI(B),

We n o w

the b o u n d a r y

corresponding

near

of n o t a t i o n ,

we

value

rotating

on B

.

problem.

yj

We

there-

into

itself.

= {Izl

= I}

We

yj

assume

j = 1 a n d YI

implies

lim

f A(8)

Im

(~"((1-r)eiS)e2iS)d@

= O

r÷O 7 1 , r Since put

~' £ H] I (B) , ~'

r = O after

O = Im

has

boundary

an i n t e g r a t i o n

by

values

in LI(3B) , and h e n c e

parts;

thus

w e can

f ~ ' (ei~)(iei@A ' (8) - e 1 8 A ( e ) ) d @ Y1

=Im

S ~ ' (eiS)ieiSl ' (0)d8

-Im

Y1

= Im

~'

(e1~)el~(A(O)

2~ f A ' (e) (ieie~ ' (e i0) o

-

o

2~ f ~ ' (ei~)ei~d~)d@ o 2~

+ Im

Since

the

second

t e r m in the

arbitrary

smooth

function,

(3.3.11)

along

YI

z~' (z)

- ~(z)

we

infer

O = Im(iz

or for the

~z

general

(iz@'

(3.3.11)

- i@))

boundary

7j

(e i~0) d~)

vanishes

across w.r.t

and A(8)

was

an

o f Du Bois - R a y m o n d

on YI

YI

, and thus

O , we

= -Ira z2~0(z)

curve

d

S o

f r o m the L e m m a

= const,

is a n a l y t i c

. Differentiating

(iA (0)

last e x p r e s s i o n

Im(iz@' (z) - i@(z))

Therefore,

+~A' (~)dS)d~

YI

,

obtain

@ is s m o o t h

on YI

25

(3.3.12)

O = I m { (z - zj)2%0(z)}

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

~ can be a n a l y t i c a l l y

N o w let fj(8)

=

Furthermore,

for z E 7 j

(z - zj)2~(z)

from

(3.3.9)

continued

on y j . By

and

across ~B

(3.3.12),

fi_ is real on yj

.

(3.3.8)

2~ (3.3.13)

o~ fj (0)d8 = 0

2~f.(9)

(3.3.14)

o

Since

=

]

27 ~fj(8)

Denote

the

sin @d@ = 0

o-

f. is 2 7 - p e r i o d i c , (3.3.13) a n d 3 ~ has at l e a s t 4 zeros on yj

hence

imply that f

(3.3.14

3

and

zeros of ~ on ~B by z s and let

B : = B~/(B P s Let

cos @ d 8

p be so s m a l l

D B(Zs,p))

(p > O)

t h a t the b a l l s

B(Zs, p) c o n t a i n n o zeros of ~ o t h e r

than

z s The n u m b e r o f zeros o f ~ i n s i d e B

is n o n n e g a t i v e

and given by

P n = ~I

(3.3.15) Here, As

~B

P in such a w a y t h a t Bp is to the left.

is o r i e n t e d

P p ÷ 0 , the c o n t r i b u t i o n

is -I/2

for e a c h

l e a s t 4k zeros B(z

~HS d log

z

s On yj D Bp

27i

Since

s

zs

,p) is a l w a y s

by

(Note t h a t the o r i e n t a t i o n

I d log ~jRBp

(z - z j ) 2 ~ ( z )

S d log

, since

in

there

(3.3.15) are at

of ~Bp is such t h a t

is g i v e n by

((z - z j ) 2 ~ ( z ) )

is real o n yj

-

, the

S d log YjnBQ

(z - z j )

2

= -2

for j = 2 , . . . , k

= 2

for j = I

(because of the o r i e n t a t i o n we have

in

of ~B ). P as p ÷ O

(3.3.15),

(z - z j )

first integral

the s e c o n d one gives

YJ

Altogether,

at least -2k

to the right).

, the c o n t r i b u t i o n

p + O , while

~B(Zs, p) D B to the i n t e g r a l

, or a l t o g e t h e r

tends to z e r o as

28

1 n = 2--~

which

is a c o n t r a d i c t i o n ,

The only

possibility

~o --

i.e.

f d l o g ~ S - 2k + 2(k-1) 3B P

0

since

which

- 2 S -4

n is n o n n e g a t i v e .

avoids

this

contradiction

is t h a t

,

that

T is w e a k l y

conformal.

Thus

l~xl2 = 1~yl 2

~d

(3.3.16) Tx

almost be

, Ty >

everywhere

in B

= O

(so far,

these

expressions

are o n l y k n o w n

to

in L I ( B ) ) .

(If S is p a r a m e t r i z e d metric

tensor

by

local

coordinates

(gi~),j the e q u a t i o n s

•

(3.3.16)

I

, T

take

2

with

the

corresponding

form

g..TIT 3 = g..~iY 3 13 X X 13 y y q. • I~3 = O --l3 X y for almost

all points

To r e p e a t ,

since

only

variations

domain,

we

riational namely

~

(x,y)

this p o i n t arising

can conclude problem,

.

6 B c ~

2

.)

is of c r u c i a l

from composition that

although

importance: with

T is a s o l u t i o n we work

Since we need

diffeomorphisms

of

of t h e u n c o n s t r a i n e d

in an a p r i o r i

restricted

the va-

subclass,

27

3.4.

Proof

of T h e o r e m

3.1,

continued

For notational convenience, I 2 u = T • V = T We want

to e x p l o i t

that

l i m i t of d i f f e o m o r p h i s m s o f T has Here,

the

same

with

arguments

the

the sake

Def.

3.1:

in o r d e r

difficulties

to b e of class

call

of L e m m a t a

the

local

that

everywhere

in B

arise

the

from

9.2.4,

these 9.2.5,

we p r o v i d e

G is a p l a n e

coordinates

(anti)conformal

to s h o w

C ° N H I2 , b u t

of c o m p l e t e n e s s ,

Suppose

now

T is w e a k l y

sign almost

additional

ly k n o w n

For

we

domain

of

and

the u n i f o r m

the J a c o b i a n (cf.

9.3.7.,

fact

that

problems

on S

u v - u v x y y x l.c.~.

• so far is on-

can be overcome

l.c.

all details. C I , ~0 6 C I ( G , ~ 2) ,

class

z ~ %0(~G). Then m(z,~(~G))

is d e f i n e d

to be

the w i n d i n q

number

of

the c u r v e

%0(~G)

w.r.t.z. If only

~ 6 C ° ( G , ~ 2),

then

m ( z , ~ ( ~ G ) ) : = lim m ( Z , ~ n ( ~ G ) )

for any s e q u e n c e

~ n 6 CI ($G,R 2) w h i c h

That m(z,~(~G)) ties

of w i n d i n g

Lemma Then

3.2:

where

@

numbers

G a plane

for e v e r y I

is w e l l (cf.

defined, e.g.

is I - d i m e n s i o n a l

follows

uniformly

to ~ on

from elementary

~G.

proper-

[Fe]).

I ~ 6 C ° N H2(G,~2).

domain,

x ° 6 G , there

converges

exists

Hausdorff

a set C ( x o) w i t h measure,

such

~I (C(Xo))

that

for all

= 0

,

R

C(x o)

J (£0)dx = B (Xo,R)

if B ( x

,R)

cc

G

~ m (z, ~0(~B (Xo,R)) dz ~0(B (Xo, R) )

.

O

I 2 I 2 (J(~) : = ~ x ~ y - ~y~x) Proof: formly on

We and

can

strongly

~B(Xo,R),

Since

I

find

a sequence

6 C I (D),

Dcc

G , converging

I in H 2 to ~0 , so that ~0n ÷ ~0 s t r o n g l y

if R ~ C(Xo),

H 2 (~B(Xo,R))

~n

I

functions

(C(Xo))

uni-

I in H 2 ( $ B ( X o , R ) )

= O.

are a b s o l u t e l y

continuous,

and

the

lengths

28

of ~n(~B(Xo,R))

and <0(~B(Xo,R ) are u n i f o r m l y

nal m e a s u r e

of ~(~B(Xo,R)

q0($B(Xo,R))

for a l m o s t

vanishes

(3.4.1)

m(z,~0n(~B(Xo,R))

bounded,

(R£C(Xo)).

the t w o d i m e n s i o -

Consequently,

z

all z , and thus ÷ m(z,~(~B(Xo,R))

for these

z .

Now lim J m (z,%0n (~B (Xo,R))dz n÷~ %0n(B(Xo,R) )

= lim ~ J(~°n)dX = 5 J(q0)dx n+~ B (Xo,R) B (Xo,R)

I/2 Since

]-m(z,<0n(~B(Xo,R))d z __< (meas I) I

for any m e a s u r a b l e

length

set I , we can i n t e g r a t e

(~0n(~B(Xo,R)),

(3.4.1),

and the r e s u l t

follows. Lemma

3.3:

We s u p p o s e

that • (n6~) n in H 2I to ~.

uniformly

and w e a k l y

Then J(~)

has the same

Proof:

Let B(Xo,R),

sign a l m o s t R ~ C(Xo)

en:

= max l~n(X) x£~B (Xo, R)

Vn:

= {z:d(z,~(~B(Xo,R))

3.2 t h e r e f o r e

(3.4.2)

Thus,

satisfy

the a s s u m p t i o n s

of L e m m a

3.2.

> e n} = m(z,<0($B(Xo,R))-

lim S j (<0n) = S J(<0) n÷~ %0ni - (Vn) nB (x ° ,R) B (Xo ,R)

holds

w.l.o.g.,

for a l m o s t

T is a w e a k

equations,

(3.4.3)

everywhere.

implies

Since we can a s s u m e (3.4.2)

converging

- <0(xl I

For z 6 V n , m(z,<0n($B(Xo,R)) Lemma

are d i f f e o m o r p h i s m s ,

J(~n ) ~ 0 in B(Xo,R)

all discs

solution

B(Xo,R),

of the c o r r e s p o n d i n g

~.e. -I V x = -g22(g12Ux

+ k/g Uy)

for all n , and

the r e s u l t

follows.

Cauchy -Riemann

2g

V y = g22 -l(k/g 2 - g12 ), w h e r e

(g = g 1 1 g 2 2 Since

this

ty theory Theorem

U x - gl 2 Uy) k = ±I

is a f i r s t o r d e r and

3.1,

the u s u a l which

linear

bootstrap

concern

if S is at l e a s t

of

satisfies

and

elliptic argument

the regularity

In p a r t i c u l a r , (3.4.3)

is c o n s t a n t

C 1'e,

in B b y L e m m a system,

imply

3.3.

elliptic

those

regulari-

assertions

in

of T.

then

the c o n f o r m a l i t y

T is a c l a s s i c a l relations

solution

(3.3.16) e v e r y -

where.

3.5.

Proof

In o r d e r

of Theorem

3.1,

continued

to s h o w t h e u n i v a l e n c y

of

T , we

have

to p r o v e

several

lem-

mata.

Lemma

3.4:

tions

of T h e o r e m

Proof:

Suppose

We assume

tive,

i.e.

T(z2).

Then

that

there must

We choose

Since

S is of c l a s s

3.1.

C 2'e,

i.e.

9 6 C2'~(G)

y is a h o m e o m o r p h i s m

between

T is n o t a h o m e o m o r p h i s m .

Then

in t h e n o t a -

B and

T is n o t

exist

two points

z I , z 2 , z I ~ z2 with

a shortest

segment Yn

joining

T n is a h o m e o m o r p h i s m ,

Yn:

=

T n I (yn)

Yn

' then

Tn(Z 1) a n d

is a c u r v e

S.

injec-

T(z I) =

Tn(Z2).

joining

z I and

z2 . If P n ' @

is a p o i n t

subsequence the

of

Tn c o n v e r g e

Thus,

a whole

on

(pn,~)

~B(Zl,6)

N

converging

uniformly

continuum

to

for n ÷ ~ we can

to s o m e p o i n t

T , we

is m a p p e d

see t h a t

onto

the

P6 o n

T(p6)

find a

~B(z1,6).

Since

= T(z I) = T(z2).

single point

T(z I) = T(z 2)

by T • At

interior

points,

t i m e of c l a s s

C 2"~.

we

C 1 , and consequently fined.

From

(3.5.1)

We

shall

(3.4.3)

can choose

The

we conclude

local

metric

the corresponding

A u + F11 (u

employ

again

corresponding

coordinates tensor

Christoffel

u,

I

symbols

notation

z = x + iy

r22(v

v

, this

is o f c l a s s

that u and v are harmonic,

+ u ) + 2F 2 ( U x V x + UyVy)

the complex

then

can be dee.g.

x2+

, z = x - iy

v ) = O

, u Z

I / 2 ( u x - iUy), From

(3.5.1)

and

(3.4.3)

we obtain

[Uzz I < ClUzl _<

(3.5.2) since

etc.

u

6 C 2 (B).

30

If

now

U z ( Z o)

= O

(cf.

Lemma

Wintner

for

some

3.6

z° 6 B

below),

, we

to

can

obtain

use

the

the

Lemma

asymptotic

of

Hartman-

representa-

tion

(3.5.3)

for

some

a £ C

, a ~ 0

a neighborhood

of

plies

set

that

therefore a

the u v xy

surjective

such

, and

. The

where

- u v yx map

some

-u

in B

onto

S

- z o in)

positive

latter

u v xy

- 0

gij(T(Zo))

(3.4.3),

w(z):

p

+ o(lz

is

v yx

in

not = 0

integer possible,

is

nonvoid

contradiction

. We

can

n

choose

to

the

, unless

u z --- 0

however,

since

and

open

in B

, and

the

fact

that

T is

local

it

coordinates

(3.5.4)

= u

, a

= ~ij

+ iv

and

integrating

= p(z-z

£ R , I PJ

J~J

+

loss

of

generality,

formations,

we

(3.5.5)

w(z)

= zn+1

the

boundary

is

on

can

If

z

On

a neighborhood

o

(3.5.6)

and

v

O) n+1

# O

(3.5.3),

+ o(z-z

, w° =

O

we

infer

)n÷1 + o(Jz - z o in+l) + Wo '

(u + i v ) ( Z o ) ,

in

a neighborhood

(3.5.4)

assume real in

assume

V of

= 0

on

by p =

performing I

+ o~n+l

~B

linear

, z° = w o = O

trans-

, i.e.

+ o(jsjn+l)

, we

~(z O)

~S

, ~ > 0

homeomorphic

can

introduce

local

coordinates

u,v

with

DV

again

g i j ( T ( Z o ))

Performing

the

an

that

axis

im-

in

"of z

o Without

in

that

(3.5.4)

where

z°

C 2'e

a way

Using

- z O )n

u z = a(z

elementary

the (the

upper

=

boundary x -axis

half

plane,

in

6ij

M~bius

transformation

circle

of

our i.e.

B containing

previous x

> O.

notation) We

now

put

of

the

zo

domain,

is m a p p e d

and

B

is

we

can

onto

contained

the

31

u (_z)

when when

u(z): =[_u(z) whenever Since

defined,

(3.5.1)

e.g.

again

~

holds

Applying

{x = O}

again

z = x > 0 z < 0

on a neighborhood for u

N ( z o)

, we conclude

from

(3.4.3)

(since u 6 C I ( B ) )

lUz~l ~ clUzl ~ K o n N(Zo)

Re Re

.

Lemma

3.6 b e l o w ,

we again

obtain

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

(3.5.3) , i.e.

u z = a (z _ z ° )n + o ( i z _ Z o l ) n

and w(z)

(3.5.7)

after This

= zn+1

normalization , however,

of points

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

to E.

Lemma

3.5:

phism

between

Proof:

We

of the

Under

the

B and

shall

terior

shall use

Thus, chosen We

the

follows

assumptions

.

that

This

proves

point.

formula

of L e m m a

we have

namely

3.4

obtained

a whole the

con-

lemma.

in t h e a b o v e

argument

, • is a d i f f e o m o r -

.

an i d e a of B e r g like

functional

from Lemma the a r g u m e n t

[Bg]

to s h o w

(3.5.5)

determinant

with cannot

that

, if

T is a

n ~ I cannot vanish.

hold

Thus

the

3.4. only

at boundary

points,

since

the

in-

is s i m i l a r .

we assume

for c o n t r a d i c t i o n

normalizations,

infer

z° = 0

injective,

to a s i n g l e

a representation

present

case

S

of

to t h e c o n s e q u e n c e

Hartman -Wintner

[Hz7]).

and consequently assertion

T is n o t

Heinz

homeomorphism,

We

that

is m a p p e d

(The a p p l i c a t i o n is d u e

+ o(izln+1)

is in c o n t r a d i c t i o n

from the assumption tinuum

+ ~n+1

from

(3.5.7)

that

in p a r t i c u l a r

(3.5.7)

B c

holds

{x > O}).

(with all

the

32

u ( r e i8)

and

in

=

u(rei~

for

k

For

sufficiently

If

( 3.5.8

) is

(n+1)

therefore

=

+ o(rn+1),

r n+1

(I + ~)

(_i) k

+ o ( r n+1)

is

contained

in

6 so

small,

that

{w

-I

{w

the

y

curve

u z ~ O at

~ O},

with

a diffeomorphism

closed

line

~ ~}

~ O},

lw[

the

of

the

left

hand

side

upper

{x

= O}

, then by

lwl

u(z)

~

plane

3.4

{z

{x

~ O}

a neighborhood

changes

Lemma

~ 6}

half

in

sign

, we

at

can

= x + iy:

x

of least

choose

~ O

,

= 6/2}

properties

sign

, and

3.4

sign

y of

changes z°

in L e m m a

, the

a homeomorphism

preimage

, u(z)

the

real

6 (u + iv) {x

£ (u + i v ) { x

a Jordan

in

the

{Iz - Z o l

T is

Therefore,

proved

of

Since

and

verses

a piece

r ~ e

(-I) k. curve

times.

(u + iv)

is

((n+I)8)

~ > O and

a Jordan

contains

Thus

k/n+1)

small

z traverses

zO and

is

cos

= O,I,...n+I.

which

E

r n+1

particular

(3.5.8)

of

(I + ~)

by

, that of B

stated

exactly (3.4.3)

T is

onto

once, also

above.

However,

contradicting

vz ~ O

. Since

a homeomorphism,

we

n we

as

z tra-

~ I already

conclude

that

T

S.

q.e.d.

3.6. We

Proof

are

prove

of

Theorem

now

in

a position

that

we

and

that

For

this,

such tions

that 3.3

Tis

can

find

3.1, to

continued finish

a conformal

a diffeomorphism

in

we choose a sequence n gij c o n v e r g e s almost and

3.4

and

Lemma

the

proof

homeomorphism case

(g~j)

we

Theorem

3.1

T between

, i.e. B

to

and

S

,

on

G

,

S 6 C I,~ of

everywhere

3.5,

of

find

metrics to

gij

of as

class n + ~

a corresponding

C 2'~ . By

sec-

sequence

of

33

diffeomorphisms above

and

uniform

the p r o p e r t y domain

B

, and

Furthermore,

Tn c o n v e r g e

13611

the

of the

Yu

of

uniformly

n g11

n

gn

n n = g11g22

Therefore, regularity vertible,

and weakly

as

(3.4.3),

their

-I Yn

inverses

sa-

namely

n Xv

'

n 2 (g12)

-

also

T -I s a t i s f i e s a u n i f o r m H ~ i d e r e s t i m a t e b y e l l i p t i c n t h e o r y , a n d t h u s w e see t h a t the l i m i t m a p T has to b e in-

i.e.

a homeomorphism.

S 6 C I'~,

respect

to t h e C a - n o r m

then

(Tn) , w i t h

to a c i r c u l a r I in H 2 to a m a p T ,

n

In c a s e

-I T n map

by

n

n g22 n g12 YV - g/~ Xu - g~ where

denoted

theory~ )

B n converge

diffeomorphisms,

type

n

again

domains

regularity

(3.4.3).

Tn are same

n g12

n

by e l l i p t i c

find a subsequence,

solution

since

B n is a g a i n a c i r c u l a r d o m a i n as n to (gij). T h e m a p s Yn s a t i s f y

respect

the c o r r e s p o n d i n g

the

a system

, where

with

as C a e s t i m a t e s

we can that

T is a w e a k

tisfy

: Bq + G

H I'2 as w e l l

and therefore

and

Tn

T n is c o n f o r m a l

satisfy

the metrics to

uniform

(gnj)i c a n be c h o s e n

(gij).

From

(3.6.1)

C I'~ e s t i m a t e s ,

we

to c o n v e r g e infer

that

and consequently

with the

the

limit

T is a d i f f e o m o r p h i s m .

Thus

we have

the p r o o f

3.7.

Uniqueness

Remark:

the desired 3.1

of

(and a d i f f e o m o r p h i s m

function

T I and

representation

of S , a n d

representations

it is n o t

from the process

such mappings

conformal

is c o m p l e t e .

of c o n f o r m a l

Actually,

structed phism

found

of T h e o r e m

difficult sections

in the c l a s s i c a l

T2 differ

f , if o n e of them,

T2(z) f(T1(z)). (This a s s e r t i o n is t r i v i a l

to see,

only

say

~I

that each map

3.3 - 3.4 h a s case

S 6 CI'~),

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

T con-

to be h o m e o m o r -

with

since

two

an a n a l y t i c

, is a h o m e o m o r p h i s m ,

i.e.

=

case

follows

[BJS]).

Then

b y an a p p r o x i m a t i o n one

can use

T 2 is a h o m e o m o r p h i s m ,

1) T h e s e

in the c l a s s i c a l

results

argument,

the a r g u m e n t

due

cf.

of L e m m a

if it is the u n i f o r m

are basically

case,

to M o r r e y

and

Lemma 3.4

limit

[MI].

to

the g e n e r a l 8 , p.

274

in

show that also

of h o m e o m o r p h i s m s .

34

with

Since

a conformal

three

fix points

thermore

that

automorphism

on the b o u n d a r y

if w e

a three -point

condition.

such representations

tity map

3.8.

with

require

three

In t h i s

section,

the c a s e w h e r e

~3

[M3],

the

we briefly

§9.4,

on

surface,

surfaces

class

of u n i o n s of Thm.

of

the k - c o n t o u r

of A h l f o r s

used

t o be t h e

iden-

of l o w e r

case

of Thm.

An essential

notion

for t h i s

differential,

i.e.

~ 6 L~(D,~)

÷ ~d~l 2

(3.8.1)

Thus, his

3.1

case.

to in

a k -connecof k - c o n n e c than

in t h e

since

his

approach

to

3.1 w a s cf.

approach

, represented

with

used

[AI],

in the a p p r o a c h [A2],

[AB],

is the c o n c e p t

l~I~ < I

and

of a B e l -

~ defines

a con-

the o n e

given by

by the metric

,

and a conformalmap is a s o l u t i o n

is less

completed

theory,

[BI].

laz

3.1.

curves

bound

in the c l a s s

connectivity.

we have

to T e i c h m 0 1 1 e r

on D

of Thm.

k Jordan

manifold)

of a r e a

of Thm.

problem.

1 -contour

structure

case

showed,

configuration

not correct,

Plateau

He

Riemannian

trami

from the usual

of t h e B e l t r a m i

structure

on D onto

equation

w~ = ~w z

Given

a compact

cover

of

Riemann

surface

Z is c o n f o r m a l l y

lent to a fundamental covering

existence

f r o m Thm.

3.1)

approach

could

, and

at l e a s t

and

are conformal

of

(3.8.1)

use a -priori

gives

universal

a group

automorphisms

theory

equiva-

F of o f D.

then proceeds

for g i v e n

its d e p e n d a n c e

2 , the

Z is c o n f o r m a l l y

z1(Z)

to Teichm~ller

solutions

and examining to t h i s

in D

which

approach of

Z of genus

the unit disc,

domain

transformations

The mentioned

dern

for any automor-

C 1"a t h e c l a s s i c a l

the c l a s s i c a l

the boundary

surfaces

and Bers

some applications

problem.

regular

3.1 w a s

The nonclassical

showing

thus has

least of class

if the m i n i m u m

spanning

proof

mention

Plateau

(or a h o m o g e n e o u s l y

formal

~B a n d

fur-

3.1

S is a t

Morrey

k -contour

ted minimal ted

Namely,

T 2 , TI 1 o T2 is a c o n f o r m a l

fixed points

of T h e o r e m

We call In

T I and

infer

in 3.3 a n d

of B.

Applications

solve

we

T constructed

is u n i q u e

B

identity,

representation

3.4

of

is t h e

B of t h e u n i t d i s c

the c o n f o r m a l

two

phism

of a s u b d o m a i n

~

(which

on variations

estimates

by

follows

of ~

(a m o -

and uniqueness

of

35

solutions

of

Teichm011er tures

on

of

E together

of t h o s e i.e.

(3.7.1), space

Beltrami

those

subspace

3.1,

with

represent

of t h o s e

which

to t h e

only

a theorem variations a -priori about We

of ~

come

(cf.

one -contour

and

case

to T e i c h m H l l e r

on

space

F , b y the

to a s u i t a b l e

3.1

solutions

theory

together of

in-

on E .

with

(3.8.1)

via uniqueness

on

and

o f v i e w of K o d a i r a - S p e n c e r

to d e a l

theory

under

~ , divided

differentials

of Thm. of

of the

to T e i c h m H l l e r

can be proved

have

struc-

respect

[EE]

hhe point

thus

that the

invariant

quadratic

dependance

They employ

are

with

of E a r l e - E e l l s

[ESz] ), w h i c h

structures back

orthogonal

it is s h o w n of c o n f o r m a l

is t h e q u o t i e n t

structures

of h o l o m o r p h i c

continuous

estimates.

complex

shall

the

Finally

on D w h i c h

conformal

are

space

the classical about

3.7).

s p a c e of m o d u l i

a given marking,

F i n a l l y , the a p p r o a c h needs

3.4, the

differentials

which

ner product

cf.

E , i.e.

in

only with

12.3 u n d e r

smooth

~ .

different

as-

pects. Our main

application

consist

in d e r i v i n g

phisms,

cf.

3.9.

We

section,

455 - 4 5 8 ) shall

use

Lemma

3.6:

,

in the p r e s e n t

and

regularity

chapters

6, 7 a n d

form which

the Lemma was

notation,

used

lUz~l

of h a r m o n i c

K is a f i x e d p o s i t i v e

(3.9.3)

lim u Z z+O

If

(3.9.2)

U -- O

holds

.

= O(IZl n)

• z -n

diffeomor-

8 .

of Hartman -Wintner

+

~

satisfies

= y(~U almost

lUl)

constant.

If

U(Z)

will

([HtW],

paragraphs.

i.e.

u £ C I'1 (D,~)

_< K ( I U z [

(3.9.2)

however,

in the p r e c e d i n g

Uz = 2 ~x u + i ~ u ) ,

Suppose

book,

Lemma

shall prove

the complex

(3.9.1)

where

we

in t h e

z = x + iy

3.1

existence

in p a r t i c u l a r

The Hartman -Wintner

In this pp.

of°Thm.

for s o m e n 6 ~

exists.

f o r a l l n 6 l~ , t h e n

,

then

- i~yU), every,w h e r e

etc.

36

Proof:

First

rywhere

by

Lipschitz

note

that

Rademacher's boundary

(3.9.4)

~ ~B

Suppose

we

since

Theorem.

ZB a n d

gu z = f B

a

, Uz~

closed

exists

almost

subdomain

of

eve-

D with

then

that

In

we

(3.9.4), g =

take

B

z - k ( z - z o)-I

e ÷ 0

from

(3.9.6)

=

{z:

some

e S

, z° ~ O

Izl

,

g~

Uzz-k(z

u -z zz

f

iZ-Zol = O

~ e}

, R

, we

obtain

in B

< min(1,4~), by

let-

- Zo)-1

-k

z - z

-

) -1

dz

o

(3.9.1)

2 EUz (Zo) o I -< i

(3.9.7)

~ R

. Since

B (O,R) from

6

]zf=R

2~ U z ( Z o) z-ko

hence

k

(3.9.4)

=

and

is

( U z g z + Uzz--g)dz

u z = o ( I z l k-1 ) f o r

ting

If B

g 6 CI(B,~),

(3.9.5)

and

u 6 C 1'1

lUzZ

+ x

I

z - z o)-111dz!

{]u

+

+ lul)l~l-kl~-%1

-I

dz

B(O,R) We

now

want

Ik:

to

=

estimate

I%11=1-k1=-%1 -~ dz

f B(O,R)

uniformly in

Moreover, hence the

Thus,

u

if

lemma by

order

z

z

z

(3.9.5)

assertion In

+ O o converges

(3.9.6)

lim z÷O

of

as

k

of

to

-k

that some

< n , then holds

is

the

can

be

limit

as

achieved, z

o

÷ O

then , and

the

second

integral

hence

exists.

for

trivial,

induction,

to

. If

this

k +I if

(3.9.5)

limit

. On

the

vanishes other

n = O

, and

holds

for

because hand,

the

n ~ I implies

k = n

of

, which

(3.9.2),

first

assertion

(3.9.5) implies

for the

lemma.

control

Ik

, we

multiply

(3.9.7)

By

Iz ° - zl I-I

and

, use

k = I.

first

37

(3.9.8)

f

Iz-zo l-ldz < 2R ,

if Izol

< R

and

B (0, R), _

(Z-Zo)-1(Zo a n d obtain,

z 1)

-I

i n t e g r a t i n g w.r.t,

2~

f

=

(z

_

zI

) -I

z O , IZol

((Z-Zo)-I

+ (Zo-Zl)-1)

,

< R ,

Iz-kIIz-%1-1 dz

luzl

B(O,R)

(3.9.9)

_<

z~:R lUzZ-k(z-z I)

4R

+ 4KR

-1

IIdzl +

(l~zl + lul)Iz[-klz-zll -Id~

f B(O,R)

By c h o i c e of R ( R < ~--) , and s i n c e k < assumption,

(3.9.9)

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

by

controls

induction,

n,

I k , putting (3.9.9)

i.e.

luz-kl

= 0(iz I) by

z I = z°

holds

for k = n -I

, and l e t t i n g

zI ÷ 0 , we obtain (27 - 4 K R )

f lU Z -n ldz B (O,R) z

<

(3.9.10)

4R |~ = R l U z

Z -n I Idzl +

lullzl -n dz

4KR f

Iz

B (O,R)

I

NOW

lu(z) I _< f I Z U z ( t Z ) I d t o I

f

luz-nldz _< f

B (O,R) I

Iz-n+lu

leads

Iz-n+lu

to

(tz)Idzdt

=

z

(z) Idzdt z

for n ~ 3

f

luz-nldz ~

B(O,R)

Since

f

O B (O,R)

= f tn-3 f O B (0, tR)

Hence

, which

Izl < R < I

(3.9.10)

f B(O,R)

is z-n+11 dz . z

by c h o i c e of R , we see that the s e c o n d i n t e g r a l

can be a b s o r b e d

into the l e f t h a n d side,

yielding

in

38

(3.9.11)

(2~ - 8KR)

S 2R

Since

S lUzz-nldz B (O,R)

<

Iz { = R l u z z - n I l d z ]

(2~ - 8KR)

(3.9.11) for a l l

> 0 by choice n E ~ implies

o f R, u

i t is n o t d i f f i c u l t

H 0 and hence

z

to s e e t h a t

u H 0 which

finishes

the proof. Namely,

otherwise,

c # 0 , and and

the

there would

the l e f t h a n d

right hand

and hence

side

(3.9.11)

side of

like

could

exist

c

(3.9.11)

R -n

o

not hold

z° w i t h

, c

IZol

would

, c

< R a n d U z ( Z o) grow

being

o

like

=

C]Zo[-n

independant

of n ,

for a l l n .

q.e.d.

~.

Existence

theorems

for harmonic

4.1.

A maximum

principle

for e n e r g y

We assume

that

is a n o p e n In t h i s mizing ciple

chapter,

same

4.1:

a Riemannian

BO

we

is t h e

, i.e.

Z2 a r e ZI

shall

minimizing

Suppose manifold

denotes

solve

A useful

between

twodimensional

i(Z)

i d e a as t h e o n e

~: B I ÷ B O which

of

procedure. for energy

on t h e

Lemma

ZI a n d

subset

maps

minimizing

existence

tool will maps in

the

[HI],

Riemannian injecbivity

problems

be t h e

which

is t a k e n Lemma

Suppose

that

there

manifolds. radius

of

b y an e n e r g y

following from

maximum [J6]

mini-

prin-

and based

6.

t h a t B ° a n d B I , Bo c BI N.

surfaces

maps

, are closed

exists

subsets

a projection

of

map

,

identity

on B ° and which

is d i s t a n c e

decreasing

outside

3g

d (n (x) ,n (y) ) < d ( x , y )

whenever If

h:

x,y

~ ÷

boundary

B I is

also

h (~)

if w e

c Bo

we

dh

(4.1.1)

~ for

to

fixed

, i.e.

o

is

representant

Lipschitz

almost

of

the

continuous,

every

nonzero

Sobolev

it

v

is

6 TxN

mapping

easily , x

h

seen

6 BI~

BO

.

that , and

that

have

o h)

~

= d~ , we

, which

Lemma

4.2:

< E(h)

~B ° b y

tween

Proof: Lamina

minimality

o h a.e.

than

We

on

~

from

easily

implies

pair on

the

that

manifold

N

of

of

project

such

since

BI

that

normal

the

h and

every

~

on

~ h agree

BI

that

in

, and to

conclusion

normal

, are

point

~B °

normal

same

B I ~B ° along

= 0 a.e.

inequality

, Bo c

to

geodesics

. Then

dh

n

h-1(B1TM BO). on

~

o h

by = h a.e.

claim.

B ° and

, and

, unless

Poincar~

the

geodesic

~B o

h

, and

conclude

a unique

every

bigger

the

,

Suppose

a Riemannian to

suitable

Iv]

contradicting

on

in B

respect

t

a

would

E(z

Thus

,

contained

with

I 6 W2(~,N).

o h Thus

c

BO

1 mapping W2

minimizing

are

Since

Ida(v) I <

energy

have

choose

Proof:

, x ~ y

which

h(~)

we

BO

an

values

(4.1.1)

then

6 BI~

compact

subsets

B I TM B ° c a n

that

the

be

of

joined

distance

be-

~B ° i_____~nB is I TM B ° a l w a y s as

geodesics

in L e m m a

onto

4.1

holds.

~B ° a n d

apply

4. I .

q.e.d.

Another

Lemma

consequence

4.3:

Suppose

of

BO

Lemma

is

4.1

is

a geodesic

ball

with

radius

s and

center

D.

40

s ~ I/3 m i n nal

(i(p),

curvature

z/2<),

of N a n d

If h: Q ÷ N is e n e r g y map

g:

~ ÷ B

h(~)

o

, and

i(p)

<

2

is the

minimzing

is an u p p e r

bound

injectivity

radius

among

if h(~Q) c

B

maps

, then

o

which

for the of p

sectio-

.

are h o m o t o p i c

to s o m e

also

c BO .

(for a s u i t a b l e

Proof:

where

representative

By a s s u m p t i o n ,

of h

we c a n

, again).

introduce

geodesic

polar

coordinates

(r,%o) o n B ( p , 3 s ) (0 ~ r ~ 3s). We define

a map

z(r,%o)

=

(r,%o) = (q)

(Here,

fies

ifr~s

(~(3s-r),%o)

ifs~r~3s

we have

the

way:

(r,%o)

= p

in g e o d e s i c Using

~ in t h e f o l l o w i n g

if q 6 N ~ B ( p , 3 s )

identified

polar

Rauch

a point

in B ( p , 3 s )

with

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

coordinates.)

comparison

the assumptions

theorem,

of L e m m a

it is e a s i l y

seen,

that

~ satis-

4.1.

q.e.d.

4.2.

The Dirichlet

We now want [HKW3],

Theorem M < ~

to p r o v e

using

ly, m o s t

problem,

of

the

it a l r e a d y

4.1:

Suppose

, where

there

<

2

exists

and h minimizes Vice

versa,

each

of c o n t i n u i t y

following

follows ~

~ @

result

, B(p,M)

such

map

with

energy

h:

minimizing

I) H e r e ,

we

can define

coordinates

I H 2(~,B(p,M)) o n B(p,M)

map

with

of g

(Actual-

radius

curvature

of

boundary

boundary

values

g

values. The modulus

I , i(Z1),

M

, ~ ,

.

unambiguously

given

-Widman

4.2.

an extension

is h a r m o n i c . of

-Kaul

in Z2 w i t h

and admits

to t h e s e

disc

[M2].)

of the G a u s s

in t e r m s

of c o n t i n u i t y

and Lemma

work

~ ÷ B(p,M)

respect

of h c a n b e e s t i m a t e d

a n d the m o d u l u s

3.1

is a d i s c

bound

in a c o n v e x

of H i l d e b r a n d t

-Lemma

is c o n t i n u o u s

a harmonic

the energy

is c o n t a i n e d

from Morrey's

~ 0 is an u p p e r

a n d E(g)

global

image

the C o u r a n t - L e b e s g u e

B(p,M), and g:~ + B(p,M) I I) 6 H 2(~,B(P,M)) . Then

if the

by eXpp

with

t h e h e l p of the

,

41

Proof:

(the i d e a

Since M

the cut

locus

< M I < z/2<

We

take

class

v~

Applying

of t h e

Lemma

actually

maps

By Thm.

2.1,

4.2

~ into

arc

fore,

we can apply

Suppose c 2 are

in B ( p , M ) ,

and

the

the

geodesics

vectors

N B(p,M) q

converging

, minimizes

energy

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

in its

integral.

we conclude

that h

by a unique

of c o n j u g a t e

theorem unit

vectors

by arclength

points.

in the

and

geo-

There-

following

way.

in TqZ 2 , a n d c I , starting

at q with

Then

(d(c1(c),c2(c)),

exists

c° > O w i t h satisfy

and every

6 ~ there

= {v 6 H~(~,

6 B(p,M).

= B(p,M)

6 B(p,M)

s o m e M I,

2.1).

B(q,c) ,

> rain

there

a n d BI:

find

~ d(c1(t),c2(t))

c2(t)

d(c1(t),c2(t))

in V:

c a n be j o i n e d

is free

comparison

parametrized

on B(p,M)~

Consequently,

arc

we can

B(p,M).

v I and v 2 are

(t<)

l o n g as c 1 ( t ) ,

Therefore,

each x

this

Rauch

[ H W I].)

(cf. Thm.

B I = B(p,MI),

ball

in

a subsequence

by h

in B ( p , M )

v I , v 2 resp.

IVl-V21< -I s i n

every

smaller

that q 6 B(p,M),

tangent

as

the

energy

semicontinuity

two points

4.1

set,

a disc

has

limit,denoted

lower

to B ° = B ( p , M ) ,

every

desic

for t h e

Such a sequence

in H i , a n d t h e

o f Thm.

p is a c l o s e d

B ( p , M I) is s t i l l

sequence

= g}.

because

from the proof

of a p o i n t

for w h i c h

a minimizing

B(p,MI)), weakly

is t a k e n

exists

E ~ co

I V l - V 2 1 < -I s i n

the p r o p e r t y

the a s s u m p t i o n s • Lemma

a sufficiently

3.1

small

(2M<)),

t h a t Bo:

of L e m m a

then

implies

p > 0 with

= B(q,c)

4.2

for

that

for

the p r o p e r t y

that

h(B(x,p)

for

some q

(which

N ~)

c B(q,s)

6 B(p,M).

is b o u n d e d

p depends

on

b y the e n e r g y

c , ~ , i(E1) , the e n e r g y

of g), a n d

the m o d u l u s

of h

of c o n t i n u i t y

of g . Therefore, easy

Lemma

to see

h is e n e r g y

4.2

(cf.

implies

[HKW3])

minimizing

that h actually

lies

the

continuity

t h a t h is a w e a k

in V a n d o n the o t h e r

in the

interior 01

h in t h e d i r e c t i o n gularity (cf. a l s o

follows [G] a n d

of a n y ~ 6 H 2 n L

e.g.

of h

from

[Hi2]).

[LU] a n d (We s h a l l

of V

. Furthermore,

solution hand,

of

T(h)

h(~) c

= O

it is , since

B(p,M),

, and the variation

so

of E a t

oo

(~,Z 2) has classical derive

to v a n i s h .

linear

higher

Higher

elliptic

order

re-

theory

42

estimates

in t h e

following

chapters

in m u c h m o r e

detail,

however).

q.e.d.

4.3.

Remarks

Thm.

4.1

about

holds

the higher -dimensional

in a r b i t r a r y

Kaul -Widman

[HKW3].

was

given

[J4].

one

(which p r o v e s

to the

in

following

Proposition bounded

While

an manifold.

proof,

conceptually

the regularity maximum

4.1:

domain

dimensions,

A different

Suppose

Assume

disjoint

to the c u t

If f ( ~ )

c y

, where

the h e a t

to H i l d e b r a n d t

of p

-

flow method,

complicated

than

of e n e r g y )

the

first

it l e a d s

[J5]).

f: ~ c N is a h a r m o n i c

c B(p,M),

locus

due

more

(cf.

Riemannian

f(~)

again using

of t h e m i n i m u m

principle

~ in s o m e

situation

manifold where

mapping

from a

into a complete

again M < ~/2K

Riemanni-

and B(p,M)

is

.

Y c B(p,M)

has a smooth

convex

boundary,

then

also

f (~) c

One

y

can also

tinuity, ball

derive

in a r b i t r a r y is m u c h

lows:

First

condition function

shall

where

is s u p p o s e d ,

can gain

and

exhibit

more

constructed

and can

theorem

have

the argument these I)

The

easier,

is d u e

version

instead

T h e m e t h o d of p r o o f

of t h e

strictly

is d u e

using

convex that,

on B(p,M). if o n e

is s t r i c t l y to W i e g n e r

con-

[Wi]

points,

is g i v e n

[GJ]. A

the

in

fact that

for t h e p r o o f

convex assigned

the more

squared

The general

argument

at b o u n d a r y

He u s e s

[J~K2]

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

for M < ~ / 4 K c a n c o v e r

functions

[Sp].

and Kaul

are

fol-

stronger

6.5.

of the proof

to S p e r n e r

5.2)

is d u e

how-

distance

on B(p,M).

d2(-,q)

as

if t h e

squared

to c o n c l u d e q

to i t e r a t e

in s e c t i o n

outlined

strictly

such an argument

by J~ger

(cf.

some more

idea

the

convex

The proof,

easier,

iteration

informations

of

[GH].

convex

is s t i l l

enough

a minimum

then

of c o n -

in a s t r i c t l y

or

somewhat

is s t r i c t l y

d2(.,q)

for t h e m o d u l u s

c a n be r o u g h l y

are

to p

geometric

functions uniqueness

idea

[HJW]

b y an

so on.

approach

e.g.

to be c o v e r e d

a version

different

e.g.

contained

because

somewhat,for

it is s o m e w h a t

A somewhat

cf.

the e s t i m a t e s

then has

the domain

image

involved~)The

for q close

this one

estimates,

with

f r o m a n y q 6 B(p,M)

vex on B(p,M) We

more

M < ~/4K

that

shrinks

maps

dimensions,

of all,

case M < z/2<

From

a -priori

for harmonic

ever,

first

.

on the whole point.

general

distance.

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

case

Still

of B(p,M)

Therefore, if o n e

other

[HW2].

the

of t h e i r

uses

proofs

43

were p r o v i d e d

by E l i a s s o n

For a general

account

silin e a r

elliptic

we refer

to the lecture

More

general

appr o a c h mizing these

systems

(of.

maps

the m e t h o d s

supports

shows

problem,

one can prove coordinate

as one m i g h t

boundary

tible anyway. a well

geodesic

stri c t l y

inside,

convex

The D i r i c h l e t sectional

of the b o u n d a r y

cf.

Prop.

for the case,

was p r e v i o u s l y

like c o n v e x i t y

on the image

4.4.

mappings.

namely

as we

that

shall

the

see

maps b e t w e e n

For this purpose,

map w i t h

az 1 ~ ~

is c o n t r a c -

homotopy

geodesics

group,

(cf.

[O])

a nontrivial

to the e x i s t e n c e

of the

the image has n o n p o s i t i v e

solved

by Hamilton,

conditions

homotopy

to prove

needed

using

the heat

energy.

has to va-

sections.

and Sacks - U h l e n b e c k

to attack

the general

Z2 denote boundary.

We denote

are h o m o t o p i c

ture b o u n d on ~2

the e x i s t e n c e

existence

problem

for

surfaces,

a~ 2 = ~

, but

surfaces.

nonempty

s = I/3 min

restric-

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

group of the image

compact Let ~:

by

ZI ~ Z2 be a c o n t i n u o u s

[~] the class

to ~ and c o i n c i d e

(i(E 2)

, ~/2K),

' and i(E 2) is the

where

of all c o n t i n u o u s

w i t h ~ on ~Z I , in case

.

We c h o o s e

function

is not as strictly

one does not need any g e o m e t r i c

second

let ZI and

having

finite

maps w h i c h

with

5.1.

in the f o l l o w i n g

We are now in a p o s i t i o n

ZI p o s s i b l y

is c o v e r e d

last c o n d i t i o n

would yield

w h e re

The only c o n d i t i o n

The T h e o r e m of L e m a i r e

harmonic

image

a manifold

convex

and

of G i a q u i n t a -

([Hm]).

tions

nish,

the

have some n o n v a n i s h i n g

however,

nature,

This

since

a strictly

In two d i m e n s i o n s ,

of h a r m o n i c

[J6]).

with

one can solve

convex b o u n d a r y

the m e t h o d s

provided

in c o n t r a d i c t i o n

function,

problem

curvature

flow m e t h o d

(cf.

hypotheses

can be o b t a i n e d

techniques,

from the theory of c l o s e d

and the strict c o n v e x i t y closed

U s in g

first think,

if it w o u l d

known argument

their

This

of energy m i n i -

geometric

image has a s t r i c t l y

and s u p p o r t i n g

For,

With

of par-

[E]).

singularities

suitable

function.

chart

maps,

by the a p p r o a c h

The best r e s u l t

the same result,

restrictive convex

the p o s s i b l e

if the

by a single

for h a r m o n i c

[SU2],[GGI],[GG2],[J6],[JM],

occur.

convex

that one

can be Q b t a i n e d

of Schoen - U h l e n b e c k .

a strictly

for the class of qua-

by Hildebrandt.

that under

cannot

[To]

results

includes

[Hi2]

results

[SUI],

and then

the D i r i c h l e t

Giusti,

which

first c h a r a c t e r i z e s

singularities

and T o l k s d o r f

notes

existence

tial r e g u l a r i t y

[Es]

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

K

2

injectivity

~

0 is an u p p e r curvaradius

of ~2

"

44

Let

vature

of

where

27

E(~)

" E(%0) I/2

is the

(4.4.2)

energy

d ( X l , X 2)

Let 0 < 6 < 6o i = 1,...m

E(q0) w . l . . o . g ,

I/6o )-I/2

of ~

exists

a finite

for w h i c h

for all

n

We now

for the

cur-

we have

defined

0 Z1 ) n e a r

have

,

< s/2

of p o i n t s

B(xi,6/2)

minimizing

for X l , X 2 6 ~ZI

x i £ ZI

cover

sequence

71

in

'

.

[~0], E ( u n)

<_

. (4.4.1) and (4.4.2), for e v e r y

U n ( ~ B ( x 1,rn, I))

~(B(x,r)

_-< s/2

number

the discs

energy

Lemma 3.1 a n d u s i n g

(4.4.3) (Here,

bound

, and

rn, 1 , ~ < rn, 1 < / ~ , a n d Pn,1

=

a lower

< /60 ~ d(~(Xl),~0(x2))

• There

= m(6),

(log

let u n b e a c o n t i n u o u s

Applying

(_12 b e i n g

Z I) s a t i s f y

(4.4.1)

We

i/~2)

60 ~ m i n ( 1 , i ( z 1 ) 2

6 Z2 w i t h

n ,we can

the p r o p e r t y

find

that

c B ( P n ,1,s)

B(x,r)

=

{y C ZI:

d(x,y)

< r} a n d

thus

~B(x,r)

the b o u n d a r y . )

two p o s s i b i l i t i e s :

either 1) T h e r e

exists

some

~ , 0 < ~ < ~

, with

the p r o p e r t y

that

for any

O

x 6 ZI

' some

Un(gB(x,r)) UnlB(x,r)

r

cB(p,s)

for

is h o m o t o p i c

(4.4.4)

on x a n d n w i t h

some

p6Z

to the

g: B(x,r)

+ B(p,s)

gl~B(x,r)

= Unl~B(x,r)

(The e x i s t e n c e by Thm.

, depending

5.1,

solution

this

every

by Thm.

is n o t n e e d e d

sufficiently

of the D i r i c h l e t

harmonic

o f g is e n s u r e d

but

2 , and

6 < r < /~ and w i t h

and

4.1;

in the

energy

n

problem

minimizing

g is a c t u a l l y following

large

unique

constructions)

or

2) P o s s i b l y of p o i n t s

choosing x n 6 ~I

a subsequence , and radii

U n ( ~ B ( X n ' r n )) = B(Pn'Cn) (using

Lemma

the s o l u t i o n

3.1), of

but

of

rn > O

for some for w h i c h

the D i r i c h l e t

the u n , w e

find a s e q u e n c e

, x n + xo E Z I ' r n ÷ 0

Pn £ ~2

' Pn ÷

UnlB(Xn,rn)

problem

can

p 6 Z 2 , en + 0

is n o t h o m o t o p i c

(4.4.4).

, with

to

,

45

In c a s e I), w e r e p l a c e let problem (4.4.4) for and , u s i n g solution

of

u n on B ( X l , r n , I) by x = xI

and

the i n t e r i o r m o d u l u s (4.4.4)

(cf. Thm.

< ~ . By Lemma

(4.4.5)

r = rn, I

.We can a s s u m e

of c o n t i n u i t y

4.1)

1 d e n o t e d by u n , c o n v e r g e

maps,

the s o l u t i o n of the D i r i c h -

estimates

rn, I + r I

for the

t h a t the r e p l a c e d

uniformly

on B ( x 1 , ~ - ~ ) ,

for any 0 <

4.3

E(uln) < E ( u n)

B y the same a r g u m e n t

as above,

we

then find r a d i i r

n,2

< rn, 2 <

'

/~ , with I Un(aB(x2,rn,2)) for p o i n t s P n , 2

c B(Pn,2,s)

C Y2 "

1 we r e p l a c e u n on B ( x 2 , r n , 2) b y the s o l u t i o n

Again, problem

(4.4.4)

2 u n . Again,

for x = x 2 and r = rn, 2 . W e d e n o t e

w.l.o.g.,

converges

0 < ~ < ~ , w e see

that the b o u n d a r y

the e s t i m a t e s

(4.4.4)

the m a p s U2n c o n v e r g e < ~ . by L e m m a

E(U 2) < E(Uln)

<

by the f i r s t r e p l a c e m e n t

o n B ( x 2 , r 2) S B ( x I,~-~/2) values

points

(4.4.6)

if

for o u r s e c o n d r e p l a c e -

of c o n t i n u i t y

(cf. Thm.

4.3 a g a i n

and

for the s o l u t i o n

4.1), w e c a n a s s u m e

u n i f o r m l y on B(x1,~-~)

U B(x2,~-n),

that

if 0 <

(4.4.5)

E(U n)

In this way, w e w e r e p e a t the r e p l a c e m e n t sequence u

step,

on aB(x2,rn, 2) n B ( x 1 , ~ - ~ / 2 ) .

for the m o d u l u s

at these b o u n d a r y

Furthermore,

that,

uniformly

ment step converge uniformly Using

the new m a p s by

rn, 2 ÷ r 2 .

If we take i n t o c o n s i d e r a t i o n I in p a r t i c u l a r un

of the D i r i c h l e t

m =: v n , w i t h n

E ( v n) < E ( u n)

argument,

u n t i l w e get a

of

46

which

converges

hence on all

uniformly

of Z1

on all balls

, since

these balls

We denote

the l i m i t of the v

homotopic

to ~ .

Since E ( v n) S E(~)

by

w e a k H 21 c o n v e r g e n c e (4.4.6),

hence

(4.4.6),

and since

u minimizes

In particular,

energy

u minimizes

it is harmonic

Observing

we can a s s u m e

that

if ~2(Z2)

~2(Z2)

= O

= O

coincides

by L e m ma

ZI and

with

a harmonic

~ o__nn ~ZI

in case

also

sequence

by

class. to small balls,

into Z 2 are

surfaces,

map w i t h

ZI + Z2 w h i c h

and

4.1.

from a d i s c

Z2 are compact

map u:

is

of the energy w.r.t.

4.3 and Thm.

, any two m a p s

u

that the v n c o n v e r g e

energy w h e n r e s t r i c t e d

. I f ~ : ZI ~ Z2 is a c o n t i n u o u s

then there exists

convergence,

semicontinuity

in its h o m o t o p y

, and

ZI

the Vn are a m i n i m i z i n g

and regular

homotopic, we o b t a i n T h e o r e m 4.2: Suppose

cover

by u . By u n i f o r m

n

I in H 2 to u , and by lower

weakly

B(xi,~/2) , i = 1 , . . . m

~Z2 = g

' and

finite energy,

is h o m o t o p i c

to ~ ,

~ZI @ @ and is energy m i n i m i z i n ~

among all such maps. Theorem [L2])

4.2 is the f u n d a m e n t a l

and Sacks - U h l e n b e c k

A different

proof was

The p r e s e n t

proof

existence

([SkU],

theorem due to L e m a i r e

in case

given b y S u h o e n - Zau

is taken

from

~I

= ~ )"

[SY2].

[J6].

In the case of the D i r i c h l e t

problem,

that Z2 is compact,

that it is h o m o g e n e o u s l y

b u t only

sense of M o r r e y

[M2],

that the fixed b o u n d a r y

dedness

convergent

The D i r i c h l e t Solution,

In this

[L2].

of the e q u i c o n t i n u o u s

uniformly

4.5.

cf.

it is a c t u a l l y

have

to o b s e r v e

section,

and Brezis

problem,

we w a n t

a n d Coron

domains

are treated).

Theorem

4.3:

Looking

seauence

subsequence

Suppose

ZI

regular

imply

proof,

w e only

the u n i f o r m boun-

as in the case of a c o m p a c t

values

in the

v n , and h e n c e we can select a

is h o m e o m o r p h i c

image.

to S 2

are nonconstant.

to show the f o l l o w i n g

[BC2]

not n e c e s s a r y

at the p r e s e n t values

if the image

if the b o u n d a r y

([L1],

result

(in the latter paper,

is a c o m p a c t

only

of J o s t simply

two - d i m e n s i o n a l

[J7] connected

Riemannian

47

manifold

with

nonempty

boundary

homeomorphic

to S 2

tinuous

map,

not m a p p i n g

tinuous

extension

Then u:

there

are

Proof:

(the s t a n d a r d

ul~

first

a single

point

ZI to Z2 w i t h

4:

~ZI ÷ Z2 is a con-

and

admitting

finite

different

, and b o t h m a p p i n g s

investigate

en < = s/2

more

for all

and

for x = x n

, r = r n is c o n t a i n e d

U n I B ( X n , r n)

and

manifold

a con-

energy. harmonic

minimize

maps

energy

in t h e i r

classes.

c B ( p , 2 e n)

Since

Z2 is a R i e m a n n i a n

2-sphere),

onto

from

, and

two h o m o t o p i c a l l y

I = ~

homotopy

We

~Z1

to a m a p

at l e a s t

El ÷ ~2 w i t h

respective

~ZI

is n o t

closely

n

and

case

thus

in B ( p , 2 e n)

homotopic

to g

2). W . l . o . g .

the

solution

by L e m m a

, it has

B ( P n , £ n)

g of

(4.4.4)

4.3.

to c o v e r

Z2 ~ B ( p ' 2 e n ) "

If we d e f i n e [u on ~I ~ B ( X n ' r n ) Un =I n ;g on B ( X n , r n) , t h e n we

see t h a t

(4.5.1)

lim E ( u n)

> lim E(UnIZ1TM B ( x n , r n ) ) > lim E ( u n)

since

E(g)

÷ O as n ÷ =

+ Area

(Z2)

+ lim E ( U n l B ( x n , r n ) ) ,

, because

2z 2d fl gs(rn,8) I 8 ÷ 0 o as n ÷ =

. (cf.

(Furthermore,

E(v[B)

and

(3.2.3)). it is e l e m e n t a r y

>_- A r e a

equality

holds

that

(v(B)) ,

if and o n l y

if v is c o n f o r m a l ) .

We n o w d e f i n e

E : = inf

for a h o m o t o p y and

E:

= rain E

{E(v):

class

v

£ e}

~ of m a p s

with

vi ~Z 1 = ~ t

48

We

first

topy

show

class

E

choose

Assuming

< E

that

2)

contradict

an

let

Now

~ an to

mains

to

metric

energy

we

can

in

on

not

find S2

troduce

local

to

a conformal

C from

an

(4.5.4)

for We

x now

above . In v is

homo-

e with

as

n

above.

Therefore,

map

in

a

clearly

as

(cf.

class

with

in ~

, i.e.

some

+ Area

(Z2)

show

that to

homotopy

we

can

complete

conformally use

a constant

shown

[BCl]

map, the

assume

south

factor.

find

above,

for

we

a similar

= E. ~ ~ ~ with

a harmonic

proof,

By

also

u

it

is n o t

map

thus

of

ZI

is

the

the

image

by

on

Po

Taylor's

" d~(Po) theorem,

O(e2),

[~ o ~(x)

- d(z

o ~)(Xo)(x

maps

of

standard for

of

mi-

only

re-

for

which

south

~

is

the

- Xo) I = O ( e 2)

the

form

Po

hence

~ 0

" We

. in-

projection identity

o ~l~B(Xo,e

i.e.

and

d ~ ( x o)

pole

on

image.

map,

stereographic then

metric

the

a constant

~ ( x o)

pole

interior

the

domain

that

order

conformal

to

a parameter

of

at

E(u) class

the

equivalent

S 2 as

x ° in

can

error

in

£ ~B(Xo,£). look

Since

.

can

the

map

v

order

coordinates

S2 ÷

to

a homotopy

< E(~)

we

, we

z:

up

~

harmonic

a map

a point

Rotating

map

in

n

define

minimizing

e

Z2

thus,

~ is

we

~ be

construct

, and

Since

let

arguments

energy

S2

u

(4.5.1) , h o w e v e r .

construct

nimal

The

any

(Z2)

minimzing

E(v)

the

in

,

(4.5.3.)

Then

map

,

energy

= E

want

harmonic

(Z2)'

holds,

would

argument).

and

a minimizing

sequence

+ Area

> E

E~

of

+ Area

E (~n)

obtain

We

< E

a minimizing

E(Un)

this

existence

~ with

(4 • 5.2)

We

the

) is

map

up

a linear

49

w

The is

= az

+ b/z

restrictions given

,

of

a,b

such

6 f

,

a map

a = a I + ia 2

to

a circle

b

p(cos

= b I + ib 2

8 + i sin

e)

in

by

u =

(alP

bI + -~)

cos

~ +

b2 (~

- a2P)sin

8

v =

(a2P

b2 + -~)

cos

8 +

(alP

bI - -~)

8

where

,

w

= u

Therefore,

+ iv we

,

.

can

choose

this

circle

coincides

This

map

nonsingular

is

sin

a and

with

b

any

in

such

a way,

prescribed

that

w

nontrivial

restricted

linear

to

map.

if

2

2 b2 # 2 2 aI + a2

4

bl

P

+

W.l.o.g. 2 b] (4 5.5)

p4

2 b2

+

<

"

=

2

2

aI + a2 (otherwise Hence

w

can

circle ty

the

perform

an

be

extended

8

+ i sin

p(cos

holds

and

we

in

(4.5.5),

exterior

is

inversion as

at

a conformal

8)

onto

the

then

this

image

the

the

unit

map

from

exterior

completement

circle).

of

the its

is

a straight

of

this

interior image. line

line

in

of

covered

the

the

(If e q u a l i twice,

complex

plane). We

are

now

in

On

~1TM B ( X o ' e )

On

B(Xo,S-e

boundary On

we

2)

we

with

B(Xo,S)~

duce

a position put

to

v = u

choose

the

2)

polar

coordinates

f(<0):

=

g(~):

= d(~oO)(£,~)

v

map

we

.

.

a conformal

linear

B(Xo,S-s

define

_

w as

.d(~o~)

interpolate

r,~

above

which

(Xo) , a n d in

and

define

~I

d(zo~)

.-~

+ ~(g(<0)

the

put

following

(~o~) (s,~) =

(s-e2,~0)

and

t(r,~):

=

(f(~)

-g(<0))

I

-(1-e)f(<0))

coincides v = ~ way.

o w We

on

the

. intro-

50

Thus £ - £

t(r,~0) c o i n c i d e s 2 , resp.

The e n e r g y of t(r,~0)

m(t)

cj 2 r=e-~

=

with

f(~0) and g(~),

on the a n n u l u s 2 ~~(b ~=

resp.

B(Xo,£)~

for r = e and r =

B ( X o , £ - £ 2) is g i v e n by

If(q0) - g(~0).I 2 + 1 2 I ( ~ r

1-~)f, (~) +

+ (1 _ __~)g, (~0)12)rdrd%o £ £ Using

(4.5.4) E(t)

and

= O(e 3)

and h e n c e

If' (q0) I = 0(~),

Ig' (~0) I = O(£),

we c a l c u l a t e

,

also,

E(~-1ot)

= O(~3).

We p u t v = ~ - l o t on the a n n u l u s

B(Xo,£)~

B(Xo,£-£2).

Therefore

E(v)

= E(~I~]~B(Xo,£))

+ E(~-1owlB(Xo,e-E2))

+

+ E(~-1otIB(Xo,£) ~B(Xo,~-~2)) E(~)

- O(£ 2) + A r e a

s i n c e E ( ~ I B ( x ~)) -I

= 0(£2),

(E 2) + O(e3),

because

ow is the a r e a of its image,

conformal.

Thus,

and t h e p r o o f

for c o n s t a n t

In c o n t r a s t

to Thm.

Proposition

4.2:

4.3,

There

disc D onto S 2 mapping

small

Suppose

e > O , (4.5.3)

of

ow are

is s a t i s f i e d ,

[LI] s h o w e d

is no n o n c o n s t a n t

u: D ÷ S 2 is h a r m o n i c of u are c o n s t a n t ,

with respect

to v a r i a t i o n s itself,

values

harmonic

m a p f r o m the u n i t

~D o n t o a s i n g l e point.

values

~D o n t o

boundary

Lemaire

the b o u n d a r y mapping

as n and w and h e n c e a l s o n

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

-I

is c o m p l e t e .

4.6. N o n e x i s t e n c e

Proof:

d ~ ( x o) # 0 , a n d the e n e r g y

uo~

, where

w i t h u(~D)

= p 6 S 2 . Since

u is a l s o a c r i t i c a l

point

~: D + D is a d i f f e o m o r p h i s m ,

but not necessarily

b e i n g the i d e n t i t y

on ~D.

51

Thus,

one

can

again

u is a c o n f o r m a l constant the

on

whole

map

~D o n e

of

~2

interior

to i t s

constant

itself.

use

the

(cf.

can

. But

standard

[LI] or

extend then

domain

argument

[M3],

it b y

this

pp.

as

conformal

3.3

369 - 3 7 2 ) .

reflection

of definition,

in

as

map

to

Since

)D

, and

map

on

thus

that

u is

a conformal

is c o n s t a n t

namely

show

on

a curve

has

to be

q.e.d.

The by

same H.

argument

Wente

Prop.

4.2

shows

constant, By way

of

namely

Proposition

nonexistence

from

the

assumptions

energy an

Let

a torus

D be

formal tor.

of

the

map

pole

of

and

in a d i f f e r e n t

the

boundary

context

Thm.

the

non-

there

do e x i s t

nontrivial

with

constant

boundary

onto

[KW]

hypothesis 4.3

a geodesic

of P r o p . (cf.

that

to

that

with

, not

in t h e

D onto

S2

-Wood

Thm.

are

also the

hold. there

degree

4.2 w e r e

ob-

[Se2]).

boundary

Cor.

12.2

is n o one.

va-

loop.

values

will

harmonic

Finally,

are

give map

even

every

homotopy

class

contains

following

argument

shows

which

at under

an

is b a s e d

[LI ].

suppose

D and

annulus

namely

4.3

disc

mapping

the for

as

that

a sphere

generalizations

Karcher

values

4.3.

see

a 2 -sphere

map,

unit

of

the

result,

Lemaire

Furthermore,

rotations

We

of

minimizing idea

map

that

onto

to onto

dimensional

shows

that

in Thm.

easy

is n e c e s s a r y

all

on

assumption

which

[W2] a n d

4.2

constant

another

independantly

an a n n u l u s

those

by Wood

the

it is

from

Higher

tained

used

be d e l e t e d

contrast, maps

Remark:

not

that

cannot

harmonic lues,

was

[Wt].

complex

the

upper

that

(the

plane,

and

hemisphere

and

k is e q u i v a r i a n t

latter

ones

k:

D ÷ S 2 be ~D o n t o

with

leaving

the

the

respect

north

a conequa-

to t h e

and

south

S 2 fixed).

choose

the

orientation

on

S 2 in

such

a way

that

the

Jacobian

of k

is p o s i t i v e . Let

D(O,r)

be

the

plane

disc

with

center

O and

radius

r

(i.e.

D =

D(O,I)). Let

h r be

pole, bian

is

a map

there,

on D and

from

injective and

define

is for

D(O,r)

in

the

onto

S 2 which

interior

e -conformal. O < r < I the

of

We

maps

D(O,r)

introduce

mapping

~D(O,r)

and

has

polar

k r by

onto

the

north

a positive

Jaco-

coordinates

(p,~)

52

I k(]-/~

p + ~

r

ifr~@$1

, M)

=

k r (P'~)

Using

the

k r can

~hr

(p,~)

ifO~p~r

e -conformality

be m a d e

theorem

arbitrarily

it

close

to

is e a s y 6z

to

if w e

see

that

the

energy

choose

r > O

sufficient-

image

of k r

, counted

of

ly small. On

the

with

other

hand,

multiplicity.

topic

to

kr

Since

map

has

itself

nitely which

many

the

a circle

map.

Hence

By

letting

this

energy

and

given

of

in

ses

S 2 more

and

by

proof

of

an e n e r g y two

other

hand,

are

4.3.

of

related A

different

r classes

lower Hence,

which

other

proof

, however,

the

map.

to e a c h

to

a confor-

If h r

classes

have

to k

other

minimzing

homotopy

for

D onto

infi-

values

homotopic

argument.

to a m a p

conformal

it w o u l d

obtain

homo-

to be

exist

possible

we

a similar

to

Thm.

once,

this

boundary

that

map

map has

would

same

is n o t

minimizing than

they

there

the

implies

which

energy

precisely

maps,

the

On we

get

the

~ with

more

= E

, our

homotopy with

the f u n c t i o n a l

than

possibility

E~

new

, a map

the

can

from

ferent

a point

minimizing therefore

equivariant,

with

is h o m o t o p i c

are

it

(otherwise

however,

r homotopic

energy

are

This,

maps

the

in

contain

by

the

of

this

fact was

shows

that

in m a n y

[BC2].

Remark: cases, Apart

there

values

of

, and

maps

k

hence

area

conformal

h r cover

minimizing

operation

6~

is no

, then

example,

is an

to be

minimizing

-I

hemisphere

there

in D to

there

energy

degree

if has

equivariant

possible).

collapse

the

boundary

to be

mal

without

energy

homotopic

is n o t

just

Hence,

, its

conformal.

has

6~ is

E(~)

the two

proof

that

there

construction

classes = E

as

soon

, changes

determinant

of

of

Thm.

homotopically might of as

the the

sign

or

~ vanishes,

4.3

distinct

harmonic

maps.

several

homotopy

clas-

be map

v can

yield

functional if t h e r e

while

d~

two

dif-

determinant

exists

of

a point

is n o n z e r o

at

where

this

point.

4.7.

Existence

Actually, sion

of

image [LI] to The

for the

Thm.

domain

vanishes, and

this

2

but

[SkU].

in a r b i t r a r y

4.2 is

to 2

not

hold,

, and that

(The p r o o f

it

that the

given

dimensions is o n l y the

image here

necessary

second

homotopy

is a l s o also

that

the

group

dimenof

twodimensional,

immediately

the cf.

generalizes

situation).

situation

ceeds

results

. For

changes,

however,

example,

if n

if t h e

~ 3 , then

dimension for

any

of

the

(compact)

domain Y

and

exa

53

homotopy only

class if

for more cannot

details.

yield

existence They

e

6

use

above,

heat

~ is a n y

hypothesis that map

for

Later

6 e

on,

choice

uniformly result

lined

in

still

manifold,

be

map

cover

to t h e

curved,

is a K ( z , 1 )

any

diffi-

map

manifold).

the

parabolic

(0, ~)

, x

6 X

show

that

exists

, U(tn,')

under

for

all

the

problem

curvature

t 6

converges

towards

[X,Y]

does

end of

proof

showed

if of

A different

the

proof

to

4.6). are

in

there

the

approach [E]

[0,~),

and

a harmonic

the

case

In

the

was

, dim

still

at

least

for

of

case no

given

the

results

by U h l e n b e c k

The

so

unfar

no

cf.

the

is a K ( ~ , I )

obstructions, minimum

outlined

.

in h i g h e r

problem,

function

[U].

= 0

extent

solutions

topological

regu-

in a g i v e n

cases,

image

by out-

partial

map

a large

the

convex

be p r o v e d

, ~2(Y)

Dirichlet

class.

existence

regularity,

these

in a l l

closed

the

homotopy

also

the

converges

this

Y = 2

to

where

a strictly

using

that

that

u(t,-)

a harmonic

existence

proved

a given

image,

is

expect

been

of

X = 3

problem

exists

that

of p a r t i a l

observed

existence

, if d i m

not

and

and

. Nevertheless, I) fundamental. It c a n

Eells

have

there

energy

their

necessary

t ÷ ~

as

the

existence

one

as

using

yield

~ 6

whence

versal

. They

simplified

(in c o n t r a s t

regular

nonpositive

solve

t 6

(4.7.1)

t n is n o t

regarded

also

the

to b e

I)

[Ht]

results

can minimize

a

of

Furthermore,

general

at

in

tn ÷ ~

procedure,

Though

dimensions remark

map

a harmonic

nonexistence

for

in c o n t r a s t

Y

they

Y has

[ES].

homotopy

= ~(x)

to

class

The

i.e.

- T(u(t,x))

sequence

4.4.

homotopy

since

basic

.

results

solved.

method,

sequence

can

larity

a constant,

the

a minimizing

is n o t

where

is n o n p o s i t i v e l y

flow

Hartman

of

case

[EL4]

procedure The

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

this

smooth

some

in t h e

class.

in

cf.

to E e l l s - S a m p s o n

map

if Y

on Y a solution

u(.)

X,Y)

obtained

a minimizing

homotopy

is d u e

a harmonic

be

is m i n i m a l ) ,

situation,

that

to

u(O,x)

where

of

can

then

(Note

Su(t,x) ~t

(4.7.1)

energy

since,

homotopic the

of

(which

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

compact

curvature

S n ÷ Y is They

(for

map

dimensions

existence

[X,Y]

minimum

in t h i s

map

in h i g h e r

the

noted

the

a constant

Therefore,

result

sectional culty

[sn,y],

a harmonic

proved

class

~ £

~ contains

in

on

one

is k n o w n the

4.4.

uni-

54

5.

Uniqueness

5.1.

Composition

In t h i s

section,

which

shall

First

of all,

of h a r m o n i c

we

shall

be u s e f u l

maps with

display

in the

if u £ C 2 ( X , Y )

and f 6 C2(y,~ rule

theorems

where

is a m a p b e t w e e n

is a f u n c t i o n ,

A(fou)

= D2f(u

(5.1.2)

composition

property

the

Riemannian

following

manifolds,

Riemannian

chain

on X

.

i.e.

m(u)

= D2f(u

then

not difficult

We also note

dary,

e,u

(strictly)

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

by Ishihara

following

5.1:

Suppose

constant

mapping.

Proof:

From

C°(~,N)

the m a x i m u m

implies

section,

and Kaul

Theorem

5. I:

reads

function

function

as

on X

If t h e r e

principle

theorem

to p r o v e

that

. Actually,

harmonic

maps

it is

by this

Gordon

[Go]).

manifold I possibly exists

in c a s e

a strictly ~X # @

subharmonic

f has

definite

with

convex

, then

u is a

functions, second

boun-

it fol-

fundamental

is c o n s t a n t .

of J ~ g e r

for t h e D i r i c h l e t

Suppose

for

since

that u itself

we want

(cf.

is c o n s t a n t

and

on Y a n d u is

[Ih].

X is a c o m p a c t

a n d u(3X)

uniqueness

convex

consequence

f o u is c o n s t a n t ,

(5.1.2)

In t h i s

, this

that one can characterize

the

on u(X),

The

= 0

~)

a n d u: X ÷ Y is h a r m o n i c .

that

,

e

if f is a

to see,

function

form,

, T(u)> y

= y~BD2f(Ux~,UxS)

as w a s n o t e d

Proposition

f)ou

coordinates

a consequence,

property,

~(grad

frame

A(fou)

a harmonic,

+

if u is h a r m o n i c ,

A(fou)

o r in l o c a l

~) e

e

J~ger

then

e,u

e ~ is an o r t h o n o r m a l

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

5,2.

an e l e m e n t a r y

sequel.

e

lows

functions

is v a l i d .

(5.1.1)

As

convex

the

and Kaul following

problem

uniqueness

for h a r m o n i c

u. : ~ ÷ N a r e h a r m o n i c l N C 2(~,N) , ~ is a b o u n d e d d o m a i n a n d ui(~)

theorem

maps.

maps

of c l a s s

c B(p,M),

where

of

55

B(p,M) with

is a g e o d e s i c

radius

ball

M < z/2K

in N

, disjoint

(K 2 is an u p p e r

to the c u t

bound

for the

locus

of p a n d

sectional

curvature

of B(p,M)). Then

the

function

@ , q<(d(u1(x),u2(x))

0(x):

= cos

(
It~2

(1 - cos

(qK (t) : =

(

if < > 0

<2

,

satisfie~

the maximum

(5.2.1)

sup

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

if

<

=

O

),

principle

8 <sup

@

if Ul]3fi = u 2 ] a ~

, then

uI = u2 • The proof lity

in

of Thm.

(5.2.1)

for w e a k l y ven

in

[U~K2].

the

Jacobi

Lemma

We will

be

field

i.e.

follow

Furthermore,

the

that

Thm.

The proof

arguments

simpliefied,

the proof

(cf.

IY'I

show that we have

[JaK1]).

somewhat

technique

by arclength,

(cf.

lemma,

Suppose

actually

0 = const.

maps

shall

following

5.1:

along

unless

harmonic

last argument with

5.1 w i l l

5.1

of Thm.

of J ~ g e r however).

of w h i c h

strict

inequa-

also holds 5.1 w a s

a n d Kaul. Thus,

is o b t a i n e d

we

gi(The

start

by Karcher's

[K2]).

y:

[O,p]

÷ N is a g e o d e s i c

= I , a n d 0 < p < ~/K

arc parametrized

. __If X is a J a c o b i

field

y with

(5.2.2)

(x,v,

>

=

0

,

then s ' (p)

~5.2.3) (Here,

(x,x,>

we s<(t)

have

: =

Io~ ~> -s

K

- (p)

(Ix(o) 12 ÷ Ix(p) ] 2)

<

defined

~

1

sin K t

, if < > 0

<

t

, if

< = O

.

2 s~pl[X~Ol]'lxlp)l

56

Note

that

s<

s"<

+ <

Furthermore

Proof:

2

the

sK

solution

= 0

of

, s<(0)__ = 0

t = fs K o

qK(t)

, where

, s,<(0)

q<

was

=

I

defined

in

Thm.

5.1)

Let I = - s<(p)

s(t):

Then

is

s

(Ix(o)]s<(p-t)

+

IX(p)IsK(t)).

solves

(5.2.4)

2

s"

+ < s = o

, s(O)

: Ix(o) l , s(p)

=

Ix(p)1

,

and

s ~ O

on

[ O,p]

and

(5.2.5)

s' (0)

I(p)

s

([x(p)J

- s'< ( p ) I x ( o ) l )

K

s' (p)

Then,

the

g:

is

= slxl'

since

(5.2.6)

which

-

X

( s '< ( p ) I x ( p ) f

-

Ix(o)'t)

a

s'lxl

where

solves

X"

is

1 s<(p)

function

differentiable,

lated,

-

the

JXJ

second

.

Jacobi

+ R(X,y,)y'

linear

~ 0

= O

order

(Note

that

the

zeros

of

X

are

iso-

equation

,

equation.)

Moreover,

~':

slxL" I =

S

•

IxL 3 ->_ O

,

- s"

Ixi

= s(

(Ix121x' 12

)

+ <2slxl

- ~x,x') 2) -

s--[~r <x,~ix,~'),~'

> + ~2slxl

57

since sing

by a s s u m p t i o n on t h o s e

points

<X,R(X,y

intervals

T where

,

where

g' d o e s

not

2

)y'>

~ < IXl

2

. Thus

it is d i f f e r e n t i a b l e .

exist,

i.e.

g is not As was

decrea-

noted

IX(T) I = O are d i s c r e t e ,

above, and

moreover

g(T+O)

Thus,

- g(T-O)

= 2s(T) IX' (T) ] > O

g is n o t d e c r e a s i n g

IXl '(p)

= lim ~+O

on

[O,p]

IXl '(p-e),

.

, and d e f i n i n g

IXl ' (0) = lim s+O

IXl '(s)

,

we c o n c l u d e

O < g(p)

- g(O)

= s(p) IXl ' (P) - s' (p)IX(p) I-

- s(0) Ixl'(o) -- < x , x ' >

+ s'(0) Ix(0)l:

(~)-

<x,x'>

(o)-

s'< (p) s<(p)

(Ix(o) l2 + Ix(p)12)+

+F 2Ix(01 IxI ll, by

(5.2.5). q.e.d.

By a s s u m p t i o n a unique between denote

and

minimal Yl

2.4,

any

geodesic

a n d Y2 by the

this

(possibly

two p o i n t s in B(p,M),

length

modified)

y l , y 2 6 B(p,M)

c a n be

and we can m e a s u r e

of the g e o d e s i c distance

arc b e t w e e n

function

again

joined

them.

Q < ( Y I ' Y 2 ) : = q < ( d ( Y 1 ' Y 2 )) a C2 function

Moreover,

we n o t e

Ty(N×N)

for y = In the

on B(p,M)

(yl,y2)

since

I

q<(O)

= 0

that

= TyIN~Ty2N

following

x B(p,M),

6 N×N

(isometrically)

.

lemma,

we

shall

estimate

the H e s s i a n

We

by d ( Y l , Y 2 ) .

Then

defines

by

the d i s t a n c e

of Q< o n

58

B(p,M)

x B(p,M),

Lemma

5.2: v

using

the J a c o b i

I--~fYl ~ Y2

6 Ty ( N x N ) ,

' then

2Q

K

If v has the s p e c i a l

and this

D2QK(V,V)

also holds

Proof:

of L e m m a

5.1

y = (yl,Y2) , y l , y 2 : 6 B(p,M)

D2Q (v,v)

(5.2.8)

estimate

for all


(5.2.7)

field

First

Q<(y) ,v> 2 - <2Q<(y) ]vI2

(y) K

f o r m O~u or u~O

, then

> (I - K 2 Q < ( y ) )

luJ 2 ,

for Yl = Y2

"

some d e f i n i t i o n s

p: = d ( Y l , y 2) v =: v 1 ~ v y:

~ 6 TyIN~Ty2

[O,p] ÷ B ( p , M )

N ,

is the unique

geodesic

arc from Yl to Y2 w i t h IY'l

el(Y):

= -y'(O)

e2(Y): = y' (p) tan vi : = ~vi,ei(Y) > ei(Y) nor tan v. : = v. - v. 1

Then,

1

since

(i = 1,2)

1

P > O ,

grad d(y)

= e I (y)~e2(Y)

grad QK(y)

,

= SK(P) (el(Y) ~ e 2 ( Y ) )

D2QK(y) (v,v)

,

and

= < D v grad Q < , v >

!

(5.2.9) If Ys(t)

= SK(P)<e1(Y)~e2(Y),Vl~V2> is the g e o d e s i c

2 + s<(p)D2d(v,v)

arc w i t h

nor, (sv~Or) Ys (0) = eXpy I ISVl ~ ' Ys p) = eXpy 2 .

:

i

,

59

(note that Ys is unique,

if s ~ 0 is small

enough),

then

(5.2.10)

X(t):

is a Jacobi

X(O) By Synge's

= ~

Ys(t)'S=Ol

field a l o n g

nor = vI

nor = v2

, X(p)

formula

(5.2.11)

y with

(cf.

[GKM],

§4.1),

2 = ~ length ~s 2

D2d(v,v)

~(Ix'l 2

=

(ys) is=O

<X,R(X,~')y'>

-

)dt

O

(note that there X,7'>

is no b o u n d a r y

= 0

=

since

)

Since X s a t i s f i e s D2d(v,v)

term,

(5.2.6),

~(Ix'l2

we can apply

Lemma

5.1

to o b t a i n

+ <x,x">)dt

O

'

p

=<x,x > io

>

!

s<(p) (iv~Orl2 > sK (~) and thus with (5.2.12)

+

nor 2 v2 )

2

nor

I'lv2

(5.29) D2Q<(v,v)

=> S'K (p) ( < e I ~ e 2 , v l e v 2 > 2 + Iv nor I I2

21v°rlIv°rl If v = O ~ u

nor

s ip) lVl

, (5.2.12)

D2QK (v,v)

implies

> SK(p) < e 2 ( Y ) , u

>2

! + S<(p)[un°rj 2

T

= s<(p)lul 2 = = (1-K2QK(y))

JUJ 2 ,

+ Iv~°rI2)

60 while

in the general

(e1(~e2,v1~v

2

case,

we only have

>2

(Ivlanl2

< 2

tan 2) + Iv 2 l

,

and ivil2

=

vitan 2

and t h e r e f o r e

from

D2QK(v,v)

vnOr +

i

2 I

(5.2.12) 2 _

_> S'K (p) < e 1 @ ~ e 2 ' v l ( ~ V 2 >

(I -s<(p)) (Iv °r12

nor 2)2 v2

+ _-> ~(11

+ s~K(p))

>2 _ (1 -sK(p))'

<e1~e2,vlev2

(Iv112 Iv212) +

2QK(y) 1


Q<(y),v>

2

+

-


+

=

Iv21

2)

q.e.d. We now are in a p o s i t i o n We assume Then,

that

to prove

8 has a p o s i t i v e

8 is p o s i t i v e

Thm.

5.1.

maximum

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

at some

interior

point

x ° £ ~.

of x o , and log 8 > -~ in this

neighborhood. We d e f i n e ~(x):

= QK(Ul(X),U2(X))

=~(I ~

~i(x):

Then

= COS(Kd(p,uI(X)))

8 -

5% (5.2.13)

and

grad

,

-cos d2

,

(u I

Kd(u1(x),u2(x))

if K > 0

(x) fU 2 (x))

if K = 0

i = 1,2

and c o n s e q u e n t l y

log @ = grad ~

grad %01 %01

grad %02 %02

,

61

A log

(5.2.14)

0

= ~

-

A42 -tx ÷ u(x)

can make This

I grad 411

+

e1

2

2

4212

Igrad

42

Since

A41

Igrad ~l ~2

2 42

= (u1(x),u2(x))

use of the chain

6 B(p,M)

rule

x B(p,M)

(5.1.2)

in order

is also to apply

harmonic, Lemma

we

5.2.

yields 2

(5.2 . 15)

A~ => i~rad ~[ 24

-

<

24

2

2 (Idull,.

+

..Idu21 ),

since

~12 =

Igrad where

e

~

((grad

QK) ou , du(e

is an o r t h o n o r m a l

Similarly,

from

4 i(x)

(5.2.8),

frame

)> 2

on ~ .

since

= I _ < 2 Q < ( p , u i(x)) ,

we o b t a i n (5.2.16) Finally,

A4 i(x) by

=< - < 2 4 i l d u i 1 2

(5.2.13), 2

(5.2.17)

_1 Igrad ~I 2 ~2

> -

grad

Igrad

4 I12

Igrad

+

4212

+ 2 41

to2 2

I log 0 , ~ grad

grad 42

grad 41 log 0 + 41

+

42

>

Putting k(x):

I = ~ grad

grad 41 log 0 +

grad 42 +

41 and p l u g g i n g

(5.2.15),

(5.2.16),

42 and

(5.2.17)

tain A log @ +

~grad

log @ , k ( x ) >

> O

into

(5.2.14),

we ob-

62

Therefore

log

Thm.

5.1

tion

that

8 is a s u b s o l u t i o n

follows

from'E.

8 has

a positive

of

Hopf's

a linear

maximum

maximum

elliptic

principle,

in t h e

interior

problem,

if t h e

equation,

and

since

the

assump-

leads

to a c o n t r a -

diction.

5.3.

Uniqueness tive

In this

and

section,

5.1

we

the

which

Yau

in

5.2:

show

that,

assumption [Hm].

shall

calculations

and

Theorem

skip

to

to Hamilton

[Sy3],

(those

Schoen

Dirichlet

we want

can

is d u e

Yau

the

We

enable

in t h e

if N h a s ui(~)

shall

has

us

nonposi-

sectional

in T h m .

an observation

to carry

context

nonpositive

c B(p,M)

use

over

of Thm.

the

proof

5.2 were

5.1. of

also

This

Schoen

of

Thm.

given

by

[SY3]).

Suppose

N has

nonpositive

sectional

u.: ~ ÷ N , i = 1,2 , a r e h a r m o n i c m a p s , l u i 6 C2(~,N) n C°(~,N). If u 1 1 ~ = u21~ topic,

image

curvature

curvature, result

for

~

curvature,

is a b o u n d e d

, and

u I and

and

domain,

u 2 are

and

homo-

then

uI ~ u2 . Proof: N has

Let

denotes

the

isometrics

thus

=

N be

the

sectional

distance

on N

~(x,y)

and a

~ and

nonpositive

universal

covers

curvature,

function

of N

~2 x N

is

of

~ a n d N,

smooth

. ~(N)

on N

acts

resp. x N

Since , where

as a g r o u p

of

x N via

(~(x),~(y))

induces

for

~

6 ~i (N)

,

a function

"a': N x ~/~r 1 (N) ->- ~ . Let

F: ~

u I (x)

÷ N and 7

x

and

[0,1 ] ÷ N b e F(x,1)

thus

obtain

6 ~I (~) ' t h e r e

(5.3.1)

for

all

6 ~

.

for

liftings

exists

Ul (Y(x))

x

a homotopy

= u2(x)

~

x Ul

6 ~

between . We

= F(-,O)

u I and

choose and

u2

u 2 , with

a lifting = F(-,I).

6 71 (N) w i t h

= ~Ul (x)

and

u2 (Y(x))

= ~u2(x)

F:

F(x,O) ~ ×

Then

=

[0,1 ] for

any

63

~: ~ ÷ N x N , d e f i n e d

Thus by

(5.3.1)

induces

b y ~(x)

a harmonic

U : f~ ÷ N x N I z I (N)

=

(~1(x),~2(x))

, is h a r m o n i c

and

map

.

Then

O : = 1/2d2ou

is a s m o o t h proof

function

of Thm.

(5.3.2)

5.1

sup

on ~

, and we can

to s h o w t h a t

8 ~ sup

carry

e satisfies

over

the

arguments

the m a x i m u m

of t h e

principle

@

q.e.d.

5.4.

Uniqueness sitive

The

arguments

uniqueness

Theorem

5.3:

a)

Suppose

homotopic

the

curve

, t £

maps

[0,1]

ht(x),

t 6

We construct

5.2,

From

(5.2.15)

TxM

assuming

T

we

sectional

N

. From we

s~ive, (i.e.

X'

a function

infer

for

has

nonpo-

see

following

[SY3].

without

to u u n l e s s

that

curvature,

, then

maps

and with

8 in t h e

u I and u 2 are again

boundary r

u(M)

Then

is c o n t a i n e d

that

8(x)

exists

therefore

between

that D2d(v,v) since

the Jacobi

the

a smooth

length

arc,

= 0

s a m e w a y as

Yl a n d Y2

field X defined

in

of

is a

where

' then we

v = v1~v

curvature

.

maps.

by the maximum

v i = d~i(ee),

, where

of x

in the p r o o f

harmonic

constant

sectional

fa-

~arame-

independant

= I/2 d 2 ( u 1 ( x ) , u 2 ( x ) )

yi = ~ i ( x ) ,

and moreover

and u I and u 2 are

h O = u I and h I = u 2 , and

homotopic

i = 1,2

and

there

with

on M and

5.2,

(5.2.11)

= O)

the in

is for a n y x £ M a 9 e o d e s i c

, a n d y = Y x is the g e o d e s i c of Thm.

shown

manifold

homotopic

to a r c l e n g t h

that

function

If w e p u t

the proof

image

a n d u: M ÷ N is h a r m o n i c .

f r o m M to N

[0,1]

Proof:

subharmonic

map

, of h a r m o n i c

proportionally

Thm.

also yield

as w a s

M is a c o m p a c t

ha~ nonpositive

trized

ciple.

if the

.

harmonic

ht

[Ht],

curvature,

harmonic

of N

If N o n l y

mily

solutions,

section

of H a r t m a n

sectional

is n o o t h e r

in a g e o d e s i c b)

for c l o s e d

of the p r e c e d i n g

theorem

N has negative there

results

curvature

e see

prin£ from

2 6 TyIN

of N is n o n p o -

(5 2.10)

is p a r a l l e l

64

"'<X,R(X,~')~' 2 :

(5.4.1)

Since

G is c o n s t a n t ,

0

d ( ~ 1 ( x ) , ~ 2 ( x ) ) =:

p is i n d e p e n d a n t

of x

, and we

define

ht(x)

= Yx(~)

t 6

[O,I ] and x

6

J

i

Since

X

= O and

(5.4.2)

where

dht(x) ( e )

Yx

= d~2(e

).

~I(N)

exists

htoo

and

e

Yx

by

6 ~I(N)

M ÷ N

From

Yx(O)

since

on N x N

= d ~ 1 ( e ~)

and Yx(p)

X is p a r a l l e l ) .

, we

see

that

for

o

£ z1(M),

with

[O,1 ]).

induced

maps

,

(5.4.2)

and

vector

e(h t) (x) = I/2 TxM)

since

Yx is p a r a l l e l ,

is i n d e p e n d a n t

of t

, and

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

any

the h t have

would with

all

exist

Sampson

proved.

For

sectional Since

a),

we o n l y

curvature,

Yx

independant

infer

that

maps

h

, homotopic than

have

Namely,

dht(x) (e ~)

is a

Thus,

to note

that

if N has

implies

that

X and

, and

t

>

energy.

the

same

energy.

otherwise,

b)

u 2 are h o m o t o p i c arc

also

to u I ,

theorem

of E e l l s

is c o m p l e t e l y

strictly

= O by c o n s t r u c t i o n ,

in the g e o d e s i c

there

negative

Yx! are p r o p o r t i o n a l .

to Yx' ' for any e

if u I and

same

to h t and thus

have

<X,Tx

the

h t by the e x i s t e n c e

is a c o n t r a d i c t i o n .

is p r o p o r t i o n a l of x

harmonic

h t have

(5.4.1)

hand

to be c o n t a i n e d

the m a p s

to be h a r m o n i c .

energy

which

on the o t h e r

therefore

map

smaller ~S],

all

two h o m o t o p i c

a harmonic

strictly

we

f i e l d a l o n g the g e o d e s i c f r o m ul (x) to u2(x). T h u s n [ Idht(x) ( e ) I 2 (where e is an o r t h o n o r m a l b a s e of ~=I

Therefore,

have

Yx w i t h

hO = u I , h I = u 2

parallel

and

along

is p a r a l l e l ,

(t £

we o b t a i n

,

isometries

= ~oh t

Therefore,

ht:

field

(Note t h a t acts

holds,

= Yx(t)

is the J a c o b i

Because there

(5.4.1)

6 TxM

X = O

, and

. Thus,

harmonic

Yx

maps,

is they

y = Yx q.e.d.

65

Remark:

In the t w o d i m e n s i o n a l ,

o r e m of E e l l s - S a m p s o n

5.5.

Uniqueness

which

we can replace

w e do n o t p r o v e

and nonuniqueness

the a p p e a l

in t h i s

for h a r m o n i c

maps

to t h e

the-

b o o k b y Thm.

4.2.

between

closed

surfaces We now

look more

between

closed

carefully

surfaces,

t h e n u is n e c e s s a r i l y (unless

it is c o n s t a n t , the o n l y

12.1.

a family

case

Therefore

ZI ÷ Z2

a conformal

quently Cor.

at u n i q u e n e s s

u:

which

maps,

is the

or a n t i c o n f o r m a l

of h a r m o n i c two -sphere branched

maps S2 ,

cover

is t h e

c a s e if X ( Z 2) ~ O , a n d c o n s e is Z2 = S 2 ), as w e s h a l l see in

of i n t e r e s t in e v e r y

of harmonic

properties

" If ZI

homotopy

which

class

depends

on

e 6

[$2,$2],

we obtain

(2 + 4 d e g e) p a r a -

meters. If ~I or ZI o r

Z2 is a f l a t

Z2 a g a i n

O n the o t h e r tries

gives

hand,

homotopic

formal

with

torus

a surface

to the

respect

with

ter

I , nonuniqueness

We

the

also observe

also

isometries

domain

cannot

homotopy

(or c o n f o r m a l

cause

class

any

A-priori

6.1.

Composition

In t h e c a s e ,

Lemma

situation,

of nonzero

6.

position

Proof:

of

be c a u s e d Theorem sense

by

that

if the g e n u s

if Z2 has isometries 12.1),

its d e g r e e

cf.

of h a r m o n i c

genus

to

grea-

of the

image

any map

from

is zero.

Lea%Ma 6.1)

maps

be con-

and had

of Z2 is g r e a t e r

automorphisms,

isome-

~uld

structure

(cf. Thm.

any

t h a n one, of

the

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

degree

between where

surfaces

of g e n u s

the h a r m o n i c

map

at least

two,

in a g i v e n h o m o -

is n o t u n i q u e .

C1'~-estimates

where

of harmonic

maps

X is a s u r f a c e ,

with

conformal

we have

the

maps

following

useful

com-

property:

6.1:

Suppose

M ÷ X is a c o n f o r m a l again

not have

s u c h an i s o m e t r y conformal

g > I . Thus,

nonuniqueness

in the c a s e o f m a p s

do n o t k n o w

topy

in t h e

case,

an i s o m e t r y

class.

Actually, we

since

by K n e s e r ' s

in t h e n o n t r i v i a l

u with

g > I does

since

cannot

S 2 o r T 2 t o Z2 is t r i v i a l Thus,

of g e n u s

underlying

identity,

that

composing

map.

identity,

to the

coincide than

T 2 , then

a harmonic

harmonic,

u

6 C2(X,Y)

map

between

a n d E(u°k)

We only have

is h a r m o n i c the

and dim X = 2

surfaces

M and X

= E(u)

to n o t e

that

(1.3.1),

namely

If k:

, then uok

is

66

u i) I /Y

~ ~x ~

remains valent

(~yy y~B

valid, one,

(1.3.4)),

if we

say

and

replace

l(x)yes(x)

the

same

the m e t r i c

, l(x)

is true

> O

%~ B(x)

by a c o n f o r m a l l y

, in case

dim X = 2

equi-

(cf.

also

for

~u i ~u j

I

E(u)

+ ya~F i ~ uj ~ uk = O jk ~x ~ -~x 8

~x ~

= ~ ~ ¥~Bgij

/~ dx I dx 2 ~x ~

q.e.d.

Cor.

We

6.1:

On

shall

prove

later

on e x p l o i t

a -priori

for the derive

case

a -priori

taking

6.1

conformal

in the

for h a r m o n i c

maps

maps

between

the d o m a i n

is the u n i t

estimates

from below

for

maps,

us to a p p l y

the e x i s t e n c e

i.e.

these

theorem

the

maps

We

shall

surfaces,

disc.

first

T h e n we

conformal

estimates

for c o n f o r m a l

way.

functional

in p a r t i c u l a r

special

are h a r m o n i c .

following

where

harmonic

enable

domains,

Lemma

estimates

special

of u n i v a l e n t will

twodimensional

shall

determinant maps,

which

to the g e n e r a l

case,

of c h a p t e r

3 into

account.

6.2.

A maximum

The p u r p o s e mates not

of the p r e s e n t

for g e n e r a l

only

could

appeal

harmonic

to Thm.

however.

dary values

map

5.1

mates

obtained

trary

twodimensional an e x i s t e n c e

Schauder

degree

following

dimension. sional general

case

4.1

chapter

domains

estimates

all

for

4.1,

simplicity

for

that

the d o m a i n

for

be

stated

image and

of e x p o s i t i o n ,

we

conditions

that

goal

in

the b o u n -

the C 1 ' ~ - e s t i -

D and extended

9 will

this

these

a different

energy,

eventually

assumption

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

only minor

under

ball,

controlled

Of c o u r s e ,

to r e q u i r e

finite

hold

4.1.

esti-

in a c o n v e x

we a l r e a d y

We h a v e

we h a d

with

without

instead

requires

of Thm.

to s h o w

a -priori

contained

for w h i c h

in c h a p t e r

theorem

of t h e m w i l l

is to o b t a i n

image

minimizing.

theory

Some

ones,

and

in T h e o r e m

to p r o v e

The

ones,

an e x t e n s i o n

in this

now

with

in the p r o o f

is e n e r g y

While

possess

maps

minimizing

of c o n t i n u i t y

every

chapter

harmonic

for e n e r g y

the m o d u l u s

mind,

principle

to a r b i -

enable

using

us

Leray -

methods.

manifolds proved

although

modifications.

of a r b i t r a r y

only

They

for t w o d i m e n -

the p r o o f also

hold

in the for

87

domains cf. We

e.g.

arbitrary

with

Jost

Lemma

the

[J1],

6.2:

, e.g.

B(p,M)

If

is

Suppose

~ is is

max x£~ is

Proof:

which

For

by

c in V w i t h

the

proofs

are

different,

principle

which

can

be

for

in B ( p , M )

~

p,q

(cf.

6 V

q

cut

locus

6 B(p,M)

d(u(x),q)

M

boundary

< z/2<

, and

of

p

, then,

is

a convex

if

,

,

= B(u(x),M)

. By

Lemma

, c(I)

2.4),

d(u(x),c(t))

the

some

Lipschitz

c B(p,M),

.

, V:

= p

with

u(~)

to

deduced

[Td].

domain and

disjoint

< max x6~

6 ~Q

c(O)

Tolksdorf

a bounded

is

on

x

assumption,

(6.2.3)

or

harmonic,

d(u(x),q)

any

and

= q

we

N B(p,M)

2.1,

we

. Since

can

this

find

set,

a geodesic

geodesic

arc

is

infer

<_- f o r

all

x

6 ~

for

t 6

define

~(t):

~(t)

have

= max x6~

d(u(x),c(t))

is

a continuous

We

now

~(t)

claim

= z/2<

function

~(t) for

of

< ~/2<

some

t

is

for

subharmonic

subharmonic

on

max x6~

d (u (x) ,c (t) ) < m a x x6~

(6.2.3).

We

conclude

that

in p a r t i c u l a r

strong

maximum

principle

max x£~ unless

u

d(u(x),q)

is

constant

= max x6~ on

~

~

, and

we

2.2

can

get

~ ~/2<

and

~ M

if w e

< ~/2<. would

(5.1.2), the

maximum

contradiction

=< M

, and

< max x6~

~(0)

Indeed,

apply

the

subharmonic

d(u(x),c(1))

.

assumption

Lemma

to

.

[O,1].

d(u(x),c(t))

~(I) for

by

t 6

by

functions,

by

[0,1]

, and all

, then,

principle

2<

t

for

d2(u(x),c(t))

the

here

maximum

c B(q,M)

constant

arc

We

~ +Y

a ball

(6.2.2)

unique

but

[Sp],

u:

u(~)

u

following

Sperner

(6.2.1)

unless

dimension,

[JKI].

start

from

~

of

applying

functions,

this we

time

obtain

d(u(x),c(1))

q.e.d.

68

The

continuity

Widman

[HWI].

principle monic

6.3.

argument With

[St],

maps,

one

cf.

Interior

harmonic, disjoint

using

locus

is d u e

to H i l d e b r a n d t

Stampacchia's

principle

-

maximum

for w e a k l y

har-

above.

the

we o b t a i n

~ is a t w o d i m e n s i o n a l

of p

. Then

same

the m o d u l u s

on

I

(where

(injectivity

o f the

domain,

, a n d B(p,M) is a b a l l ,

only

i(~)

of c o n t i n u i t y

Lemma

6.2,

M < ~/2K

of ~),

We can use

a maximum

mentioned

in d e p e n d a n c e

curvature

the modulus

Proof:

Suppose

c B(p,M),

to t h e c u t

proof

and using

of c o n t i n u i t y

of L e m m a

6.1:

c a n be e s t i m a t e d for t h e

can also prove

modulus

u(Q)

argument

the r e f e r e n c e s

As a n a p p l i c a t i o n

Proposition

in t h e p r e c e d i n g

a similar

boundary

argument

6.2 as t h e a p p r o p r i a t e

as

-l

< , M

is

, E(u), a n d

ul~.

in the p r o o f

maximum

~ ÷ Y

which

of c o n t i n u i t y of u 2 is a l o w e r b o u n d

radius), values

u:

principle

of Thm. this

4.1,

time.

q.e.d.

6.4.

Interior

estimates

In the i n t e r i o r , of Prop.

6.1

we can

by v i r t u e

Lemma

8'):

Lemma

6.3:

Suppose

and

u(D)

c B(p,M),

locus

of p

, with

we have

the

remove of

the

the d e p e n d a n c e following

Lemma

u: D ÷ Y is h a r m o n i c , where M

< ~/2<

B(p,M)

on E(u) of E.

where

is a b a l l

. __If B(Xo,P)

in t h e e s t i m a t e

Heinz

f I

2--~ B ( X o '

)

Idul2dx -<-

I

< ctg(<M)

disjoint

~ D , then

log

I

p/pl

We define

f(x):

= I/2 d 2 ( u ( x ) ,U(Xo))

disc,

to the c u t

for O

max d (u (x) ,u (x O) ) x 6 ~ B (Xo,P)

Proof:

([Hzl],

D is the u n i t

inequality

I

(6.4.1)

for the e n e r g y

< pl

< p ,

69 Then,

using

Lemma

2.2b),

; log ~ - - T l d u l B(Xo,P)

i

2

log ~ B(Xo,D )

I KM ctg(KM)

S (f(x o + pe i~) ~B(Xo,P)

-

2~

< = using This

Lemma easily

Af d x

KM c t g ( < M )

f(Xo))dR0

-

_-<

max d(u(x) ,u(x O)) x6~B,Xo,P~,

< ctg(KM)

=

,

2.2a). implies

(6.4.1).

q.e.d.

Remarks:

A corresponding

in a n y d i m e n s i o n , for w e a k l y

cf.

harmonic

[JK1],

maps,

statement Bsp.

not necessarily

At the boundary,

we have

does

not pertain

to t h e b o u n d a r y .

6.5.

Boundary

In t h i s ven

in

Proposition again

6 > 0

(cf.

6.2:

also

for a n y d o m a i n

it is e v e n v a l i d

continuous.

in a d i f f e r e n t

way,

since Lemma

to p r o v e

the b o u n d a r y

regularity

result

Suppose

u: D + B(p,M)

M < Z/2K

on ~

then

, K , M

is h a r m o n i c ,

and disjoint for e v e r y

to t h e

cut

d(u(y),U(Xo))

If g is H O l d e r

(6.5.2) e and

continuous

d(u(y),U(Xo)) c

depend

gi-

g > 0 we can

, the m o d u l u s

where

B(p,M)

locus

of p

find

some

of c o n t i n u i t y

of g

s , for w h i c h

(6.5.1)

6.3

[J1]).

is c o n t i n u o u s ,

, depending

a n d on

where

we want

is a b a l l w i t h

I_~f g = u I Z D

to a r g u e

holds

continuity

section, [GH]

actually

2. F u r t h e r m o r e ,

on ~

< s

with

=< c

for y

some

0 B(Xo,6)

exponen t

Jy-XoJe

, K , M

6 D

8 , then

for y 6 D

,B, a n d

JgJc8

"

N B(Xo,6)

.

,

70

Proof:

We n e e d some d e f i n i t i o n s :

D(Xo,R):

= D N B(Xo,R).

If x ° 6 ~D , let c: c(I)

[0,1] ~ B(p,M)

= g(x o), p a r a m e t r i z e d

Pt:

= c (t)

vt:

= d 2 (u(x) ,pt )

Furthermore,

be the g e o d e s i c w i t h c(O)

proportionally

to a r c l e n g t h ,

= p ,

and

,

let wt, R be the s o l u t i o n

Awt, R = 0

of

on D(Xo,R)

Wt,RI ~D(Xo,R) If G R is G r e e n ' s

= vti~D(Xo,R)

function

on D(Xo,R),

we d e r i v e

from Green's

represen-

tation formula

(6.5.3)

~

Avt(x)

GR(X,y)dx

= Wt,R(y)

- vt(Y)

D (Xo,R) From

the d e f i n i t i o n vt(Y)

(6.5.4)

of v t a n d wt, R , we have

= d2(u(y),pt ) <

(1+t)2M 2

and

(6.5.5)

W t , R ( X o) = v t ( x o) = d 2 ( ~ ( X o ) , P t ) < (1-t)2M 2

We n o w w a n t

to e x p l o i t

~D(Xo,R) Namely, r(e,R)

t h a t the b o u n d a r y

of wt, R on

are a i v e n by d 2 ( g ( x ) , P t ), i.'e. c o n t r o l l e d given

e > O and R > O,R ~ R

( d e p e n d i n g on c , R , M

d 2 ( g ( x ) , P t ) on ~D N ~D(Xo,Ro)

(6.5.6)

Wt,R(y)

for all y 6 D(xo,r). found,

values

e.g.,

in

o

by a s s u m p t i o n .

, there exists

, and the m o d u l u s w i t h the p r o p e r t y

~D n

some n u m b e r

of c o n t i n u i t y

r = of

that

=< W t , R ( X o) + e This

is a r e s u l t

[GT], Thm.

If d 2 ( g ( x ) , P t ) is H ~ i d e r

from potential

8.27).

continuous,

w e even h a v e

theory

(and can be

71

(6.5.7)

where

Wt,R(Y)

e

, ~ depend

We n o w w a n t

we

e

to a p p l y

~:

=

~

e:

= M2((I-

with

for

~ is the

some

radius

where

r is the s a m e

Then,

with

=

by L e m m a

R

smallest o

=<

r as in

(5.1.2),

by

a n d if m. _~ ~ / 2 < l

(6.5.3),

a

I . Furthermore,

(6.5.6)

; Idu[2G (x,y)dx D(Xo, R i - i) Ri_ I

(6.5.4),

, t h e n by

_~ v t ti+1

+ ~ M ~ z/2<

(6.5.8)

,

i

mi+ I < z/2< by induction,

_~ ~ / 2 <

and again

~

(6.5.6).

w t . , R " (y) =< (I - t i ) 2 M 2 + e _~ M 2 1 1

m I _~ z/2<

m

with

< I and define

Furthermore,

Therefore,

integer

(i=I .... p),

2<m. c t g ( < m i ) 1

(6.5.8)

i.e.

and put

max (vt. (x)) I/2 x 6 D (Xo, Ri_ I ) l

2.2b,

v

I glc 8

procedure,

I ~ i _~ p - I

R i = r ( e , R i _ I)

mi:

, 6 , and

(1 _ ~ ) 2

t : = I, w h e r e start

, K , M

an i t e r a t i o n

for y 6 D ( X o , r ) ,

12e

I

2M<

t. : = i ~ l and

-~ W t , R ( X O) + ~ I Y - X o

from

, (6.5.8),

and

(6.5.6)

(y)

+ v

=<

ti

f o r all y 6 D ( X o , R i)

72

Vl (Y) --< W 1 , R

(y) < W l , R

for all y 6 D ( X o , R This

gives

boundary,

(Xo) + e =

).

the d e s i r e d

estimate

o f the m o d u l u s

@ = R U the b o u n d a r y d a t a a r e H ~ i d e r

In c a s e

e

of c o n t i n u i t y

a t the

putting

continuous,

we

use

(6.5.7),

to

get

d(u(y),U(Xo))

_-< ( v 1 ( y ) ) 1 / 2

< cly-Xo

la

q.e.d.

Remarks: weakly

The proof works

harmonic

maps,

in a n y

dimension,

if one a p p r o x i m a t e s

a n d is e v e n

Green's

valid

function

for

via molli-

fications. The

corresponding

cated,

cf.

Corollary we

[HJW]

and

6.2:

can c a l c u l a t e

dulus

iteration

@

, depending

> 0

d(U(Xl),U(X2))

This

6.6.

Interior

We now prove

Theorem

6. I:

where

D is

where

B(p,M)

curvature

I

Suppose

the u n i t

~ , w

the p r o p e r t y

from Prop.

c

o

6.1,

Prop.

case of Thin. 3.1

6.2,

of

o

xI , x2

and Lemma

6.3 t h r o u g h

[JKI]

u: D ÷ Y is a h a r m o n i c

disc

in the c o m p l e x

is a d i s c w i t h on B ( p , M ) ) .

IdU(Xo) I A c

= c

, and the mo-

for a l l

considerations.

radius

M

Then,

plane,

map between

(~,<,M)

max o x6B(Xo,R )

surfaces,

a n d u(D) c B ( p , M ) , 2 2 -~ _-< K _~ ~ are

< ~/2<

(here

if B ( X o , R )

c D

d(u(x),U(Xo))

with

e > 0

, < , M that

mate

(6.6.1)

for e v e r y

-estimates

a special

bounds

6.2,

,

elementary

C

compli-

-< e •

follows

straightforward

o f Prop.

only on

o f ul 8D , w i t h

~ @

is m o r e

.

the a s s u m p t i o n s

d(Xl,X2)

Proof:

in the i n t e r i o r

Under

of c o n t i n u i t y

6 D with

[GH]

procedure

, we have

the e s t i -

73

Proof:

The e s s e n t i a l

We p u t

R : = R/2 o

~:

There

=

i d e a of

the p r o o f

max (RO - d ( X , X o ) ) x£B(Xo,R o )

exists

x

is d u e to E.

Heinz

[Hzl]

and

6 B(Xo'R)o

- Idu(x) l

with

I (6.6.2)

=

(R O - d ( X l , X o ) )

IdU(Xl)) I

d: = Ro - d ( x 1 ' X o ) and

(6.6.3)

Idu (Xo) I < ~ / R O

Furthermore,

=d

By Prop.

(~):

6.1

as d e s i r e d

~ -_< ~

6.3

choosing

~6

o

max d(u(x),u(xl)) x£B(Xl,dS)

and Lemma

by

8o d e p e n d s

=

_~

only

on ~o

' ~

of d , s i n c e

leaves

invariant

E(u)

we

only

the

center

require

6

= R/2), 6 c a n be m a d e as s m a l l o small, in o t h e r w o r d s

o

independant

u(x I) as

(p = R,p I = R

8 sufficiently

, < , and M

. Note

by a h o m o t h e t i c

by L e m m a

6.1,

we

~ M

in p a r t i c u l a r

of the

can a c h i e v e

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

o

that

change

d =

h of L e m m a

8o is

domain

which

I ° We

take

2.5.

For a moment,

.

Then

(6.6.4)

~//d =

IdU(Xl) [ =

i

<

S

- ~d282

I

we have

Lemma

8" in

Now

lh(u(x))

f

- h(u(xl))lldxl+

l~houl d ( x , x I)

u s e d an e a s y

[Hzl],

=<

~B (Xl ,d~ )

+ 2-~ B ( x l , d ~ )

Here,

Id(hou) (Xl) I

or Lemma

consequence 2.3 in

of G r e e n ' s

[JKI]

formula,

for a m o r e

general

cf.

e.g.

statement.

74

(6.6.5)

lh(u(x))

- h ( U ( X l ) ) I _~ ci"~

(6.6.6)

l~hou(x)

< c2- Idu(x) I2

2

where

c I and c 2 can be

(6.6.4)

-

2cI~ ~ - d~ -

u/d

or,

(6.6.6)

if 8 _-< @

(6.6.7)

on ~

Then

above,

(2.6.2).

+ b/2

~U 2

can

choose

we

for all 8 ~ ~o

> O so s m a l l

o

(depending

only

that

abe(8)

(6.6.9a)

and

d(1-~) 2

~(~o.) ~

~ a/2

noted

(6.6.7)

(2.6.1)

~ 28 + c2

, < , a n d M)

(6.6.8)

from

imply

=< 1/2

o

~

As we have

calculated

=< c 2 d 2 ( i _ @ ) 2

_-< a b d ( 8 o )

< 1

for

implies

that

for 8 ~ 8

~ 8 _->I/b

(1 +

1-~-/~-6)

~8-
(I - / ~ )

all ~ ~ 8 °

, either

o

or

(6.6.9b)

Since ~

is d e f i n e d

that only

independantly

the s e c o n d

(6.6.9b)

holds

(6.6.10)

~

for

for a l l

possibility

@ = 8

o

, and

of 8 , w e of

8 ~

see,

(6.6.9)

o letting

can hold.

8 tend

t o zero,

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

therefore

a~(~ o) 8o

The

result

then

follows

from

(6.6.3).

q.e.d.

Remark: and

An e s t i m a t e

for a n y d o m a i n ,

along

the

to p r o v e

same

cf.

lines,

a suitable

like

(6.6.1)

[JKI],

needing

version

of

Thm. a much

actually 3.1. more

(6.6.4),

holds

The proof refined

however.

in a n y d i m e n s i o n basically

geometric

procceeds argument

75

Other

interior

[GH],

Choi

6.7.

[Ci],

Interior

Corollary

C I -estimates and Sperner

obtained

by G i a q u i n t a -

Hildebrandt

[Sp].

C I' a - e s t i m a t e s

6.3:

(6.7.1)

were

Under

the

assumptions

lUlc I

_~ c I

of Thm.

6. I,

,

' ~(B (Xo, R/2) ) c I = C I (~,<,M,R) Proof: u + Fi

(6.7.2)

jk a n d we t a k e

as

local

x I 6 B (x ,R/2)) o corresponding

~uJ

~uk

~x ~

~x ~

ccordinates

those

given

Christoffel

(6.7. I) is a d i r e c t

= O

on a n e i g h b o r h o o d

from Lemma

symbols

consequence

2.5.

of u(x I ) (for any

Since

the m a g n i t u d e

F ijk is c o n t r o l l e d

of

(6.6.1)

by L e m m a

and potential

of the

2.5,

theory.

q.e.d.

6.8.

C1 -

In this

and

C 1'~-estimates

section,

we

shall

6.2:

Suppose

at the b o u n d a r y

treat

the

twodimensional

case

of Thm.

3.2 o f

[JKI].

Theorem where

D a~ain

disc w i t h

is the

radius

u:

unit

M < z/2<

D + Y is a h a r m o n i c

disc,

a n d u(D)

. Suppose

map between

c B(p,M),

where

surfaces,

B(p,M)

is a

uI~D = g £ CI'~(~D,B(p,M)).

Then

(6 . 8.1)

IUlcI(D) <

c 2

c 2 = c 2 ( ~ , < , M , IglcI,~,T), way

that B(p,M+T)

Proof:

By Thm.

is still

6.1

,

where

0 < T < ~/2<

a disc,

and Prop.

max d ( u ( x ) , u ( x )) d (x ,x O) <-R o

_~ c a

i.e.

6.1,

- M is c h o s e n

disjoint

to the c u t

it is e n o u g h

to p r o v e

in s u c h

a

locus

of p

.

78

in c a s e

d ( x O, BD)

= R

,

or

(6.8.2)

d(u(x2),U(Xl))

~ cR °

for d ( X o , X 2) ~ R , d ( X o , X l ) This

can be achieved

is to r e d u c e schitz

We

that

v is i n d e e d

desic

that any

arc i n s i d e

If u(x)

T ° Using

Prop. c D of

we

6.2

in B ( p , M + T )

b y Thm.

continue

and possibly

(6.8.5)

B ( x 1,6)

N D c Do

(6.8.6)

Vx 6 B(Xo,Ro):

(6.8.2),

v(x)

Lemma

where

to g u a r a n t e e ,

first have

to r e d u c e

the

can be

joined by a unique

geo-

2.1.

q(x)

arc

from

6 B(p,M+T)

shrinking

R , we

u(x) with can

to u ( x I) b e y d(q(x),u(xl))

=

find a closed

sub-

we

take

for s o m e

~ > 0

u ( D O) c B ( q ( x ) , ~ / 2 < )

an~° x 2 6 B ( X o , R ) ,

assume

w.l.o.g,

u ( x 2)

2.2b,

and

of Lemma

3 , we

compare

same boundary

values

o n ' D ° as v

(5.1.2)

= d2(u(x),q)

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

in D o

the harmonic

. As

function

i.e.

(6.8.7)

In o r d e r

and put q = q(x2).

(6.8.6),

with

to a L i p -

= d2(u(x),q),

class C 2 with

c Do

u(x I),

we

the g e o d e s i c

to a p o i n t

B(Xo,R)

By

v(x)

idea

6.2.

(6.8.4)

To prove

at the b o u n d a r y

B(p,M).

a subharmonic,

two points

coming

region

o

(6.8.2)

. The p r i n c i p a l

,

B(p,M+T)

~ U(Xl),

u ( x I) u n t i l

D

[HKW1]

function

outside

u s i n g Prop.

T ~ ~/4<

and note

point

of

w.l.o.g.

(6.8.3)

ond

chosen

suitably,

assume

estimate

for the s u b h a r m o n i c

q is a s u i t a b l e

domain

= R , x I £ 8D .

the m e t h o d

the L i p s c h i t z

estimate

however,

with

Ah = O

in D

o

in

[HKWI],

h with

proof

the

v ,

77

(6.8.7)

By

h(x)

the m a x i m u m

(6.8.8)

= d2(u(x),q)

for x £ ~D o

principle

v < h

in D o

Now

(6.8.9)

d(U(Xl),U(X2))

= d(u(x2), q)

- d(U(Xl), q)

_~ I/2T

(d2(u(x2),q)

_~ I/2T

(h(x 2) - h ( X l ) )

by

choice

- d2(u(xl),q))

by

of q

-~

(6.8.7)

and

(6.8.8). Thus,

(6.8.2)

is r e d u c e d

h at

~D . This

is

to a L i p s c h i t z

a standard

result

bound

for

the h a r m o n i c

from potential

function

theory.

q.e.d.

Corollary

6.4:

(6.8.10) where The

lu ICI 'e(D)

c 3 depends

proof

regular

assumption

thermore,

we

and

D will

be

the

even

6.2,

constants

for all derive

taking

statements

[JKI].

domain

as c 2 °

6.3,

In

hold

later

is the

unit

regular

Thm.

into

in any

sections, disc by use

for the

shall

and

remove

of c o n f o r m a l domains.

(cf. Thms.

functional

account.

dimension

we

twodimensional

C 2'~ e s t i m a t e s

from below

6.2

10.3,

Fur10.4).

determinant

of

functional

de-

di f f e o m o r p h i s m s inequality

chapter,

terminant

cf.

estimates

A Harnack

In this

same

(6.8.10)

A - priori

7. I.

disc

that

of Thm.

~ c3 ,

corresponding

shall

harmonic

the a s s u m p t i o n s

as of Cor.

domain,

and prove

7.

same

Again,

for any the

on the

is the

Remark:

maps

Under

we

obtain

of h a r m o n i c Z a surface removed

o f E. H e i n z estimates

diffeomorphisms with

boundary.

in c h a p t e r

9 .

from b e l o w

for the

u: D + Z , w h e r e The

assumption

that

D is the

unit

the d o m a i n

is

78

The

results

interior

f r o m this

estimates

First

o f all,

Lemma

7.1:

plane

chapter

which

we have

Hilfssatz

Suppose

disc

(7.1.1)

B(XoR) IAfl

Furthermore,

(7.1.2)

we

with

]?fl

taken

from

on the

4 of

[JKI].

ideas

[Hz6],

f 6 C2(B(Xo,R),R)

of

which

We

start with

the

[Hz6]. we

formulate

is a f u n c t i o n

as

defined

in the

I R ~ 2+24-------~ , s a t i s f y i n q _

~ a(IVfl 2 +

assume

O <

are

are b a s e d

IfI2) 1/2

f(Xo)

= 0 and

~ Y < ~

Then

(7.1.3)

l?f(Xo) I ~ 2y e x p

)3)

(-1174(iog

for all x 6 B ( X o , R / 2 ) . Lemma the omit the

7.1

provides

differential the

proof

since

representation

which

makes

inequality

7.2.

Interior 7.1

together

a suitable maps.

L e m m a 7.2: o out D , w h e r e

with

[Hz6].

The p r o o f

case.

We n o t e

on

functions

to a s u i t a b l e

that L e m m a

of

We

is b a s e d

for p s e u d o a n a l y t i c

the a s s e r t i o n

functions.

a geometric

analogue

Suppose J(x)

consideration

of H i l f s s a t z

5 of

[Hz6]

u: D + Z is h a r m o n i c ,

is the

7.1

Harnack is v a l i d

functional

and

determinant

now

enables

us to

for u n i v a l e n t

that

J(x)

o f u at x

log

{49

for some

log

e

10 2 c}

IJ(Xl) L

p < I ,

. If

then

9+96A2c 2 ~ 3

1-p

log

{49 log ~

har-

~ O through-

l?u(x) I ~ c

6 B(O,1/2(1+p))

(7.2.2)

in

for s o l u t i o n s

twodimensional

estimates

monic

for x

found

estimate

in the

dimensions.

prove

(7.2.1)

gradient

of Bers - V e k u a

to r e d u c e

for h o l o m o r p h i c

in two

type

(7.1.1)

it c a n be

theorem

it p o s s i b l e

only

Lemma

a Harnack

inequality

10 2 }

79

for Xl,X 2 6 B(O,p), Proof:

if

IKI =< A2

The idea consists

function

(K is the Gauss curvature of Z).

in constructing

a suitable

real v a l u e d

f , with

IVf(x) I -~ J(x) satisfying

,

the inequality

IAf I _-< alf I (note that this is stronger than the assumptions be o b t a i n e d by composing This geometric

u with the distance

idea is the main difference

of Lemma

7.1).

from a suitable

f will

geodesic.

to the proof of Hilfssatz

5

in [Hz6]. We define (7.2.3)

(7.2.1)

O~: =

implies

(7.2.4)

I- p 2+24A2c 2

for x £ B(Xo,P~)

d(u(x),U(Xo))

Consequently,

if h measures

taken w i t h a negative B(Xo,P~).

=< p~c < z/4A the distance

sign on one side,

Using the assumptions

and

from a geodesic

through u(x O),

then hou is differentiable

IVh I = I , we obtain

on

for x 6

B(Xo, p*) (7.2.5)

0 < IJ(x) l ~ IV(hou) (x) I _~ c c

We now choose (7.2.6)

the geodesic through U(Xo)

IV(hou) (Xo) I =< IJ(x o) I1/2

(7.2.4), Lemma 2.4, (7.2.7) Because

in such a way that also

(5.1.2),

IA(hou)(x)l of

(7.2.5) and

(7.2.1),and

IVhl = I imply

< A2c21 (hou)(x) I

(7.2.7), we can apply Lemma 7.1 to hou (7.2.6)

for x 6 B(Xo, P~/2)

, and we obtain with

80

IJ(Xo) L c

_>- 4 e x p

2

{-2348

(log ~ ) 3 }

i.e. 10 2

(7.2.8)

10 2

< 3 iog(49 log _ ~A ) 3

iog(49 i o g ~ )

Covering (7.2.3)

B(O,Q) of

by balls

p~ t h a t

o f radius p~/2

(7.2.8)

implies

, we

(7.2.2)

see for

f r o m the d e f i n i t i o n any X l , X 2 6 B ( O , Q ) .

q.e.d.

Lemma J(x)

7.3:

Suppose o # 0 throughout. D

Proof:

We

can use

u:

D + Z is a u n i v a l e n t

the s a m e

argument

as in

harmonic

the p r o o f

map.

Then

of Lemma

3.5.

q.e.d.

( L e m m a 7.3 is

Theorem

7.1:

u(D)

B(p,M)

c B(p,M)

being

(7.2.9)

~

[Hz5].)

a disc with

(u(B(o,~)))

, O < ~ < 1

= ~(M,~,<,~, u,r).

By

f B(O,O)

is a b i j e c t i v e

M < ~/2<

~ I,

ij(x) i > ~-I ,

Proof:

D ÷ u(D)

that

for x 6 B(O,r),r

(7.2.1o)

u:

,

assume

meas

for some

to

Suppose

Furthermore,

Then,

due

(7.2.9)

IJ(x~Idx ->

~ ~

>

O

harmonic

map,

where

8~

Therefore,

there

(7.2.11)

exists

IJ(x°)I

Putting

p: = m a x

(7.2.12)

x O 6 B(O,o)

with

_-> ~/~

(r,o),

IVu(x)l

we h a v e

< c

for x 6 B ( O , 1 / 2 ( 1 + Q ) )

by Thm.

6.1

,

c = c(M,~,<,p). Lemma

7.3 a n d

(7.2.12)

enable

us to c o n c l u d e

from Lemma

7.2

for x

6

B(O,p) 10 2

9+96A2c2 I p

log

10 2

log

a n d thus

(7.2.13)

by

{49

log ~

10 2 }

=< 3

9+96A2c2 1-p

log

{49

10 2 log ze ~c }

(7.2.11).

Since

on the o t h e r

c we

log

2

can

the

_>-~/~ replace

desired

hand

by

(7.2.11)

, c 2 by ~/z

on the

left hand

side of

(7.2.13)

to o b t a i n

estimate. q.e.d.

Corollary

7. I:

with

u(D)

c B(p,M),

Suppose

r 6

(0,1)

B(p,M)

that being

u: D ÷ Z is an i n j e c t i v e a disc with

M < ~/2<

harmonic

map,

. Then,

for any

in a d d i t i o n

on s o m e

a n d x 6 B(O,r)

IJ(x) l > 6 -I =

where

6

= ~(M,~,<,r,meas

kind

of n o r m a l i z a t i o n

or f i x i n g

the

image

u(D),E(u)) like of 0 or

and ~ d e p e n d s

a 3- point

condition

on the b o u n d a r [

82

= ~(M,~,<,r,meas

For any ~

Proof: perty

u ( D ) , lul ~DIc~).

< meas

u(D),

we

can

find some

e > O with

the p r o -

that

(7.2.14)

meas

{q 6 u(D):

Since

on

nuity

o f u o n D b y E(u)

the o t h e r

by univalency

hand,

of u

we or

, ~u(D)

d(q,~u(D))

~ E} ~ ~

can e s t i m a t e lu[~Dl

•

the g l o b a l m o d u l u s

(cf. s e c t i o n s

3.3,

Ca = u(~D), we can calculate

of conti-

6 . 3 - 6.5)

and,

~ 6

(O,1) w i t h

estimate

from below

d ( u ( x ) , ~u(D) ) -<

if

d(x,~D)

Therefore,

S I - O.

since

u(B(O,o))

and by

> {q 6 u(D):

a n d Cor.

(u(B(O,a))

7. I f o l l o w s

Boundary

> E}

,

section,

f o r the

functional

Theorem

7.2:

B(p,M)

Suppose

we have

we want

determinant

of

the b o u n d a r y [JKI].

u: D ÷ E is h a r m o n i c ,

is

~u(D)

= u(~D)

a disc with and

radius

t h a t g:

M

a n d u(D)

< ~/2<

= u I ~D:

c B(p,M),

.

~D ÷

~u(D)

is a C 2 -

with

0 < b ~

furthermore the

7. I.

to d e r i v e

Suppose

dif f e o m o r p h i s m

Assume

f r o m Thm.

again

that

(7.3.1)

>_-~ ,

estimates

In t h i s

where

d(q,~u(D))

(7.2.14)

meas

7.3.

u is u n i v a l e n t

dg(~)

d~

t h a t g(~D)

following

f o r all ~0 6 ~D .

is s t r i c t l y

estimates

for

convex

the @ e o d e s i c

w.r.t,

u(D),

curvature

of

and

that

g(~D)

83

(7.3.2)

O < a I <
f o r all ~ 6 ~D .

< a2

Then

(7.3.3) where

IJ(x) I => ~i I

~I = ~ 1 ( ~ ' K ' M ' T ' a 1 ' a 2 ' b ' I g l c 1 , e )

Proof:

We d e f i n e h(q):

(7.3.1)

and

(7.3.4)

(7.3.2)

A(hou)

This will enable h ou at b o u n d a r y Hopf.

for all x 6 ~D ,

This

(T is g i v e n

= -d(q,3u(D))

imply

for q

6

~u(D)

in Thm.

6.2).

for q 6 u(D). (cf.

(2.5.1),

(2.5.2),

(5.1.2))

=> alb2

us to g e t a l o w e r b o u n d p o i n t s w i t h the a r g u m e n t

assertion

in turn i m p l i e s

for the r a d i a l of the b o u n d a r y

(7.3.3),

taking

derivative

of

l e m m a of E.

(7.3.1)

into

account. The constants

a 2 and < c o n t r o l

then d e t e r m i n e

how

a n d free o f d o u b l e p o i n t s . fore

find a neighborhood

C 2 function with Suppose

x

o

Using now

x ° 6 3B(Xl,rl),

Defining

Taking

V ° of

strictly

Cor.

a n d Cor.

6.4, w e can c h o o s e

2 rl = ~ - - a152(1

function

for x 6 B ( X l , r I) y(x)

(x_x I )2 2 ) rI

-

f

we have

= -I/2 alb2

and consequently

by

A ( h o u + y)(x) Moreover,

we can t h e r e that h is a

some d i s c B ( X l , r 1) c D ,

in s u c h a w a y t h a t

the a u x i l i a r y

A¥(x)

the p r o p e r t y

convex

level c u r v e s on U(Vo).

A(hou) (x) _~ 1/2 alb2

~(x):

and a I and strictly

6.2 into a c c o u n t ,

~D in D w i t h

convex

of u(~D),

curves of h r e m a i n

6 ~D . (7.3.4)

(7.3.5)

focal p o i n t s

long the level

,

(7.3.5)

> O

on B ( X l , r l ) .

via

84

(hou) (Xo) + Y(Xo)

= O

and

(hou) (x) + ¥(x)

=< O

on

~B(x

,rl), I

since

by a s s u m p t i o n

sumes

nonpositive

The m a x i m u m

principle

the d i r e c t i o n

and

of

the

(hou + y ) ( x

~r

u is m a p p e d

values,

o

onto

the

side

a n d y l ~ B ( X l , r I) = O

now

controls

outer

normal,

) > O =

the

of

~u(D),

where

h as-

e

derivative

of h o u + 7 at x

o

in

namely

,

thus

(7.3.6) (7.3.6)

~--~ and

(hou)

(7.3.1

(7.3.7)

) > rl alb2

(x o

by

-~-

definition

of y

.

imply rI ~ - alb3

IJ(x o)

=:

~i 1

q.e.d.

Corollary where

u(D)

Suppose Then

7.2:

c B(p,M),

that

for all

(7.3.8)

Assume

g:

a n d B(p,M)

= ul~D

6 C 1'e

is a d i s c w i t h , and

that

harmonic

radius

7.3.1)

and

mapr

M < z/2<

.

(7.3.2)

hold.

x 6 D

IJ(x) I ->~

,

~2 = 62(~'<'M'T'a1'a2'b' Proof:

u: D ÷ ~ is an i n ~ e c t i v e

(7.3.8)

Igl I,~ )" C

follows

from

Cor.

7.1,

Thm.

7.2,

and

Cot.

6.4.

q.e.d.

7.4.

Discussion

In c h a p t e r determinant

9, we

of the shall

situation obtain

of u n i v a l e n t

ry t w o d i m e n s i o n a l

estimates

harmonic

domains.

in h i g h e r

maps

dimensions

from below not only

for the

on D

functional

, b u t on a r b i t r a -

85

We n o w w a n t functional

to d i s c u s s

the q u e s t i o n

determinant

dimensions.

First

of all,

the a r g u m e n t

quite

general

and obviously

(note

that we

did not even have

ded only The

that

u maps

situation

wing (cf.

[ELI],

$7,

not

D onto

is q u i t e

consideration

one

[EL2],

restricted

the c o n v e x

from below

to the

7.2 is

twodimensional

case

we nee-

side of u(~D)).

in the i n t e r i o r ,

For

o f Thm.

the

in h i g h e r

t h a t u is u n i v a l e n t ;

on an o b s e r v a t i o n 7.2).

can a l s o e s t i m a t e

maps

of the proof

to a s s u m e

different

is b a s e d

and

whether

o f univalent h a r m o n i c

however.

of Eells

any c o m p a c t

M

The

follo-

and Lemaire

, its A l b a n e s e

map

~: M ~ A(M)

into

its A l b a n e s e

A(M)

where

= H*/F

H* is t h e

I - forms, is

and

where

Tn

denote

g(t)

ut:

we

(Tn,g(t))

onto

differ

Now

÷

g(O) can

since

space

of harmonic

of ~I(M).

one-forms.

are metrics

Thus,

=go

by

go

a l s o J(e)

Thus,

A(M)

B y an i n v e s -

g on the n - t o r u s

has

' and construct

a n d g(1)

find harmonic

zeroes

for

a smooth

= g . B y the e x i s t e n c e

family theorem

maps

(Tn,go)

harmonic

maps

o n l y b y an i s o m e t r y can

derive

, independant

[0,1]

uniform

of

equation

for w h i c h

u O is a n i s o m e t r y .

u t and v t from

(Tn,g O) b y Thm.

a -priorl

of t , just using

o f the L a p l a c e - B e l t r a m i

the s e t o f t 6

empty,

zeroes.

two homotopic

the i m a g e is flat, w e ut

f o r the h a r m o n i c

of A(Tn,g)

with

Sampson

any

the m a p s

image

+ A(Tn,g)

Furthermore,

tions

the r e a l v e c t o r

for n >_- 3 , t h e r e

wi have

the metric

(Tn,g)

of

homomorphic

,

[C],

some

of m e t r i c s

of Eells-

space

is a b a s i s

of Calabi

(Tn,g)

Here,

Furthermore

= ~01A...A~O n

, for which

e:

We

conjugate

F is a s u i t a b l e

(~i)i=1,..n

tigation

is h a r m o n i c .

,

a f l a t torus.

J(e)

t o r u s A(M)

J ( u t)

on

•

-1,~

~

5.3.

estimates

the e s t i m a t e s

Since for

for solu-

(Tn,g(t)).

> 0 on T n is o p e n

If w e w o u l d

(Tn,g(t))

have

and not

a lower bound

f o r the

88

functional defined

determinant

on s o m e

u t would

J ( u t)

subset

enable

nant of those

of

u t with

harmonic

(Tn,g(t)),

us t o o b t a i n

> O on T n w o u l d

[0,1],

of u n i v a l e n t

J ( u t)

then

maps with

the

lower bounds

uniform

f o r the

> 0 1) . C o n s e q u e n t l y

also be closed,

and

small

image,

C a- estimate

functional

for

determi-

the s e t o f t w i t h

therefore

coincide

with

a n d in p a r t i c u l a r

J(u I ) > 0

On the

.

other hand,

isometry

of

8.

existence

The

again

(Tn,go),

by

which

uniqueness,

u I differs

is a c o n t r a d i c t i o n ,

of harmonic

diffeomor~hisms

from

a o n l y b y an

since

J(~)

has

which

solve

zeroes.

a Dirichlet

problem 8.1.

Proof

of

the e x i s t e n c e

a convex We

can n o w e a s i l y

Theorem ~

ball

8.1:

on s o m e

theorem

and bounded prove

the m a i n

Suppose

surface,

by

~ is

and that

upper

in s o m e

schitz Then

disc

curvature class

there

and

prescribed

image,

and

then

with

8.1

and

M

surface. imaqe, < ~/2<

t h a t the

by

is a h o m e o m o r p h i s m

is e v e n

the

contained

in

Lipschitz We

assume

boundary that

that

~(~) is con2 (where < ~ O is an

curves

~(~)

are of L i p -

~(~).

mapping

% which

is

[J3].

a harmonic

u: ~ ÷ B ( p , M )

with

the b o u n d a r y

between

~ a n d its

in the i n t e r i o r .

a C2 -diffeomorphism

u is a d i f f e o m o r p h i s m

Theorem

and

image

domain with

its

radius

on B ( p , M ) )

a diffeomorphism if ~I ~

onto

the curve

of

a compact

convex w.r.t.

exists

values

Moreover,

B(p,M)

bound

result

Z is a n o t h e r

~: ~ ÷ Z m a p s ~ h o m e o m o r p h i c a l l y tained

in c a s e

a convex

between

C 2 -curves,

up to the b o u n d a r y .

uniqueness

theorem

of J ~ g e r

and

Kaul

(cf. Thm.

5.1)

imply

Corollary

8._!:

map which

solves

into

a geodesic

Under the disc

the a s s u m p t i o n s

Dirichlet B(p,M)

problem

with

radius

of Thm.

defined M

by

< ~/2<

8.1,

each harmonic

~ and which , is

maps

a diffeomorphism

I) T h i s p a t c h i n g t o g e t h e r o f l o c a l e s t i m a t e s w o r k s as f o l l o w s . B y u n i f o r m c o n t i n u i t y , f o r a n y e > 0 t h e r e is s o m e ~ > 0 w i t h u t ( B ( x , ~ ) ) c B ( u ( x ~ , e ) . T h u s , u t l B ( x , 6 ) h a s a r b i t r a r i l y s m a l l i m a g e , a n d if w e have estimates for maps with small image, we have corresponding ones for ut °

87

in ~ .

Proof

of Thm.

8.1:

would

consist

o f at l e a s t

Therefore, B(p,M)

we

with

could

lar statement).

ted,

o f all,

two

Since

map

curves,

both

which

is

of them

closed

argument

a geodesic

and M < ~/2K

to be a p o i n t , and since

~g is c o n n e c t e d .

find a nontrivial

an e a s y A r z e l a - A s c o l i

of a harmonic y has

First

2.2.b

y in

(cf. L e m m a

2.1

~ is h o m e o m o r p h i c

to

~(~),

we

4(~)

5.1

Therefore,

conclude

4(~). c

for a simi-

as a s p e c i a l

a n d Prop.

a contradiction.

4(~)

convex w.r.t.

geodesic

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

, Lemma

Otherwise,

imply ~

case that

is c o n n e c -

t h a t ~ is a disc,

topologically. Therefore,

we have

the p l a n e

unit

m a l m a p k:

to p r o v e

disc

D ÷ ~ a n d the

For the

only

convex,

titative

but

only

for t h e

case w h e r e

the e x i s t e n c e

theorem

3. I for a c o n f o r -

composition

rest of this

C2 -diffeomorphism

the t h e o r e m

D , taking

section,

between strictly

property

we assume

curves

of

convex,

class

and

Lemma that

C 2'a

6.1

4:

that we have

into account.

~D + 4(~D)

, that the

~ is

4(~)

is a

is n o t

following

quan-

bounds d2

1" 7 4( )1

(8.1.1)

b1

and

f o r ~ £ ~D

4( )I b21

d

(8.1.2)

and

(8.1.3)

O

< a I < Kg(4(~D))

These assumptions which

we

shall

By virtue riation into

can b e

indicate

of Thm.

3.1,

of b o u n d a r y

4 preserves

we now want map

u

we may of

m a p k:

to d e f o r m

D ÷ 4(D). this

B y a va-

conformal

map

. assume

~D . N o w

o f 4(]9) b y

arguments

8.2.

is a c o n f o r m a l

harmonic

the o r i e n t a t i o n curve

l a t e r on b y a p p r o x i m a t i o n

in s e c t i o n there

los.s o f g e n e r a l i t y ,

o f the b o u n d a r y

(8.1.4)

removed

values,

a diffeomorphic

Without

-~ a 2

that

let y b e

arclength.

~o(q~,l): = y(Iy-I (k(q~)) +

the b o u n d a r y

We

value map

the parametrization

set

(I - l ) y - l ( 4 ( ( ~ ) ) ) , £

~D ,

~, £

[0,1]

88

deformes scribed Since

the b o u n d a r y

by

we

values

of k into

the b o u n d a r y

values

pre-

~ .

assumed

that

application

o f Thm.

(8.1.5)

~($,~)

(8.1.1)

and

3.1 y i e l d s

(8.1.2)

hold

and

that

~(~D)

6 C 2,e

,

that 22

are

continuous

(8.1.6)

Let now lues

ul

functions

u I denote

[O,1]

n By Cor.

~(~,I)

~-~ ~($,1)

~(-,I),

£

, ~

~($,1)

,

does

not vanish

(the e x i s t e n c e

converges

~

of I

the h a r m o n i c

be a s e q u e n c e

6.4,the

and

map

f r o m D to B(p,M)

of ul

follows

converging

Arzela-Ascoli

map

f r o m Thm°

to some

Theorem

to the h a r m o n i c

for a n y # 6 ~D and i 6 [O,1].

I

and the

ul in the

with

boundary

4.1)

and

va-

let

6 [O,1]. uniqueness

theorem

C I'B ~- t o p o l o g y ,

5.1,

O < B

< ~ •

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

p(1):

depends We

L:

conformal

Cor.

=

p(1)

Since

p(1) u I is

arguments

8.1,

> O

feomorphism

of

3.1,

(8.1.5)

O 6 T; (U O is the

Since

we

assumed

and

(8.1.6)

we

can a p p l y

I , (8.1.7)

implies

L =

for I 6 L

on

and

and a d i f f e o m o r p h i s m

consequently

lifting

theorem. 8.1

is c o m p l e t e ,

in the n e x t

a global

except

[O,1].

between

the

diffeomorphism

f o r the

approximation

section.

arguments

we h a v e

convex,

. By Thm.

o f ul).

that

of Thm.

Approximation

the J a c o b i a n

L is n o t empty.

implied

diffeomorphism

described

In s e c t i o n

> O}

therefore

of D a n d UI(D),

the p r o o f

p(1)

continuously

a local

the h o m o t o p y

strictly

(J(u I) d e n o t e s

which

> Po

depends

boundaries

8.2.

and

(S. 1.3),

7.2 to the e x t e n t

Thus,

Thus,

on I

{I 6 [O,1]:

m a p k),

and

(8.1.7)

by

IJ(un ) (x) 1

continuously

define

(8.1.2)

= inf x6D

and,

class

assumed

that

in a d d i t i o n , C2

. In this

the b o u n d a r y

that

of the

the b o u n d a r y

section,

we

shall

image

values prove

is

are a dif-

the t h e o r e m

89

also

for

the case

that

the b o u n d a r y

that

values

of

the b o u n d a r i e s .

We

shall

present

only

fication

of the

183.

reasoning

pp.

The

351 - 352,

handled

convex,

of class

a metric

~A is even

[Hz4],

§4. M o r e o v e r ,

on A t o g e t h e r Keeping

with

values

given

so c l o s e

above

to gij

first

of

Cor.

ori b o u n d

of

the

by virtue

of

7.1,

Cor.

6.4,

we

u(x).

classical

solution,

in the

morphisms,

i.e.

interior,

since

it is a d i f f e o m o r p h i s m

, according

to

as n +

solves

of

on A such

derivatives,

consider

the m a p un(x)

the D i r i c h l e t

to gij

which

problem

of u n is o u a r a n t e e d values

of n

the

b y the arn , w h e n gij is

are satisfied.

r < I , there

of u

n

of the

with

map w.r.t,

Euler

the

Moreover,

Moreover,

functions

first

deriva-

to u s t r o n g l y

the m e t r i c

equations. theory

it is the

is an a - p r i -

(x) f r o m below.

un converge

regularity

harmonic.

and

the b o u n d a r y

of m e t r i c s

a Subsequence

corresponding

a diffeo-

uniformly

D together

harmonic

elliptic

is

,~(D) is only

to c o n v e r g e

B(O,r),

can c h o o s e

of the

[Hz3],

way:

n to gij

conditions

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

C I , linear

we

determinant

a weak

phism

second

disc

on

178 -

boundary

to be

to w h i c h

{g~j}

for l a r g e

uniformly

pp.

from

the i m a g e

following

respect

and w h i c h

on e a c h

u is a w e a k l y

solution

with

and

over

a general

assumed

respect

existence

H 2I • T h e r e f o r e ,

of class

with

geometric

functional

converges

to a m a p

g~j

. The

the

of

still

in the

~ fixed,

- at least

that

By v i r t u e

(x) w h i c h

~

~ are

is a s e q u e n c e

values

[Hz4],

curves.

can be c h o s e n

in the ~ e t r i c

boundary

guments

image

of

and

It is a m o d i -

in

taken

case

convex

a homeomorphism

argument.

can be

The

argue

convex

their

to i n d u c e

the b o u n d a r y

values

{g~j}

the b o u n d a r y

is h a r m o n i c

that

to be

by E. H e i n z

case

6 C 2'e.

there

strictly

given

by s m o o t h

the

supposed

approximation

one

C 2. T h e n we

is convex,

is o n l y

supposed

second

~(D)

gij on

that

tives

for the

the b o u n d a r y

~(D}

n

first

let us s u p p o s e

A:

u

the

corresponding

in case

while

morphism

with

are o n l y

by a n a p p r o x i m a t i o n

Therefore,

Given

the b o u n d a r y

Since

implies

limit

, i.e.

u is a l s o

that

u is a local

uniform

gij

in

u is a

diffeomorof d i f f e o -

in the i n t e r i o r . q.e.d.

Remarks: don't

even

1) A c t u a l l y , have

We n e e d o n l y

to a s s u m e

that

they

l i m i t of h o m e o m o r p h i s m s .

using

a further

that

the b o u n d a r y

are c o n t i n u o u s ~he

approximation values

and monotonic,

corresponding

harmonic

argument,

we

are homeor~grphic. i.e.

a uniform

solution

of

the

g0

Dirichlet 2)

problem

In the n e x t

chapter,

non-variational

8.3.

Remarks:

I) In the mains

still

Rad6 a n d K n e s e r

both the

[Rd],

use a d e g r e e

8.1,

i.e.

necessity

~ and

~(~)

assertion

[Knl]

of

argument

image

image

the m e a n

nic

not b e l o n g i n g This

is in e s s e n t i a l

vexity

of the i m a g e

contrast

is a d i f f e o m o r p h i s m solution

to the

case

is n o t n e c e s s a r y (cf. Thm.

o f a free b o u n d a r y

3.1).

value

Note

values

~(~) then

of h a r m o -

points

points

of D w i l l

between

p and q

to ~(~) .

of c o n f o r m a l

to g u a r a n t e e

maps

that

the

that a conformal

problem

8.1 to

following

property

some

onto

that

near p and q ,

value

functions,

be m a p p e d

by

showed

the

If the b o u n d a r y

are c o n c e n t r a t e d by

do-

for Thm.

has

8.1

obtained

also

is n e c e s s a r y

the

4.1.

connected

already

Choquet

a

Thm.

of Theorem

simply

8.1 w a s

[Cq].

shape.

to o b t a i n

involving

are b o u n d e d

Suppose

interior.

the h y p o t h e s e s

of Thm.

of the

following.

in the

one not

and C h o q u e t

o f the b o u n d a r y

hold. The r e a s o n is the

a diffeomorphism

shall

of Thm.

domains,

case w h e r e

in the plane,

the c o n v e x i t y

we

proof

Plane

remains

instead

where

con-

solution map

is a

of a Dirichlet

pro-

blem. 2) The

size

chapter

11

9.

restriction (cf. Thm.

C 1'e- a-priori existence

9.1.

C 1'~-estimates section,

domains

(cf.

[JKI].)

Theorem

9.1:

Suppose

again

estimates

u(~)

we

shall

8.1 w i l l

be

for a r b i t r a r y

domains.

Non - variational

u:

removed

in

6 . 3 a n d 6.4

for arbitrary

~ ÷ Z is a h a r m o n i c

map between

surfaces.

where

are c u r v a t u r e

on ~

bounds

for e v e r y

lul

surfaces

Corollaries

c B(p,M),

prove

curvature

(9.1.1)

in Thm.

on a r b i t r a r y

a n d -w 2 _~ K <~K 2 are

Then

image

proofs

In t h i s

pose

on the

11.2)~

B(p,M) bounds

is a d i s c w i t h on B ( p , M ) ,

.

~ cc o

C1'c~(~o)

-

ci0

ci0 = c 1 0 ( e , < , M , l , A , i ( ~ ) , d i a m

, ~,d(~o,~)).

while

radius _%2

Sup-

M

< 7/2<

< K~

< A2

,

91

Proof: with

First

the p r o p e r t y

discs each

B(xi,R)

by Thm.

3.1.

there

' where

is a c o n f o r m a l

Thm.

then

P < I s i n c e k.

directly

bounds hand,

terms

of

proof

of Thm.

e.g.

implies

and

that

the

x i on ~ , smaller

i(~) ;I/2 d ( ~ o , ~ ) ) .

For

k i of B(xi,2R) ,

by ki(O) Lemma

= x i) 6.1),

k.

the C o u r a n t -

of c o n t i n u i t y

is b o u n d e d

I , A , R , since

of p o i n t s

c11(e,K,M,p)

the m o d u l u s E(k i)

(I/2

(using

is a d i f f e o m o r p h i s m ,

1

the o t h e r

are d i s c s

R = min

(normalized

6.1

number

representation

Id(uoki) IB(O,p ) d

for e v e r y Moreover,

is a f i n i t e

all B(xi,2R)

~o

D ÷ B(xi,2R)

(9.1.2)

3.1

that

cover

i , there

ki:

On

of all,

by

of k i by E ( k i)

the g e o m e t r y

is m i n i m i z i n g

Lebesgue

among

of

Lemma (cf.

3.3).

~ (actually

diffeomorphisms

in (cf.

1

(9.1.3)

as

3.1)). Thus,

we

can

calculate

some

p < I , for w h i c h

ki(B(O,p) ) D B(xi,R )

in the p r o o f

of Cor.

7.1.

Furthermore

(9.1.4)

J(ki ) >= ~-I

= 6(X,A,R,p) (9.1.2)

-

(cf.

(9.1.4)

(9.1.5)

on B(O,p),

Thm.

then

imply

IdUlB(xi,R)

If w e n o w

introduce

balls

of d o m a i n

since

the e q u a t i o n s

7.1).

_-< c 1 2 ( e , < , M , I , A , R ) .

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

and image,

given

(9.1.1)

follows

for h a r m o n i c

maps

~

pi

from Lemma from

in l o c a l

2.5 on s m a l l

linear

elliptic

coordinates

take

theor~ the

form

(/T ¥

Theorem u(~) where

9.2:

c B(p,M). we

assume

~B

ui

Suppose Suppose

) + ¥~B

B

- 0

u: ~ ÷ E is a h a r m o n i c

B(p,M+T),

curvature

~u j ~u k

jk

bounds

T > O

i

map between

, is a d i s c W i t h

surfaces{

M+ T < 7/2<

_ 2 < = K =< K2 on B ( p , M + T ) .

,

92

Furthermore, of ~ , a n d

let _X2 suppose

~ K~

~

~ A 2 be

6 C 1'a

, g:

curvature = ul~

bounds

on a n e i g h b o r h o o d

6 C l'a

Then

luk

c14

<__ c 1

C I ,~(~)

4

'

= c14(e,<,M,T,I,A,diam

~,i(~),IgIc1,a,

The p r o o f

proceeds

along

therefore

omitted,

in o r d e r

Remark: where

Thms.

we

shall

9.1

and

derive

geometric

quantities

9.2.

Estimates

for

Cor.

7.1

Theorem

u(~)

for a n y ~

(9.2.1)

c B(p,M), o

6 3 = d3(M,e,<,meas u(~), A 2 for

u:

the c u r v a t u r e

be

considerably bounds

9.1

improved

which

determinant

and is

in 10.5,

depend

for

arbitrary

on a r b i t r a r y domains

~ ÷ Z is an i n j e c t i v e

where

cc ~ and x 6 ~

;J(u) { ~ ~ I

as the one of Thm.

repetition.

on

the

as the C 1'a b o u n d s !

the f u n c t i o n a l

Suppose

lines

C 2'e a - p r i o r i

can be p r o v e d

9.3:

surfaces, Then

7.2

same

to a v o i d

9.2 w i l l

even

same

and

the

I~ic1,a).

B(p,M)

radius

map

between

M

< ~/2<

(-12

~ K~

.

,

o

> 0

in ~o

i(~),X,A

diam

of ~ ) , a n d

as well.

harmonic

is a d i s c w i t h

surfaces

'

~ , d(~o,~),

the d e p e n d a n c e

E(u)),

on E(u)

can be r e p l a c e d

by IuI~L Ca If m o r e o v e r

(9.2

2)

g:

= ul~

O < b <

"

£ C 1'a

and

dg(~)

=

for

all ~ 6 ~

d~0

and

(9.2.3)

O < a I ~
_-< a 2 ,

then

Ij(u) I ~ 641

where

6 4 depends

> 0

on

the same

in

quantities

as ~

and on

I gIc 1,a,

al

' a2 '

93

b,

a n d T , w h e r e T is g i v e n

ture b o u n d s

Proof:

We

the p r o o f Since

again

of Thm.

j ( u o k i)

provides

in Thm.

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

compose

9.1.

interior

u with

Cor.

= J(u)

7.1

• J ( k i)

gradient

6.2

(also -I 2 ~ K~ ~ A 2 a r e

curva-

of ~).

suitable

and , the

bounds

conformal

7.2 b o u n d assertions

k 1 as in

maFs

IJ(u~ki) ] follow,

•

from below. since

Thm.

6.1

for k. 1 q.e.d.

9.3.

A non- variational

W e are n o w Thms. dure

9.1

ready and

was

to g i v e

9.2.

involved

which

of c o n f o r m a l

refined

geometric

section we had sent 9.2

situation, together

Actually, because

we

tension

with

Theorem

9.4:

with

Proof:

= g

however,

given

degree

that

in

in 4,

in the p r e -

o f Thms.

9.1

and

the b o u n d a r y

than Thm.

values

admit

4.1, an e x -

energy.

~ is a b o u n d e d

g(~)

c B(p,M),

5% < ~/2K

. Then

where there

domain

B(p,M) exists

on a surface,

is a d i s c a harmonic

on map

g:~

÷

the s u r f a c e h:

~ ÷ B(p,M)

.

as in 8.2,

the Leray-

Schauder

degree

the g e n e r a l

involving

theory

as in

[HKW2 ] . define

g(~,t)

=

S

O (2t-1)g(~)

(4 6 ~ ) ,

given

in c h a p t e r

[LU],

statement

argument

(9.3.1)

of

b y u s e of a m o r e

theory.

stronger

lows by an approximation

we

map

can d i s p e n s e

While

estimates

g 6 C 2 , since

Therefore,

using proce-

to the o n e

e.g.

w.l.o.g,

to use

One

compared

the a - p r i o r i

to r e q u i r e

4.1,

for a conformal

can assume

We want

We

results

a slightly

~

radius

h I~

prove

finite

proofs,

proof,

Leray - Schauder

don't have

3.1

a n d 9.2.

t h a t i t is s e l f c o n t a i n e d .

can use

shall

is c o n t i n u o u s , with

we

9.1

of Thm.

some variational

[JKI]).

the p r e s e n t

is

theorem

in t h o s e as in

proof

is s t i l l

of Thm.

to r e g u l a r i t y

with

we

maps,

of

4, h o w e v e r , to a p p e a l

there

the e x i s t e n c e

argument

advantage

4.1

a non- variational

in the p r o o f s

the u s e

The main

of T h e o r e m

(Actually,

through

used

proof

for 0 =< t =< I/2 for

I/2

=< t =< 1

Cor.

case 6.2.

[HKWI]

and

fol-

94

We

cover

< ~/2<

B (p,M) o

by n o r m a l

, a n d M O is

We d e n o t e

the

chosen

(q 6 G ( B ( p , M ) ) ,

=

G centered

a way

Christoffel

~2t

at p

t h a t B(p,M) symbols

, where

is s t i l l F ijk

by

, and

F ijk(q)

for O =< t _< 1/2

F i (q) jk

for

I/2

M < M

o

a disc. define

< t < 1 = =

i , j , k = 1,2)

If we e x t e n d

Fi as jk

to the w h o l e

plane

into

in such

corresponding

F iJ k ( q ' t ) :

(9.3.2)

coordinates

continuous ~2

, we

and b o u n d e d

can

define

the

functions following

from G(B(p,Mo)) map

of

C 1 (~,~2)

itself:

U + F(u,t)

where

,

v = F(u,t)

is the s o l u t i o n + ye~

Avi(x)

of the

F~k(U(x) ) ~ J ( x )

~uk(x)

~xa

(9.3.3) vl~ where

(yeS)

responding

Lemma

Dirichlet

a) F o r e a c h

b) On e v e r y

Proof:

of ~ in l o c a l

Laplace -Beltrami

9.1:

- O

on

~xB

= g(-,t),

is the m e t r i c

ly c o n t i n u o u s

problem

t 6 [O,I],

transformation

ball

F(',t):

of C I (~,R2)

{ l U I c I =< R},

A consequence

coordinates

and A is the

cor-

operator.

of

F is u n i f o r m l y

linear

~ ÷ F(~,t)

into

elliptic

is a c o m p l e t e -

itself. continuous

in t .

theory. q.e.d.

Lemma type

9.1

implies

on b o u n d e d

gree

t h a t u ÷ ~(~,t):

open

subsets

deg(~(u,t),Y,O)

We n o w

define

Y: = map

with

a suitable

u: ~ ÷ B(p,M)

respect

set Y

{u 6 C I (~,~2) ,

image

(9.3

Idul

2c14}

we

value

can d e f i n e O

(cf.

a de-

[LS]).

.

with

lu(x) I = d(u(x),p)

<

to the

is o f L e r a y - S c h a u d e r

Thus,

-u is the c o o r d i n a t e

(9.3.4)

5)

= ~ - F(u,t)

Y of C I (~,~2).

< M

O

in ~ a n d

representation

of

some

95

c14 is t h e c o n s t a n t the m e t r i c s chosen

Lemma

with

independantly

(9.3.6)

9.2

deg

(9.3.2)

implies

= O

in Y

Thm.

9.4,

representation

of Lemma

values

(9.3.1)

(note t h a t c14

and

~an b e

of

.

since

of

t £ [0, I]

the s o l u t i o n

a harmonic

of

(9.3.6)

m a p h: ~ ÷ B ( p , M o ) ,

is

the co-

and Lemma

c B(p,M).

9.2:

(~(~,O),X,O)

for all

is a s o l u t i o n

t h e n implies h(~)

Thus,

= I

there

#(~,I)

ordinate

Proof

f o r the b o u n d a r y

symbols

of t)

(¢(~,t),Y,O)

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

6.2

9.2,

9.2 :

deg

Lemma

in Thm.

Christoffel

Since

¢(~,0)

= ~

and

0 6 Y

, we

see

that

= I

we only have

to s h o w t h a t

deg

(¢(~,t),Y,O)

is i n d e p e n d a n t

of

t . The

fundamental

that ¢(~,1) In o r d e r have

= O has

to s h o w

to exclude

(9.3.7)

has

theorem

a solution

that deg

seen

~ 6 ~Y

from Lemma

a harmonic

map with ~ 6 Y

in a n y

then implies

.

(~(~,t),Y,O)

of

(9.3.7)

(5.1o2)

values,

then Thus,

63)

is i n d e p e n d a n t

of t , we only

.

2.2 a n d

boundary

If ~ £ ~ is a s o l u t i o n

imply

p.

= 0

I f ~ £ ~ is a s o l u t i o n

vanishing

in Y

([LS],

that

¢(~,t)

a solution

of Leray - Schauder

of

and

that

t Z I/2

for some

values

, then

l~I 2 is s u b h a r m o n i c

consequently

(9.3.7)

boundary

for some

~ H 0 t ~ I/2

g(',t).

Lemma

it is e a s i l y on ~ with

. , then ~ represents 6.2 a n d Thm.

9.2

.

case,

(9.3.7)

cannot

have

a solution

~ £ ~Y

.

q.e.d.

9.4.

A non-variational

In t h i s 8.1.

section,

It p r o c e e d s

proof

we want as in

of Theorem

to p r o v i d e

[J3]

8.1

a degree

a n d is b a s e d

theoretic

on a r g u m e n t s

proof

o f Thm.

o f E. H e i n z

96

(cf.

[Hz3])

Let us first formulate the assertion again

T h e o r e m 9.5:

Suppose under the assumptions of Thm.

9.4, that the

b o u n d a r y values e x t e n d to a h o m e o m o r p h i s m ~ of ~ onto its image, that ~(8~)

is convex w.r.t.

and

~(~).

Then we can find a h a r m o n i c solution of the c o r r e s p o n d i n g D i r i c h l e t p r o b l e m w h i c h is a h o m e o m o r p h i s m on ~ and a d i f f e o m o r ~ h i s m in the interior.

Proof:

As in the p r o o f of Thm°

8.1, we see that ~ is t o p o l o g i c a l l y

a disc. We can therefore again assume that (8•1.3) h o l d for ~ 6 ~

(8.1.1),

, since these assumptions

(8.1.2)

can be removed by

the a p p r o x i m a t i o n arguments of 8.2. Finally, by Thm• c o n f o r m a l maps k1:

D ÷ ~ and k2: D + ~(~),

and

3.1, there exist

and therefore also a con-

formal map k: = k2ok~1: ~ + ~(~). We can again d e f o r m the b o u n d a r y values of k into ~13~

as in 8.1, and

As in the proof of Thm. We put B: = G(~(~)),

(8•1.5)

and

(8.1•6) pertain.

9.3, we r e p r e s e n t B(p,M) by n o r m a l coordinates.

and let again ~ denote the coordinate r e p r e s e n t a -

tion of u .

D e f i n i t i o n 9.1:

Let I be an e l e m e n t of [O,1]• Then K(1) is the set

of all functions ~ = ~(x), ~: ~ ÷ B , with the f o l l o w i n g properties: a) ~(x)

6 C2(~,B)

D C°(~-,B), and u(x) maps ~ h o m e o m o r p h i c a l l y onto B

w i t h n o n v a n i s h i n ~ functional determinant, b) u(x) is h a r m o n i c in ~ , i.e. ~ = ~(x)

A~i+ Y eS_i ~jk (~)D ~jDB~k = O

(9.4.1)

c)

u(})

= ~(},~)

Let K: =

is a solution of the system

(i = I ,2)

(~ c ~ n ) .

U K(1) 1610,1]

We now define the t r a n s f o r m a t i o n ~ + H(~,I) by the r e q u i r e m e n t that H(~,I) (9 4.2) •

is the s o l u t i o n v of Avi(x) + 7eB Fi jk(U(X))

v]~

~J(x) ~x ~

= G(eC•,l))

By e l l i p t i c r e g u l a r i t y theory

~u--k(x) _ O xB

on

97

Lemma only

9.3:

A function

if ~ = H(~,I)

where

~ £ C I ( ~ , ~ 2) is

a n d the

functional

contained

determinant

in K(1),

if and

of ~ v a n i s h e s

no-

in

We n o w

define

Y

: = {~ 6 C I (~,~2),

~ represents

u with

o (9.4.3)

l:(x) I

(9.4.4)

Idu]

(9.4.5) Here, and

=

>

is d e f i n e d

6 3 can be

< M

< 2c14

IJ(u) l c14

d(u(x),p)

chosen

o

in

1/2631 } in Thm.

9.2,

uniformly

and

6 3 in Thm.

for the

family

9.3

(note

of b o u n d a r y

t h a t c14 values

~(.,~)). Lemma

9.4:

The

into itself

transformation

is of L e r a y - S c h a u d e r

~ ÷ ~(u,l)

= ~ - H (5,1J

of

C I (~, 2)

type.

Moreover,

(9.4.6)

deg

(~(~,l)Yo,O)

for all

= I

I 6 [0,1]

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

(9.4.7)

has

~(~,I)

a solution

Lemma

9.4

Proof

of L e m m a

type,

follows

if ~(u,l) Y

o

in Y

implies

We now

c Yo

lution

o Thm.

9.4:

show

That

that

' and o

~ 6 To

, and

~ 6 Y

Consequently,

(~(~,l),Yo,O)

~ 6 K(1)

thus

9.3.

the t r a n s f o r m a t i o n

9.1.

deg

= 0 for s o m e

~ 6 ~Y

9.5 v i a L e m m a

from L e m m a

. Consequently,

K(1)

= O

o

, then

by L e m m a

is of L e r a y - S c h a u d e r degree

is i n d e p e n d a n t

is w e l l of I

definedh Indeed,

]J(u) I > 0 in ~ by

definition

9.3.

9.3 i m p l y

. Therefore,

the d e g r e e

the

Thms. ~(u,l)

is i n d e p e n d a n t

9.2 a n d

= O cannot of ~

.

have

of

a so-

98

It o n l y

remains

(9.4.8)

deg

If we

define

~(~,0)

where

the

By Lemma

with

(~(u,O),Yo,O)

= 1

, then

= ~(u, 1),

transformation

¢ was

defined

in 9.3.

therefore

(~(U,O) ,Y,O)

= I .

that Y

c Y (Y w a s d e f i n e d in 9.3, too). o h a n d , b y the u n i q u e n e s s t h e o r e m of J ~ g e r - K a u l

On the other 5.1),

that

g(~0) = k(%0) for R0 6 ~

9.2

deg

We note

to s h o w

any

solution

the one

~

6 Y of the e q u a t i o n

solution

Consequently,

any

in Y . Hence o o f the L e r a y -

Schauder

we know

solution

(9.4.8)

already,

~(~,O)

namely

in Y o f ~ ( ~ , O )

follows degree

9.2

(cf.

p.

(of. Thm.

to c o i n c i d e

the c o n f o r m a l

= 0 is a c t u a l l y

from Lemma [D1],

= 0 has

and

map k

.

contained

the e x c i s i o n

property

67).

q.e.d.

Remark:

Actually,

preceding values,

10.

proof

cf.

by

one

using

can

dispense

a more

of the c o n f o r m a l

geometric

variation

m a p k in t h e

o f the b o u n d a r y

[J1].

Harmonic

coordinates.

C 2' ~ - a - p r i o r i

estimates

for harmonic

maps 10. I. E x i s t e n c e In t h i s

chapter,

coordinates functions. optimal bounds

we want

on s u r f a c e s , It w i l l

regularity

coordinates.

to p r o v e i.e.

turn out

Riemannian

bounds.

properties.

normal

involve

Actually,

in

This

coordinates,

metric

with

is o n l y H 6 1 d e r

continuous

these

and regularity

with

be

displayed

(cf.

one

with

coordinate possess

can o b t a i n ±n t e r m s the

Ca -

only of

fact that

f o r the

for

Christoffel

[KI]).

the

continuous

in n o r m a l

of harmonic

coordinates

symbols,

contrasted

even L-bounds

HSlder

harmonic

harmonic

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

derivatives

[JK1 ], t h e r e w a s

twodimensional

existence

Christoffel

should

curvature

C I 'a - e s t i m a t e s

coordinates

that

f o r the c o r r e s p o n d i n g

curvature

symbols

of harmonic

following

example

curvature,

coordinates,

which

of

a

itself

but not better:

99

ds 2 = d r 2 + G 2 ( r , ~ ) d ~ 2

w i th

G2(r'~)

F o r this

{~

2( I + r 2 s i n ~ ) 2

=

forO

Grr

F'| -6r~ine~_e = ~ I + si,L ~

G

reason

ordinates

[JKI]

f o r O =< ~0 _~

does

the

not involve

detailed and

a disadvantage

the s i z e

increases.

is d i f f e r e n t

the e x i s t e n c e

Theorem

theorem

10.1:

being

of

class

C 1'e

Ah = 0

noted

by

in

coordinates

of these

Suppose

. Then

respect

c a n be

to ~

balls

in t w o

.

f o u n d in regularity

de T u r c k - K a z d a n

[JKI].

In h i g h e r

is t h a t

decreasing

dimensions,

co-

dimen-

(at l e a s t w i t h only

as the d i m e n s i o n

since here we

can

use

8.1 t o o b t a i n

the t o p o l o g i c a l

h: B ÷ D , D b e i n g

of G with

a v a i l a b l e ) Dne can p r o v e t h e i r e x i s t e n c e

on s m a l l b a l l s , This

for K in n o r m a l

t h a t the o p t i m a l

were

explicit

of h a r m o n i c

presently

formula

coordinates

be mentioned

coordinates

made quantitively

techniques

the

of harmonic

It s h o u l d

of harmonic

is t h a t

any derivatives

discussion

[JK2].

and were

sions,

for z _~ ~ < 2~

for this phenomenon

properties [dTK]

o~< 1)

for ~ < ~ _~ 2~

O

A more

(0<

~ z

metric

K = -

The

<~

there

B is a b o u n d e d t y p e o f the exists

the u n i t

domain

disc,

a canonically

disc

on a s u r f a c e

and having

i n the p l a n e ,

Z

, B

a boundary

~B o f

defined

coordinate

map

which

is h a r m o n i c ,

i.e.

.

Furthermore

(IO.I .1)

lhlc 1,~ <=

c 4 depending

on ~

o f ~B

c4 ,

, < (-e 2 _~ K ~ <

2

on B),

diam B

, and

the r e g u l a r i t y

.

Remark:

The

regularity

result will

be improved

to C 2 ~

in the n e x t

section.

Proof:

If w e

take

any map ~ which

to a r c l e n g t h ,

then we can extend

B ÷ D b y Thm.

8. I (h%01~B = ~0I~B).

maps

~B o n t o

~ to a h a r m o n i c

~D proportionally

diffeomorphism

h

:

100

In local

coordinates,

since

I _3___(yaB ~yy ~ ~x a

Choosing

h i) : 0

coordinates

then

to g e t

orientation.

operator

Clearly,

e.g.,

H~ider

if R(@)

choosing

any

•

on B

, the c o e f f i c i e n t s

continuous,

linear

coordinates

the p r o p e r t y

In f o r m u l a e ,

elliptic

canonically

described

is the

~

(10.1.1)

of for

theory. defined,

above)

rotation

such m a p

and

with

we

aver-

a fixed

o f the p l a n e

by an

,

271-

1

= 2--~ ~=0 S

is i n d e p e n d a n t

2.5,

are

of s t a n d a r d

~0 (with

of @ , then,

h(x):

(i = 1 2)

as in L e m m a

the h a r m o n i c

age o v e r all maps

angle

flat

'

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

In o r d e r

is

~x B

the L a p l a c e - B e l t r a m i h

the i m a g e

R(-~ )hR(~) oq0 (x)d@

of the p a r a m e t r i z a t i o n

the e s t i m a t e

(10.1.1)

,

~0 .

holds

for h

, since

it h o l d s

uniformly

hR(g)ot p

for all

q.e.d.

Remarks: maps.

The

haviour

10.2.

Other ones

chosen

can be m u c h

a result

due

B ° onto

its

Suppose

on a s u r f a c e

(10.2.1)

class

Z

We

their boundary

be-

coordinates estimates

for h a r m o n i c

coordinates,

~ D is a h a r m o n i c d i f f e o m o r p h i s m o f o ~ 6 -I and [dh I ~ c (Here, B ° is a b o u n d e d

, and D again

is the

unit

disc).

If 3B ° and

, then

,

B O , 6 , c , l h I ~ B o l c 2 , a , I~BO[c2,

first need

for the

that

conformal

[JKI].

I h ] c 2 , ~ ( B o ) _~ C 5

C 5 = C 5(e,<,diam

the advantag~

by

h: B

J(h)

C 2,a

can be p r o v i d e d

controlled.

C 2'a - a - p r i o r i

image with

hi ~B O are of

, and

have

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

10.2:

Proof:

coordinates

for h a r m o n i c

to p r o v e

Theorem

B°

here

better

C 2'~ - e s t i m a t e s

We n o w w a n t

domain

harmonic

some

corresponding

calculations, metric

tensor

) h defines

coordinates

(gi~)3 w e h a v e

on

101

g ij =
(10.2.2)

and consequently (10.2.3)

dxgij

= < D X grad h i , grad h J >

+ ~grad

h i , D X grad h J >

and in particular (10.2.4)

~ D g r a d h k grad h i , grad h j > = i/2(dgra d hk gij _ dgra d hi gjk + dgra d hj gik)

Since h is harmonic, (10.2.5) where e

i

D 2 1. i h k = 0 e ,e is an orthonormal

Differentiating 0 = =

(10.2.5)

,

h = 1,2

frame.

w.r.t,

~ D J,ei grad h k , e

e j , we obtain >

e ,e

+ K < grad h k , e J >

,

i.e. (10.2.6) From

Agrad h = K • grad h

(10.2.3)

(10.2.7)

and

(10.2.6)

Ag ij = 2Kg ij +

< D k grad h i , e l > e

= 2Kgij + gms grt
/ D

hs , D

grad h m

< e I , D k grad h j > e grad h i , grad h r >

grad h J >

!

grad h t and therefore (10.2.8) using

[AgiJ[

(10.2.4)

If k: D + B(p,R)

=< 21KI "ldhl 2 +

(Here, ~ denotes

~21d~121d~l 2

the matrix

c Z is conformal,

then

,

(gij)).

from

(10.2.8)

•

102

(10.2.9)

IA (giJok) I

21~11~12.n

_<

+

~2nl~121d~l 2

if

(10.2.10) We d e f i n e Putting

J(k) i: = ~ok:

-1

> 0 .

D + ~3

in case B ( X o , 2 R o) c D

~:

=

max (R - d ( X , X o ) ) I d l ( x ) I x6B (Xo, RO) O

there exists

x I 6 B ( X o R O) w i t h

(10.2.11) where

_~ q

~ = d. ldl(Xl)

I

,

d = R O - d ( X l , X o)

and

Id:](xo)

(10.2.12)

P l =< ~--

o AS in the p r o o f of Thm.

i

=<

d

1

6.1, we o b t a i n

{ ll(x) ~ B ( x 1,d@)

~d2~ 2

- l(Xl)I

ldxl

IAl[ 1

[

+ 2--~ B(Xl,da) Using for

(10.2.9),

Idkl

d (x,xi ~ dx

the a s s u m p t i o n

Idhl

__< c

, and a s s u m i n g

, i.e.

(1o.2.13)

f~l

-~ z o

,

and defining

= ~(8):

=

max d ( l ( x ) , l ( x 1 )), x 6 B (x I , d@)

we o b t a i n

as in the p r o o f of Thm.

(10.2.14)

a _ (~o) _ + b 8 2 ~ =< 2 ~ 2

6. I

'

an u p p e r b o u n d

103

and

arguing

as there,

a6 (~o) (10.2.15)

since

~ _ < 8o

again

ficiently

small,

By Thms. fying Then

~ (@o)

3.1,

by

(10.2.10)

(1o.2.16)

Going

can be m a d e

6.1

back

to

and

7.1, w e

Idhl

small,

_~ c

can a l w a y s

(10~2.13)

(10.2.15)

Id~I

arbitrarily

(10.2.13)

and

and

(IO.2.12),

,

in discs

and

@o > O suf-

.

find a conformal

interior

(10.2.10)

choosing

to D

map k satis-

.

imply

=< c 6

10.2.8),

we

infer

by a r e s u l t

from

linear

elliptic

the-

ory

(10.2.17)

IgiJlcl, ~ < c 7

(In o r d e r that

to a p p l y

the C e - ~ o r m s

tor A on B ordinates i.e.

J(h)

of the

are b o u n d e d .

o on B ° g i v e n

a C I -bound

(g13),

this

and

on

result

from elliptic

coefficients This

by h

, for w h i c h

g:

= det(glj)

we have

the L a p l a c e

is no p r o b l e m ,

the c o e f f i c i e n t s

for w h i c h

of

theory,

we

since

already

of the

is c o n t r o l l e d

-Beltrami

we

by

opera-

can use the co-

proved

inverse

to e n s u r e

(10.2.16),

metric

tensor

the a s s u m p t i o n

~ 6 -I )

Since

gij

= < grad h i

, grad h j>

, (10.2.17)

implies

(10.2.1).

q.e.d.

10.3.

Bounds

Corollary , has

on

10.1:

Suppose

a boundary

ordinate

map

~B of

B,a bounded

class

lhlc2,~ < c 7

c 7 = c7(e,<,diam In p a r t i c u l a r ,

C 2'~

Conformal

topological

. Then

there

coordinates

d i s c on a s u r f a c e

exists

a harmonic

co-

,

B,I~BIc2,~) for

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

representation,

(10.3.2)

symbols.

h: B + D w i t h

(I0.3.1)

dinate

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

IF ijkIC ~ < c 8

,

s~mbols

in the

corres~ondin~

coor-

104

c 8 = c8(e,<,diam Proof:

The

B,

I~BIc2,~)

result

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

o f Thms.

IO~I,

10.2,

9.1

and

9.2. (10.3.2)

follows,

since

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

symbols

are g i v e n b y

Let

T: B ÷ S b e the

conformal

D2h

, cf.

(2.6.3) .

Corollary

10.2:

structed

in Thm.

3.1.

We assume

representation

t h a t S is of c l a s s

C 3 . If B

con-

cc

o

B

,

then

(1o.3.3)

c9

,

C2'~Bo ) C9 = c9(~,~,diam In p a r t i c u l a r ,

10.4.

symbols

and boundary

Proof:

The

Higher

Corollary

~B))(_2

in i s o t h e r m a l

the C h r i s t o f f e l meter

S , d i s t ( B O,

Fi jk

~ K ~ <

coordinates, ' depending

2

on S).

we have

only

interior

on c u r v a t u r e

bounds

on

bounds,

dia-

distance.

assertion

regularity

10.3:

again

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

of harmonic

S u p p o s e h:

I f the

curvature

o f B is of

Ck+2'~

or ~ + 3 , B

, resp.,

curvature

is o f c l a s s

of Thms.

a n d 9.1.

coordinates

B ÷ D is a h a r m o n i c

class

10.2

C k c r C k'B

and we have

coordinate

, t h e n h is o f

corresponding

map.

class

a-priori

esti-

mates. If the

(At the b o u n d a r y ~

these

C~ o r r e a l a n a l y t i c ,

results

hold provided

then

so is h

.

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

regu-

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

Proof:

we have

Higher

such regularity

regularity

follows

results

for conformal

by differentiating

maps.

(10.2.7).

q.e.d.

10.5.

C 2'~ - e s t i m a t e s

Using harmonic full order

of

for harmonic

c~ordinates,

one

differentiation,

maps

can n o w i.e. w e

improve can p a s s

Thms.

10.3:

Under

the a s s u m p t i o n s

of Thm.

a n d 9.2 b y

a

f r o m C I'~ to C 2'~ - e s t i -

mates. Theorem

9.1

9.1,

105

(10.5.1) where

lUIc2,~(~o ) =< c15

, as Cio and on n o t h i n g else.

c15 depends on the same q u a n t i t i e s

Proof:

For any x ° £ ~o

B(Xo,R),

' we can introduce h a r m o n i c

R: = m i n ( i ( ~ ) , d ( X o , 8 ~ ) ) ,

by Cor.

troduce harmonic c o o r d i n a t e s on B(p,M). u in these coordinates, I ~ /--~ ~x ~

(10.5.2)

coordinates on

10.1. Likewise, we can in-

If we w r i t e the equations

(/~ yeB ~u i ) + yeB F i ~uJ ~uk ~x B jk ~x ~ ~x B

O

(10.3.1) imply

(i = 1,2) '

we see i m m e d i a t e l y from linear e l l i p t i c regularity theory, and

for

i.e.

that

(9.1.1)

(10.5.1). q.e.d.

In a similar way, we derive

T h e o r e m 10.4: C 2'~ and ~

If under the assumptions of Thm.

6 C 2'e

(IO.5.3)

lul

9.2 g: = u I ~

£

, then

C2'~(~)

~ c16

,

where c16 depends on the same q u a n t i t i e s as c14 and on

IgIc2, e a n d

[ ~ I C 2 , ~ and on n o t h i n g else. 10.6. H i g h e r r e g u l a r i t y of h a r m o n i c maps Using Cor.

10.3, we finally o b t a i n

T h e o r e m 10.5:

If under the assumptions of Thm.

of domain and image is of class C k or C k'B or C k+3'B

, then u is of class ~ + 2 , ~

, resp.

C o r r e s p o n d i n @ a - priori estimates in Thm.

9.1, the c u r v a t u r e

de~end on the q u a n t i t i e s m e n t i o n e d

9.1 and on the C k or C k'B - n o r m s

of the curvatures.

The same assertions h o l d at the b o u n d a r y under the assumptions of Thin. 9.2, p r o v i d e d g and ~

are s u f f i c i e n t l y regular.

If the data are C ~ or real analytic,

then so is u .

106

11.

The

11.1.

Harmonic

The m a i n

existence

diffeomorphisms

result

Theorem

of

11.1:

boundary,

of h a r m o n i c

this

that

~:

diffeomorphism

of l e a s t

energy

11.1

was

among

proved

Shibata

[Sh]

and was

therefore

H.

Sealey

and w a s of

able

[JS],

Corollary positive phism

[SYI]

11.1:

11.2:

morphism.

Then

homotopic

to ~ .

Then

4:

then

4:

there

Corollary

exists

to ~

it w a s

exists

a

u i_~s

.

first

claimed

mistakes,

paper of

completely

in h i s

by

however,

in an e s s e n t i a l

imply

the

thesis

the m i s t a k e s .

different

following

[Se]

The proof

lines

than

the

w a y on Thm.

corollary,

8.1.

proved

[Sa].

the a s s u m p t i o n s harmonic

a surface

map

harmonic

Z~

of Thm.

11.1,

homotopic

w

Z2 has

non-

to a d i f f e o m o r -

back

to

i.e. map

surfaces a local

u:

without

diffeo-

EI ÷ E2 ,

the m e t r i c

to ZI

By Thm.

, homotopic

compact

map,

covering

can p u l l

isometry.

covering

ds 2 of Z2

and w i t h m e t r i c

11.1,

there

the i d e n t i t y ,

is

via

~ds 2

a harmonic

u: = ~ o u

then

map.

11.1

11.1:

to S 2 , then we

diffeomorphism

Z2 are

, diffeomorphic

E I ÷ E2

of T h e o r e m

of T h e o r e m

and

a harmonic

We

E 2 + E 2 is a l o c a l

desired

t h a t ZI

11.2:

is the

morphic

there

without

. Furthermore,

several

ZI ÷ E 2 is a c o v e r i n g

!

Proof

surfaces

Then

to ~

but

Shibata's

depends

every

Suppose

that

u':

Proof

and

and Sampson

diffeomorphism

11.2

11.1)

diffeomorphic.

and

to o b t a i n

(Theorem

we h a v e

boundary,

of

compact

homotopic

[JS],

(but not all)

along

If u n d e r

Corollary

Proof

some

approach

is i t s e l f

Z 2 are

contained

examined

proceeds

curvature,

Furthermore,

proof

5.3 i m m e d i a t e l y

by S c h o e n - Y a u

and

diffeomorphisms

His

to c o r r e c t

and

surfaces

rejected.

however,

11.1

ZI

E I ÷ E 2 homotopic

all

carefully

Shibata -Sealey Thms.

u:

surfaces

is

by J o s t - S c h o e n

in 1963.

then

that

closed

between

ZI ÷ Z2 is a d i f f e o m o r p h i s m .

harmonic

Thm.

between

chapter

Suppose

and

diffeomorphisms

homotopic

(following can

find

to ~

[JS]).

a conformal

, since

any

If Z I a n d Z2 are (and h e n c e

two m e t r i c s

homeo-

harmonic)

on S 2 are

107

conformally ZI

equivalent

a n d Z2 are

handled

by p a s s i n g

w.l.o.g, Let

by

to two - s h e e t e d

to ~ the ciently

the c l a s s energy

large

that

lift

every

map

which

are

discs,

to c o n c l u d e

is b o u n d e d

by K

empty.

topologically,

the

semicontinuity

Thus

case w h e r e

is s i m i l a r l y

we c a n a s s u m e

f r o m ZI , where

o n t o Z2 h o m o t o p i c

K is c h o s e n

S i n c e ~ 2 ( E i) = O

and

the u n i v e r s a l

apply

s u f f i -~

(i = 1,2), covers

the a r g u m e n t

of

we can

the

Zi

of s e c t i o n

'

3.3

closure

in ~ K h a v e

of

DK with of

energy

respect

the e n e r g y bounded

to w e a k

w.r.t,

by K

H 2I c o n v e r g e n c e .

weak

. Let

By

H 2I c o n v e r g e n c e ,

all

(tn)n£ N be a s e q u e n c e

in

with

E(tn)

w.l.o.g. sume

÷ inf E(t) t6 D K (or more

that

topology

Bu i.e.

to show point

the o p e n

again

the

Ascoli

and uniformly class,

Theorem.

can as

using

By the

in the C°

the w e a k

lower

semi-

again,

a sequence uniformly

that

u

o

of d i f f e o m o r p h i s m s

un 6 D K which

converge

to u °

is a h a r m o n i c

x ° 6 Z I and

disc

diffeomorphism.

We c o n s i d e r

an

define

in Z2 c e n t e r e d

ourselves

injectivity

is an

nO:

the A r z e l a -

homotopy

we

o : = B ( U o ( X O),O)

restrict

than

find

in H 1 a n d

arbitrary

in the H I2 t o p o l o g y

u ° in the p r e s c r i b e d

the e n e r g y

to a s u b s e q u e n c e )

= inf E(t) t6 D K

w e can

We w a n t

by p a s s i n g

weakly

of H 2I and of

E ( u O)

weakly

precisely,

to a m a p

continuity

Also,

for n +

tn converges

compactness

We

The

space

D K is e q u i c o n t i n u o u s .

the

elements

diffeomorphisms

D K is n o t

L e t D K be lower

coverings.

in D K to a map b e t w e e n

that

projective

(i = 1,2).

of all

of w h i c h

of R i e m a n n - R o c h .

to the real

that z2(Zi ) = O

D K be

DK

the T h e o r e m

homeomorphic

in the s e q u e l radius

upper b o u n d

-1 = u o (B a)

at U o ( X o) w i t h

for

to v a l u e s

of Z2 a n d s m a l l e r the

curvature

radius

of ~ w h i c h t/%an z/2<

of Z2

" We

~

.

are s m a l l e r 2 , where

define

108

fin:

1

=

(Bo)

W.l.o.g.,

we

can

formly

u°

. Let

to

F

be

: D÷~

n

proof

the

since

instead

of

point

(and

sary

for

Since

F

the

is

n

which

in as

is

n

a

values

prescribed

class

and

all

disc

unique

is

maps

n

in

, since the

the

complex

u

converge

n

plane

uni-

and

order

in

0

F

at this to

curve

onto

harmonic n

point,

guarantee

of

a homeomorphism

u

x

but

the

that

3.1, interior

is n o t

equicontinuity

necesof

a mi-

3.3.

a Jordan

by

to

o is t h e s a m e as t h a t o f T h m . n boundary p o i n t s , w e c a n f i x an

of

three

~D homeomorphically

exists

In

for

unit

direction

proof)

: = ~2 F

the

fixing

a tangent

morphism), maps

D be

existence

sequence n

x o £ fn

mapping

of

nimizing

assume

n

a conformal

The

(n £ N)

of ~B

Vn:

, and

n

D onto

. By

mapping oF

class

v

CI

(because

~

, and

Thm.

D +

n

8.1

B

minimizes

n

is

n

a diffeo-

therefore

and

which

u

Cor

8.1,

assumes

energy

u

in

oF

n

there

the its

n

boundary

homotopy

a diffeomorphism.

particular,

(11,2.1)

E D ( V n)

< ED(UnOFn

) = E~

(Un)

~ K

n by

Lemma

Since

the

stays

in

6.1.

(Es(f)

un converge an

again

are

equicontinuous

can

apply

the

UnOFnl~D

therefore

which

thermore,

that now

on

I~ u

in

v ° is

oF -1 n

small

argument

assume

We

energy

uniformly

D

, are

is h a r m o n i c

define

the

arbitrarily

can

namely

is

of

the

u°

, we

to

neighborhood of

section

the a

the

v n converge

interior

of

diffeomorphism

in

f

n

in Z I

TM

fn

D

assume

of

U o ( X o).

to

the

By

show

the

that

set

S).

u n oF n (0)

Therefore,

that

boundary

the

we

values

of

and

Cor.

6.2,

uniformly

on

D to

a map

the

Cor.

interior

7.1, of

we D

see .

u n oFn

maps

10.3

Thm.

. Using in

f over

can

3.3

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

equicontinuous. that

mapping

vn

,

we v°

fur-

109

Clearly, We

can

class

Un is a L i p s c h i t z

also C 1 '~

defined If u

assume

smooth

w.l.o.g.

. Then,

on F n

do n o t

r but

in this

of the e n e r g y

Un c a n be a p p r o x i m a t e d bounded

Using

Lemma

case

minimizing

by K

3.1, we

can

on c o m p a c t

subsets

Area

=< ~ e a ( Z 1 ) ,

F maps

, a n d O is m a p p e d

that

does

not

affect

Since

a n d Vn c o i n c i d e

zing

there,

E D ( V O) Since

(11.2.3)

We

maps

define

less

than

K by u n i q u e -

that

on

the F n c o n v e r g e

. F is n o t n e c e s s a r i l y

following

of u oF and n n on

3D

, it

smooth

on

3D

,

arguments. thus

extends

follows

continuously

that also

therefore

UoOF

energy

to D

a n d Vo minimi-

preserve

ZI ~

by L e m m a

6.1,

this

implies

that

= Ezl ~ ( u ° )

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

energy

=< E ~ ( u o)

to s h o w

outside

(11.2.4)

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

n

Uo 6 ~ K a n d

v is h a r m o n i c and o in its h o m o t o p y class,

5.1)

E zl ~ ( u ° )

F o r this, rywhere

limit

E~(~o)

We n o w w a n t

is

n

< ED(UoOF)

conformal

(11.2.2)

of ~

7.2.

is s t r i c t l y

and s i n c e

(by T h e o r e m

~

of

to a c o n f o r m a l m a p F . S i n c e E D ( F n) = o D diffeomorphically onto some o p e n s e t

the

uniform

coincide

E(~ n)

to a map

to x °

u oF is the o unoFn

, then

n

are

lies t h e r e f o r e in DK in a n y case. n a s s u m e again w . l . o . g , t h a t the ~ converge n

uniformly

but

on ~

n

determinant

Fn by Cor.

TM

the u

, and u

in H~ a n d u n i f o r m l y

c Z1

that

.

map, a n d an a p p r o x i m a t i o n a r g u m e n t shows I in the H 2 n o r m by d i f f e o m o r p h i s m s w i t h

ZI w e a k l y

(~n)

in H 2I a n d E ( ~ n ) < = K

functional

on Z1

coincide

that

lies

, the

from b e l o w

ness

energy

and

(by a p p r o x i m a t i o n )

for e a c h n

and bounded

a n d v oF -I n n

n

map

. We

c

to s h o w

claim

that

uol(Z2 ~ B

)

that

u

o

and ~

o

coincide

almost

eve-

.

110

P n ( X ) : = d ( U n ( X ) , u O ( x O))

Po(X):

= d ( U o ( X ) , U o ( X O))

f o r x 6 Z1

. Let

x 6 ZI

Po(X)

= lim

Pn(X)

TM

~

>= ~

If

, then

n-Woo

-I x 6 u° (Z2~B)

Since

the

implies

PnOUnOFn

are

sufficiently

Since sets

on of

equal

to

~

on

BD

, Po(X)

<

the

> ~ > 0

large

other

x 6 ~I ~ ~

also

n

.

hand,

D , this would

sumption We

and

that

d ( F o l (x),SD)

for

equicontinuous

the

F c o n v e r g e u n i f o r m l y to F on c o m p a c t s u b n x £ F(D) = Q w h i c h c o n t r a d i c t s the as-

imply

" This

proves

(11.2.4)

have

-l(z2 ~Bo~ =uolc~B~ u uol(z2-~>

Uo and

since

joint, (SB)

the

we

sets

uol ( ~ B )

can assume

vanishes

for

cover

w.l.o.g,

one

a neighborhood

that

chosen

~

the

of x ° and

twodimensional

are

dis-I uO

measure

of

' there

exists

. If

6 uol ( E 2 ~ B ) ,

x

then

lim Pn(X)

= Po(X)

>

n-~oa

and

because

an o p e n n

. This

o Therefore follows

of

the equicontinuity

neighborhood

U of x such

of

the

that

functions

p n IU > ~

Pn

for sufficiently

large

implies

= lim n+oo

~

n

= lira u n+~

n

u° = Uo almost from

(11.2.4).

By

= u

o

on

U

everywhere the

choice

.

on of

Uo I(~2~B uo

), a n d

, we have

on

(11.2.3) the other

now hand

111

EE

(uo)

~ EZ

I Thus,

(~o) I

we

conclude

from

(11.2.2)

and

(11.2.3)

that

E 2 ( ~ o ) = E ~ ( u o) and

consequently

E D ( V O)

Since

v

energy on D

= ED(UoOF)

a n d u oF c o i n c i d e o

o

minimizing

. Therefore

phism,

the

chosen

point

11.3. With

Theorem

of Theorem

11.2: ~

a n d 5.1)

f r o m the

that v

also u

o

finishes

the p r o o f

exists

can a l s o

~ and

class

a harmonic

to

among

all d i f f e o m o r p h i s m s

satisfies

and

of

of

u oF c o i n c i d e o diffeomor-

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

of Theorem

11.1.

8.1 improve

its i m a g e

topic

o

uniqueness

is a h a r m o n i c

is a n e i g h b o r h o o d

of Lipschitz

of ~ onto

are o f L i p s c h i t z there

conclude

Thm.

8.1

L e t ~ c ZI b e a t w o d i m e n s i o n a l

consisting

homeomor~hism

Then

4.1

, which

we

, we

consequently

. This

same method,

boundar~

~(~)

in ~

x ° 6 ZI

~D

(Thms.

u oF a n d o

latter

Extension the

maps

on

curves,

~(~),

and suppose

and convex with diffeomorphism

u = ~ on

~

domain with

nonempty

a n d l e t 4: ~ ÷ Z 2 b e that

respect

to ~ ( ~ ) .

u: ~ ÷ ~(~)

. Moreover,

a

the c u r v e s

which

u is o f

homotopic

to ~ a n d a s s u m i n g

[JS].

case

is h o m o -

least energy

the s a m e b o u n -

dary values. This

result

ture was

Proof:

rise

strictly In t h i s

that

Remarks

respect

an o b v i o u s

involving

in

first

the p r o o f

arguments

As s h o w n

from

The

image

curva-

~

Thm.

~

are of

class

and ~(~)

C 2+~

and that

and that ~(~)

is

to ~(~).

proceeds change 7.2.

an d ~ ( ~ ) between

along of

The

the

the

lines

o f the p r o o f

replacement

general

case now

argument follows

of Theo-

at boundaby approxi-

as in 8.2.

about

7.4,

of non - positive

[SY1].

convex with case,

ry p o i n t s

11.4.

in

to a d i f f e o m o r p h i s m

11.1 w i t h

mation

taken

We assume

gives

rem

is

solved

the

situation

in higher

in h i g h e r

dimensions

one

dimensions

cannot expect

an a n a l o g u e

of

112

Thm.

11.1 or even of Cor.

the image was gative

flat,

curvature

still

On the o t h e r hand,

11.1

however,

remains

carrying

structure,

it was p o s s i b l e

cf.

a complex

[JY].

to be rather answer only

K~hler m a n i f o l d s s uita b l e m e t r i c

12.1.

Holomorphicity

In

and thus

ne-

i.e.

Riemannian

its

Riemannian

that the

(unique)

a diffeomorphism,

in these papers

the q u e s t i o n

seem

of the e x i s t e n c e

s e t t i n g has a s a t i s f a c t o r y

1 , since o n e - d i m e n s i o n a l orientable

conformal

compact

surfaces

with a

structure.

maps b e t w e e n

of certain h a r m o n i c

and E 2 are

Eells

surfaces

maps

closed orientable

of a surface

12.1:

and W o o d o b t a i n e d

Suppose

and an a n a l y t i c

surfaces,

Z , and d(~)

proof

X(Z)

of

denotes

is the degree

determined

of a

Theorem

topological

12.2:

surfaces,

to me-

relative

to the complex

struc-

. an a n a l y t i c a l

r e s u l t of H. K n e s e r

and furthermore

respect

,

and W o o d to give

Suppose

result:

If

or a n t i h o l o m o r p h i c

by y and g

12.1 e n a b l e d Eells

following

' resp.

IX(X2) I > 0

then h is h o l o m o r p h i c

the f o l l o w i n g

h: gl ÷ Z2 is h a r m o n i c w i t h

7 and g on ZI and Z2

x(x 1) + Id(h) l

Thm.

7.4,

.

Theorem

tures

of

strictly

Theorem

that ZI

[EWI],

trics

considered

but c o m p a c t

of h a r m o n i c

the E u l e r c h a r a c t e r i s t i c map ~

cases,

with

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

in the K ~ h l e r

dimension

are n o t h i n g

Applications

Kneser's

class

and c o r r e s p o n d i n g

12.

Suppose

however,

diffeomorphism

in c o m p l e x

compatible

to show in some

The image m a n i f o l d s

special,

of a h a r m o n i c

in the example

the image has

of K ~ h l e r manifolds,

structure

map in a given h o m o t o p y

[Si] and

Since

open.

in the c o n t e x t

manifolds,

harmonic

to hold.

the case w h e r e

again

[Kn2]

that E I and Z2 are

X(Z 2) < O

. Then

p r o o f of the

closed o r i e n t a b l e

for any continuous

map

~: ~I ÷ ~2 (12.1.1) Proof

Id(~)IX(~ 2) ~ X(ZI)

of T h e o r e m

12.2:

We i n t r o d u c e

some metrics

y and g on ZI and

113

Z2

' resp.,

Thm.

12.1,

and

find a harmonic

h is

(anti)

This,

however,

which

says

map h

. Therefore,

map h homotopic

holomorphic

in case

is in c o n t r a d i c t i o n

[d(h)]x(Z 2) = X ( ~ I ) + (12.1.1)

to the

to ~ by Thm.

By

]d(~)IX(Z 2) < X(EI). Riemann - Hmrwitz

r , r ~ O for an

must

4.2.

(anti)

formula,

holomorphic

hold.

q.e.d.

Before

proving

Corollary m a p h:

Thm.

12.1:

ZI ÷ E2 is

22.1,

we n o t e

two o t h e r

__If ZI

is d i f f e o m o r p h i c

(anti)holomorphic

(and

interesting

consequences

to S 2 , then

any h a r m o n i c

therefore

constant,

if X(Z 2)

0). This

is

due

Corollar~z then

Cor.

12.2,

12.2.

and L e m a i r e

is no h a r m o n i c

due

map

Proof

In this

[WI]

map

h:

to E e l l s - W o o d , I is

of Theorem

section,

Z I + Z 2 with

torus,

d(h)

= ±I

and Z2 to S 2 , , for any m e -

follows

a covering

f r o m Thm.

12.1,

since

any h o l o -

map.

12.1

we want

use

to the

.

of d e g r e e

shall make

[LI].

I_~f E I is d i f f e o m o r p h i c

on Z I a n d Z 2

morphic

We

12.2:

there

trics

to W o o d

to p r o v e

of some

Thm.

computations

12.1. of S c h o e n

and Yau

[SYI]

in the

sequel. It is c o n v e n i e n t If p 2 ( z ) d z d ~ nate

charts

to use

map

h -- + 2 ~ h ZZ

Lemma

complex are

12. 1:

O

cf.

Thm.

At points,

, cf.

conformal

where

~h o_~r ~h, r e s p . ,

(12.2.3)

Alog

the G a u s s

curvature

is n o n

_ f -hi 2)

l[hr2 = K I + ~2(l~hl 2 of Z, l

coordi-

on E I and E 2 , resp.,

(1.3.4)

AJ_og [~h[ 2 = ~<1 - I<2 ( [ ~ h 1 2

K. d e n o t e s l

3.1)

to

Z

(12.2.2)

where

w.r.t,

satisfies

h h- = O Z

notation.

the m e t r i c s

(for the e x i s t e n c e ,

then h as a h a r m o n i c

(12.2.1)

the

and ~2(h)dhdh

l~hl 2) , , and

zero

114

2

2

l~hr2 = ~ h z ~

, r~h[2--%~zh~

0 Proof:

0

F o r any p o s i t i v e

(12.2.4)

smooth

I • f = ~1 Af - f-~

A log

function I

f on E

I

'

fzfz

Furthermore,

(12.2.5)

Alog %

= KI

P

In o r d e r covariant

to a b b r e v i a t e derivative

D? h Z 3z (12.2.1)

=

h

ZZ

the

in the b u n d l e 2c~h +--h ~

Z

t h e n is e x p r e s s e d

(12.2.6)

following calculations,

h

h -I T2 2

we define

D as the

, e.g.

Z

as

D 3 h-- = O - Z ~z

Since

~2hzh~

=
h z } h-lTT

, 2

(12.2.7)

A~2h

zz

2

3~

< D ~

hz , ~

> , using

(12.2.6)

P I

3z

= ±2

~(", (~)

P

z

, h, (~)

,

hz

P

p where

R denotes

the c u r v a t u r e

3%-

t e n s o r of E

=-K213hl2(j(h))

~p < D ~ h z

2

+

D~_ ~ ~

>

, %)

3-'~

+

115

where

J(h)

=

I~hl 2 -

l~-hl2

is the J a c o b i a n

of h

.

Mo re ove r,

(12.2.8)

I _~_ ~z




P

1 < hz = --~

< D~

h z')

,

p using

again

D ~ hz >

hz ,

~z

(12.2.6)

and the

~

fact t h a t

the

complex

dimension

o f ~2 is

I (12.2.2) and

duced the

now

follows

(12.2.3) from

image

from

can e i t h e r (12.2.2),

(12.2.4),

be

(12.2.5),

calculated

since

I~hI2

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

=

(12.2.7), and

in the

I ~h12

as a change

same w a y

and complex

(12.2.8),

or directly conjugation

deon

of o r i e n t a t i o n .

q . e .d.

Lemma

12.2:

If h z ( Z O)

(12.2.8)

where

~ is a n o n v a n i s h i n ~

Proof:

result

By

[B2]

(12.2.1),

z = z

o'

C 2 f u n c t i o n r a n d k is h o l o m o r p h i c .

holds

w e can a p p l y or

[Hzl]),

der c o n t i n u o u s ciple

near

for

f: = h

z

h--

z

.

satisfies

-~ clfl

Therefore, (cf.

, then

l~hl 2 = ~'Ikl 2

A corresponding

Ifzl

= 0

shows

similarity

to o b t a i n

~ . An

that

the

inspection

in o u r case

the

principle

representation

o f the p r o o f

~ £ C2

of B e r s

(cf.

and

(12.2.8)

Vekua with

of the s i m i l a r i t y

[Hzl],

p.

H~Iprin-

210).

q.e.d.

Proof are

of T h e o r e m

isolated,

12.1:

unless

~h =- 0

Lemma

12.2

shows

that

the

, a n d that n e a r

each

zi

zeroes ,

z i of

l~h] 2

116 n

lahl2 = a i l z _ z i l

n

i + o(iz_zil

i) ,

for s o m e a. > O a n d s o m e n. E ]~ . 1 1 By Lemma

12.1

a n d the

residue

formula,

f K 1 - S K2(lahl 2 ~1 ~2

(12.2.9)

Similarly,

if ~h

~ O

where

m i 6 N are now

Thus,

since

X(7: 1)

I ~hl 2 -

f

K2(lahl

the o r d e r s

I~hl 2) = - Z n i

2-

I~hl 2) = - ~ m .

of t h e

zeros

l~hl 2 is the J a c o b i a n

X(~]2) =< 0

- d(h)

@h - O

,

~ K1 +

(12.2.10)

unless

,

of

of h

unless

~h - 0

unless

~ h --- 0

l~b 12

,

and X ( Z I) + d ( h ) x ( Z 2) ~ 0 a n d Thm.

12.3.

12.1

,

follows.

Contractability

of T e i c h m ~ l l e r

space

a n d the d i f f e o m o r p h i s m

group In the

following

nic maps sequel We

between are

first

Let

sections,

due

recall

surfaces

the

of genus

classical

setting

from our previous

easier

smooth 2

where

metric.

tion.

D(~)

is the d i f f e o m o r p h i s m

group

o f D(Z)

to t h e M(Z)

is the

map

space

the o r i e n t a t i o n ) . trics

compatible

of those

The

arguments

that we

manifold

by

deviate

diffeomorphisms

which

with

in this

Do(E)

is

here

endowed

(Z,g)

~ , while

without

a n d the t o r u s

a surface

denoted

group of

in the

[Tr].

of the s p h e r e

be

of h a r m o -

theory.

Z denoted

a pair will

consisting

identity

theory.

twodimensional

(The c a s e

Riemannian

applications

and Tromba

to handle). N o t e

notation, Such

[EE]

some

of Teichm~ller

oriented at least

and much

to g i v e

to TeichmHller

to E a r l e - E e l l s

Z be a compact

boundary

we want

a

sec-

is the s u b -

are h o m o t o p i c

id of

of smooth

complex

For a given with

these

structures

complex

structure

structures,

and

on we

Z

(compatible can

look

with

a t the m e -

the uniformization

theorem

117

easily

shows that

metric

of constant

o f M(E)

each

as a m e t r i c

MCZ)

defined

x DCZ)

o f M(Z)

-1

corresponds

to p r e c i s e l y

. Thus,

we

shall

think

g on ~ o f c u r v a t u r e

-1

. We have

one

o f an e l e m e n t

a natural

action

÷ MCz)

by

(g,v)

÷

(v~g)

denotes

,

where

v~g

then

v~g(x)

Then

the T e i c h m ~ l l e r

TCz)

D(Z)

element

curvature

carries

g via v

, i.e.

if x 6 Z ,

space

T(Z)

is d e f i n e d

by

the C ~ t o p o l o g y M(Z)

also

of u n i f o r m

carries

its

convergence

C~ t o p o l o g y ,

of derivatives

a n d T(Z)

inherits

of

the

topology.

The main

result

morphic

to 6g-6

theory,

we

The

Theorem

o f the m e t r i c

(x)).

= M(Z)/Do(Z) .

all o r d e r s , quotient

the p u l l b a c k

= g(v

is T e i c h m ~ l l e r ' s , where

refer to

following

12.3:

is t r i v i a l ,

[A1],

there

genus

[A2],

is d u e

The bundle

i.e.

F ÷ T(~)

M(Z)

[Tm], result

Theorem,

g is the

[AB],

to E a r l e

given by

exists

saying

of E

t h a t T(Z)

[BI],

[EE],

and Eells

the q u o t i e n t

a homeomorphism

is h o m e o -

• F o r an a c c o u n t and

o f the

[FT].

[EE].

m a p p:

F with

M(Z)

÷ T(Z)

the p r o p e r t y

that

x Do(E )

T(Z) commutes,

where

Proof: ture

We

-I

exists

. Then

~ is t h e p r o j e c t i o n

fix some

for any other

a harmonic

u(y,g)

by Theorem a-priori

conformal

onto

the

structure,

metric

first

i.e.

factor.

a metric

g on ~ o f c u r v a t u r e

y of

-7

curva-

, there

diffeomorphism

: (Z,y)

+

11.1.u(y,g) estimates

of

(Z,g)

is u n i q u e chapter

by Theorem

6, i t

follows

5.3.

By uniqueness

t h a t u(y,g)

a n d the

depends

118

continuously harmonic pends

on

Since

then

functional

show

that

determinant

also

u

(y,g)-I

of de-

context

seems

to h a v e

been

first

vou(y,v~g)

define

(12.3.2)

above,

(v £ D o ( Z ) ) ,

between

(Z,y)

and

is

an i s o m e t r y ,

(Z,g), and h o m o t o p i c

to

= u(y,g)

F is

continuous.

Furthermore,

from

(12.3.1)

and

of p

=

( p ( v * g ) , u ( y , v * g ) -I)

=

(p (g) , u ( y , g ) - 1 o v )

o

= F(g'),

(Z) by

(Z,g)

( p ( g ) , u ( y , g ) -I

F(v*g)

any v 6 D

÷

F by

=

noted

(Z,v*g)

5.3,

the d e f i n i t i o n s

o F(g')

v:

is h a r m o n i c

By Thm.

F(g)

6 D

7.1

in the p r e s e n t

construction,

(12.3.1)

If F(g)

of Thm.

for the

g .

dependance

vou(y,v*g)

u(y,g).

AS w a s

on

bounds

by Sampson).

by

We n o w

lower

diffeomorphisms

(Continuous

for

. The

continuously

noticed

also

g

then

p(g)

= p(g')

definition

of p

, a n d F(g')

= F(g),

thus

F is s u r j e c t i v e , (p(g),v)

v = id

and t h e r e f o r e

and g = g'

=

g' = v * g

(p(g),u(y,g)-1ov).

. This

shows

that

for some

v

Since

F is i n j e c t i v e .

since =

(p(g),u(y,g) - 1 o u ( Y , g ) o v )

=

(p(g),u(y,v*u(y,g)*g)

-I)

by

(12.3.1)

= F(v*u(y,g)*g)

Therefore,

F is a h o m e o m o r p h i s m ,

and

the

theorem

follows.

q.e.d.

A local

section

o(t)

o: T(Z)

= u(y,g)*g

,

÷ M(Z)

is g i v e n

by

119

w h e r e we this

can take

choice by

tractible, mannian metric

any g w i t h

(12.3.1)

since M(E)

metrics

on Z

2 - tensors

implies

Corollary

the

= M(E)/P(E), , i.e.

M(Z)

12.3:

T(Z)

12.4.

Tromba's

We now w a n t is a cell. ba's

The

to sketch

to use

the n o t a t i o n s t h e o r e m seems

Lemma

stating

12.3:

is also

Suppose

with

respect

w here

v(t)

Proof: metric

space

of T r o m b a

o f the p r e v i o u s

[T~:] w h i c h

however,

we

(Z,y) +

that T(E)

Theorem,

but Trom-

maps.

We con-

or less

for a long

from S a c k s - U h l e n b e c k

formulate

(E,g)

[SkU] .

a le~ma.

is harmonic.

Then

u(y,g)

functional

of the metric

dx

y via

family of d i f f e o m o r p h i s m s

has

shows

section.

is taken

of the e n e r g y

by a d i f f e o m o r p h i s m

[FT].)

is a cell

to have been k n o w n more

reason w h y L e m m a

the i n v e r s e

(cf.

C~ - f u n c -

12.3.

is part of T e i c h m ~ l l e r ' s

to all v a r i a t i o n s

The

sition w i t h

contractible to Thm.

= 1/2 f y~B (x)gij (u)D u l D s u 3 / ~

is a smooth

of p o s i t i v e

(E) are c o n t r a c t i b l e .

u(y,g):

a critical point

E(u,y,g)

is the space o f Rie-

o

give here

the theorem,

of

is con-

cone in the space of sym-

an easy p r o o f by u s i n g h a r m o n i c

The p r o o f we shall

Before

is i n d e p e n d a n t

fact that M(E)

M(Z)

that T e i c h m O l l e r

this

provides

following

time.

are

an a r g u m e n t

Of course,

argument

tinue

proof

-

~(t)

is the space

Corollary

and D -

where

an open c o n v e x

and P(Z)

following

since

We now use the

on Z , and P(Z)

tions on E a n d both This

t = p(g),

again.

12.3 holds

a family

of I w i t h

is that the

the same e f f e c t

v(t)*y v(O)

change

as c h a n g i n g

= id

.

of the

u by compo-

diffeomorphism.

In formulae

S YeB(v(x))gij(u(x))D~ui(x)DsuJ

= f y ~ B ( y ) g i j ( u ( v - 1 (y)))D u ( v -I

Since

uov -I (t) p r o v i d e s

a variation

(x) /y(v(x))

(y))DsU

dx

(v-1(y))

of u , the lemma

=

/ y ( - ~ dy

.

follows.

q.e.d.

Furthermore,

w e note

that E(u,oy,g)

= E(u,y,g)

for any p o s i t i v e

120

function E with

o on Z , by L e m m a

respect

all con f o r m a l l y e q u i v a l e n t 12.3 i n t o account,

by t a k i n g the class

structure We

Theorem with

12.4:

to any s m o o t h

we

Suppose

u(y,g)

u(y,g), Here,

it is e n o u g h

to

p' (¥) is d e f i n e d

conformally

equivalent

a nd then p r o j e c t i n g

to y ,

this con-

map p of the p r e v i o u s

c h a n g e y(t)

is a c r i t i c a l

branched

structure

sec-

covering.

respect

of the m e t r i c y . For any p o i n t p 6 Z , a represented

local c o o r d i n a t e s ,

by the unit disc by

and u s i n g L e m m a 6.1

that the m e t r i c in these

again,

local coordinates

form

Y~B (z) = ~eB

If we n o w c h a n g e

for z £ D .

y by a s m o o t h

7~8(t,z)

= ~ ~

Y11(t,z)

= Y22(t,z)

variation

y (t) w i t h

for z in a n e i g h b o r h o o d

of SD

and

= I for all t,z

,

then d

d-~ E ( u , y ( t ) , g )

d

= ~

both

of y in

t h a t u is a c r i t i c a l p o i n t w i t h

of p can be c o n f o r m a l l y

3.1. This p r o v i d e s

p o i n t of E ( u , y , g )

of u a n d the c o n f o r m a l

is a c o n f o r m a l

can a s s u m e w . l . o . g ,

the

are

of M(Z)

onto T(E) by the q u o t i e n t

We p r o v e d a l r e a d y

neighborhood Thm.

element

to

takina also Lemma

the e f f e c t o f a v a r i a t i o n

p' (¥) in T(Z).

of m e t r i c s w h i c h

to v a r i a t i o n s

Then u(y,g)

Proof:

. Therefore,

p o i n t of

respect

can n o w s t a t e

respect

T(E).

oy

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

element

the c o r r e s p o n d i n g

formal

if u is a c r i t i c a l

on E at a c r i t i c a l p o i n t

vary the c o r r e s p o n d i n g

tion.

metrics

if w e w a n t

o f the d o m a i n m e t r i c

i.e.

6. I. T h e r e f o r e ,

to the m e t r i c y , it is a l s o c r i t i c a l w i t h

"

~D ¥ e 8 ( t ' z ) g i J

'

D~ul DSu3

/~(t,~

dz =

= ~-~ d ~Dgij (u) (D1ui D I u j +D2Ui D2uJ + 2y 12 (t,z)

DlUi

D2uJ)

" /I-

(y12(t,z))2

dz =

has

121

2I~y

12

(t, z)gij (u) D1ul

D2u3

dz

D

d Since ~-~ E ( u , y ( t ) , g )

~-~ ~

12

(t,z)

= O

can be c h o s e n

gij (u) DlUl D2u3 If we rotate argument

(z

=

x

+

at t = O , we i n f e r

= O

D1u3

coordinates

- D2ul

the q u a d r a t i c

by an angle

of 7/4

, the same

=

D2u3 ) = 0 .

differential

(gij (u) (D1ui D1uJ

= ([u12

vanishes

and since

that a l s o

gij (u)(D1ul

(12.4.1)

arbitrarily

the i s o t h e r m a l

yields

Therefore,

at t = 0 , as shown above,

lUyl 2

-

D2 ul D2u3 - 2 i

DI ul D 2 u 3 ) d z 2 =

2i )dz2

iy) identically,

and the t h e o r e m

follows. q.e.d.

We now p r o v e Theorem

the

following

12.5:

Suppose

ture on Z , a n d define p ÷ E(U(yp,g)) is a m e t r i c

harmonic

map

T h e n E is a p r o p e r which Proof:

U(yp,g)

[SY2]:

g is a fixed m e t r i c w i t h n o n p o s i t i v e

a function

representing

(Z,yp)

to

function

minimizes

Using Theorem

E is a c o n t i n u o u s

of S c h o e n - Y a u

on T(Z)

curva-

via

,

where yp

from

result

p , and U(7p,g)

(Z,g), on T(Z)

E with

homotopic , and we

respect

of p .

corresponding

to the i d e n t i t y

of Z .

can find a p £ T(Z)

to b o t h

5.3 and the a - p r i o r i

function

is the

for

u and. p . estimates,

we see

that

122

We now

show

contains

that

if y is a m e t r i c

a closed

(12.4.2)

geodesic

E(u(y,g))

as 1 + O, w h e r e the c o l l a r isometric

> c/l

u(y,g)

theorem copy of

of

of

constant

length

1 , then

[Ke]

the region

map

homotopic

and Halpern

>= 1

o

the h o m o t o p i c a l l y > 0

~u3) ~S

e {

{z:

I <

nontrivial

?~

1

~ 2e(u)

U1 ~

(gij

s denotes

Izl

Izl = I a n d

closed

curves

on

By

an

z < z - 8} i n Izl

= e I via

(7.,g) h a v e

length

ds =

, ds => 1 o

along

a curve

(r s i n ~0) -I d r

arg

z = ~

, and using

. Noting

H61der's

Ul ~s

(gij

inequality,

implies

eI 12o ~- ({ ~ )

e1 (~I 2e(u)

integrating

w.r.t.

Since,

as n o t e d

1 e dr 2 ) = 1 5 2e(u) r sin ~ 1

e1 S 2e(u) I

above,

11. I, f o r a n y m e t r i c

u(y,g),

homotopic

w.l.o.g,

to t h e

that T has

, (12.4.2)

stays within

allows

the

length

to t h e

a Theorem space

-I

a harmonic 6.1, w e

map

can assume

. If w e n o w m i n i m i z e

that a minimizing

of

the s h o r t e s t

constant

curvature

below

by

of Mumford

[Mu],

such a set

R

c

o

closed metric

sequence

w.r.t.

ui(Yi,g)

have bounaed points

in T

under

energy,

where

( Z ) which

the natural

geodesic

with

representing

respect

p is

> O}

is a c o m p a c t

( 7. ) = M( Z )/D( 7. ) . S u p p o s e

distinct

(7)

follows.

exists

. By Lemma

bounded

ui(~i,g)

y in R

of Z

curvature

to a s s u m e

, (12.4.2)

, there

a region

{p £ T(Z):

moduli

y on ~

identity

constant us

dr . 2 r sln

drd~2 ~ 2E(u) r sin ~0

0 ÷ 0 as 1 + 0

B y Thm.

By

contains

< e I , 0 < arg

I/2 )

arclength

12 (~ - 20 ) ~-8 o 1 ~ S 8

X

(Z,y)

-

(12.4.3)

and

(7.,X)

, we have

(12.4.3) where

and

to the i d e n t i t y .

[Hp],

the P o i n c a r ~ u p p e r h a l f p l a n e , i d e n t i f y i n g 1 z + e z , a n d e + 0 as 1 + 0 . Since

-I

,

is a h a r m o n i c

of Keen

curvature

Y i is a s e q u e n c e

all correspond

projection

T

subset

on the other hand

to t h e

( 7.)+ R ( Z )

o f the that

of mutually same point = T(Z

)/

123

(D( Z ) / using

Do(Z)).

By e q u i c o n t i n u i t y

the fact that all 7i h a v e

constant

curvature

seque n c e w h i c h homotopy

classes1~y

proper

function a compact

Theorem,

of Yi" These

curvature

metric),

+ 7, ui(Yi,g) ÷ u ( 7 , g )

radius

we can choose

• u is h a r m o n i c w i t h

by c o n t i n u i t y of E as a function of y . I) w h e n c o m p o s e d w i t h the c o n f o r m a l and h e n c e Theorem

12.6 :

Proof: We

T (Z) is t o p o l o g i c a l l y

(A. T r o m b a

fix any c o n f o r m a l

g with nonpositive m o n i c map u(y,g) E(u(y,g)) Ever y

curvature.

homotopic

point

by Thm.

12.4,

fore of

d e g r e e one,

Therefore,

convergent

sub-

that E is a ui(Yi,g)

again

stays

(by M u m f o r d ' s

bounded below

a convergent respect

for

subsequence

to y an m i n i m i z i n g

ener~,y p r e s e r v i n g

map y1 + Y i

a cell.

[Tr]): structure

as a f u n c t i o n

critical

sequence

uniformly

11.2),

for their

are in d i f f e r e n t

imply

( Z ). By e q u i c o n t i n u i t y

injectivity

3.3 or

radius

(Yi,g)

assertions

on T( Z ) and that a m ~ n i m i z i n g region of T

(cf.

a uniformly

the fact that all u i

choice

all Yi now have

their c o n s t a n t 7i

metric) , the u i (yi, g) have

contradicts

within

of the ui(Yi,g)

the same i n j e c t i v i t y

p on Z and r e p r e s e n t For every m e t r i c

to the identity.

of the c o n f o r m a l

gives

and since

rise

u(y,g)

no b r a n c h

We now

structure

to a c o n f o r m a l is h o m o t o p i c

points

it by some m e t r i c

y , we can look

find a har-

at E =

represented

branched

by y .

cover u(y,g),

to the i d e n t i t y

and there-

occur.

the global m i n i m u m p of Thm.

12.5 has to be the only

criti-

cal p o i n t of E on T(Z). It is s t r a i g h t f o r w a r d p is p o s i t i v e and the result

12.5.

The

Still

another

outlined

tional

definite follows

approach

(cf.

extremal Given

a n d Rauch

two

metric

the

harmonic

finally

zing map on the m e t r i c tional p r o c e d u r e

complex

to T e i c h m ~ l l e r 1954.

structures

class

They

maps by s o l v i n g

minimum,

tried

these

conformal

dependance

to obtain

integral

and then m a x i m i z e

a continuous

theory w a s

on a surface,

of maps b e t w e e n

map o v e r all such

varia-

they m i n i structures

the e n e r g y metrics

of

with

of the m i n i m i -

to show that a r e s u l t of such a t w o f o l d v a r i a -

is an e x t r e m a l

of a s o l u t i o n

maps

[GR] in

on the image

assume

of the first v a r i a t i o n a l existence

of E at

Hence p is a n o n d e g e n e r a t e

quasiconformal

for any c o n f o r m a l corresponding

variation

theory.

of h a r m o n i c

in a given h o m o t o p y

fixed area and

that the s e c o n d

of G e r s t e n h a b e r - R a u c h

application

problems.

energy

[Tr]).

from Morse

by G e r s t e n h a b e r

TeichmHller's

mize

to c a l c u l a t e

problem

quasiconformal is p r o v i d e d

to the s e c o n d

map.

While

by L e m a i r e ' s

one is not clear,

the s o l u t i o n Theorem,

the

and the conti-

124

nuous

dependance

is n o t k n o w n fore,

this

is a l s o n o t k n o w n

for

is

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

an i n t e r e s t i n g ,

because

curved

but

uniqueness

of harmonic

image metric,

unfortunately

cf.

still

5.5.

maps

There-

incomplete

pro-

gram.

12.6.

Harmonic

L e t F:

X ÷ ~n+p

into ~n+p G(p,n)

denote

. The

Gauss

of p - p l a n e s

plane Ruh

Gauss maps

and Vilms

mersed

[RV]

12.7:

an i m m e r s i o n map

theorems

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

G: X ÷ G(p,n)

in ~ n + p

assigns

into

to e a c h

manifold

the G r a s s m a n n i a n point

X

manifold

x 6 X the normal

G: X ÷ G(p,n)

12t4:

H i_n_n~ 3 . T h e n

is

type

rallel

the G a u s s

under

12.4,

theorems

certain

is s h o w n

we

do n o t w a n t

to

[HJW]

12.7.

Surfaces

results

on

arises

from

surface, can b e

mean

curvature

into

image.

in t u r n

in

Gauss map Bern-

with

approach

This means

pa-

proves

a

manifolds

that the Gauss

that the correspon-

of ~n+p but

in this

curvature

(immersed)

This

implies

this point,

the

to p r o v e

Grassmannian

subspace

results

then

used

or submanifolds

space.

maps the

which

Gauss

field.

of constant

submanifolds

to e l a b o r a t e

refer

the

reader

in-

direction.

3- space

surfaces

of constant

Gauss

cur-

3 - space.

F is a s u r f a c e

= h11du2

is p o s i t i v e parameters positive

a minimal

These

[HOS] f o r s o m e

with

K o f F is p o s i t i v e ,

II:

Z is

is a l i n e a r

of constant

example

Suppose vature

and

i f X is i m -

X ÷ S 2 is h a r m o n i c .

for h a r m o n i c

submanifold

stead

in

G:

restrictions

Here,

vature

i.e.

to be c o n s t a n t

immersed

Another

map

if and only

curvature

field of Euclidean

theorem size

mean

for m i n i m a l

curvature

type

is h a r m o n i c ,

E is a s u r f a c e

antiholomorphic.

mean

Liouville

parallel

Suppose

If H = O in Cor. actually stein

proved

into ~n+p with

Corollary

ding

Bernstein

at F(x).

Theorem

map

and

+ 2h12dudv

definite x,y

([Db],

the s e c o n d

of

u,v

x,y are §725)

u,v

. If the G a u s s

fundamental

cur-

form

+ h22dv2

and can hence

instead

function),

and Darboux

local coordinates then

be

diagonalized

by

(i.e. II = l ( d x 2 + dy2),

called

discovered

isothermal that

introducing where

conjugated

u a n d v as

new

I is a

parameters,

functions

of x and

125

y satisfy

a s y s t e m of e l l i p t i c

fundamental not

form of F , i.e.

depending

extensively embedding

equations

on the i m m e r s i o n

by H. Lewy

depending

only on i n t r i n s i c of F into

only on the

geometric

3 - space.

[Lw] a n d E. Heinz

([Hz2])

This

first

quantities, fact was

for s o l v i n g

used

the Weyl

problem.

The s y s t e m

discovered

(12.7.1)

An +

by D a r b o u x

is the

(FII + I/2 ~

(log K ) ) ( u ~

+ (2r12 + i/2

~

I + F22

(v

+ u~) +

clog K))(ux

v2 )

+

following

Y

÷ UyVy

÷

= O

u 2 + u2 ) + 2 Av + F11 ( x y

+ (2r~2 ÷ I/2 ~

Clog K))(UxV x + UyVy) +

+ (r22 + I/2 ~ We see in p a r t i c u l a r constant,

that,

Clog~))(v~+

if the Gauss

then the t r a n s f o r m a t i o n

v ) = 0

curvature

(x,y) ÷

(u,v)

K of F is a p o s i t i v e

is harmonic.

(This

fact was p o i n t e d out to me by S. Hildebrandt). As an application, orem of L i e b m a n n

Theorem

12.8:

positive radius

a proof

[BI], p.

K into

3-space

By the Gauss - B o n n e t fundamental

x,y on the sphere

h:

By Cor.

S 2 which

a harmonic

S2 ÷ F

the-

F of constant sphere of

Theorem,

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

form of a given

diagonalize

immersion

theorem this

a sphere.

of F is p o s i t i v e

to o b t a i n

parameters

form. Since K { const.,

we

map

.

12.1, h is

the first

a closed surface

is given by a s t a n d a r d

we can use the u n i f o r m i z a t i o n

thus obtain

f o l l o w i n g well - k n o w n

.

the s e c o n d

definite,

of the

195).

The only i m m e r s i o n o f

curvature

I//K

Proof: Since

we p r e s e n t (cf.

(anti)

fundamental

form are p r o p o r t i o n a l

conformal

form.

Hence

everywhere,

and therefore

the

also d i a g o n a l i z e s

first and the s e c o n d

which means

fundamental

that the given immer-

126

sion

is e v e r y w h e r e

umbilical

and therefore

a standard

sphere

(cf.

[BI],

p.97) .

If we note

that in the s p h e r e - c a s e ,

and the T h e o r e m of R i e m a n n - Roch, is m u c h

in the s p i r i t of H. Hopf's

sphere

into

3 - space w i t h

sphere

(cf.

[Ho]).

constant

Cor.

12.1

follows

from L e m m a

then we see that the p r e c e d i n g proof,

that every

mean c u r v a t u r e

immersion

is a s t a n d a r d

1.1 proof

of a

References

[Ad]

Adams,

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[AI]

Ahlfors, L.,Some Remarks on T e i c h m 0 l l e r ' s Surfaces, Ann. Math. 74 (1961), 171 - 1 9 1

[A2 ]

Ahlfors, L . , C u r v a t u r e Properties of T e i c h m ~ l l e r ' s Space, J. d'Anal. Math. 9 (1961), 161 - 176

[AB]

Ahlfors, L., a n d L. Bers, Riemann's M a p p i n g T h e o r e m for V a r i a ble Metrics, Ann. Math. 72 (1960), 385 - 4 0 4

[Bg]

Berg, P., On U n i v a l e n t Mappings by Solutions of L i n e a r P a r t i a l D i f f e r e n t i a l Equations, Trans. Amer. Math. Soc. (1957), 3 1 0 318

[B1]

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[B2]

Bers, L., An Outline of the Theory of P s e u d o a n a l y t i c Functions, Bull. Amer. Math. Soc. 62 (1956), 291 - 331

[BJS ]

Bers, L., F. John, and M. Schechter, Partial Differential Equations, Interscience, New York, (1964)

[BI]

Blaschke, W., V o r l e s u n g e n 0her D i f f e r e n t i a l g e o m e t r i e , Springer, Berlin, 41945

[BCI ]

Brezis, H., and J. M. Coron, M u l t i p l e Solutions of H - s y s t e m s and R e l l i c h ' s Conjecture, Comm. Pure Appl. Math., to appear

[BC2]

Brezis, H., and J. M. Coron, L a r g e solutions for h a r m o n i c maps in two dimensions, Comm. ~ath. Phys., to appear

[ca]

Calabi, E., An Intrinsic C h a r a c t e r i z a t i o n of H a r m o n i c O n e Forms, in: Global Analysis, ed. by D. C. Spencer and S. Iyanaga, P r i n c e t o n Univ. Press, Tokyo, Princeton, (1969)

[ci ]

Choi, H. J., On the L i o u v i l l e T h e o r e m for Harmonic Maps, Preprint

[cq]

Choquet, G., Sur un type de t r a n s f o r m a t i o n a n a l y t i q u e g~n~ralisant la r ~ p r ~ s e n t a t i o n conforme et d~finie au moyen de fonctions harmoniques, Bull. Sci. Math. (2) 69 (1945), 1 5 6 165

[co]

Courant, R., Dirichlet's Principle, Conformal Mapping, Minimal Surfaces, New York, Interscience 1950

[Db]

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