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.

__ 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
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

=

0

,

then s ' (p)

~5.2.3) (Here,

(x,x,>

we s -s

K

- (p)

(Ix(o) 12 ÷ Ix(p) ] 2)


0


_ O

,

- s"

Ixi

= s(

(Ix121x' 12

)

+ 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 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

-

)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 sK (~) and thus with (5.2.12)

+

nor 2 v2 )

2

nor

I'lv2

(5.29) D2Q 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 S'K (p) < e 1 @ ~ e 2 ' v l ( ~ V 2 >

(I -s ~(11

+ s~K(p))

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

<e1~e2,vlev2

(Iv112 Iv212) +

2QK(y) 1

i~rad ~[ 24

-


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
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

=


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,~,-~/~ 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
6 -I =

where

6

= ~(M,~, 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
_-~ ,

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 < ~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
~

,

~2 = 62(~'

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
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,= ~

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)


~ > 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 = ~ , 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

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,

R., S o b o l e v Spaces, A c a d e m i c Press,

[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]

Bers, L., Q u a s i c o n f o r m a l M a p p i n g s and T e i c h m ~ l l e r ' s Theorem, i n : A n a l y t i c Functions, P r i n c e t o n Univ. Press, Princeton, (1960), 89 - 119

[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]

Darboux, 1894

[DI]

Deimling, K., N i c h t l i n e a r e G l e i c h u n g e n und A b b i l d u n g s g r a d e , Springer, Berlin, Heidelberg, New York, 1974

[dTK]

De Turck, D., and J. Kazdan, Some Regularity Theorems in Riem a n n i a n Geometry, Ann. Sc. Ec. N. Sup. Paris

lEE]

Earle, C. J., and J. Eells, A Fibre Bundle D e s c r i p t i o n of T e i c h m ~ l l e r Theory, J. Diff. Geom. 3 (1969), 1 9 - 4 3

G., Th~orie

1975 Space of Riemann

g~n~rale des surfaces,

p a r t I,

and

Tome III, Paris,

128

[ESz]

Earle, C. J., and A. Schatz, T e i c h m ~ l l e r Theory for Surfaces with Boundary, J. Diff. Geom. 4 (1970)~ 169 - 185

[E]

Eells, J., R e g u l a r i t y of Certain H a r m o n i c Maps, Proc. Conf. 1982

[ELI]

Eells, J., and L. Lemaire, A Report on Harmonic Maps, Bull. London Math. Soc. 10, I - 6 8 (1978)

[EL2]

Eells, J., and L. Lemaire, Deformations of Metrics and Associated H a r m o n i c Maps, P a t o d i , Mem. Vol. Geometry and Analysis, Tara Inst. 1980, 33 - 4 5

[EL3]

Eells, J., and L. Lemaire, On the C o n s t r u c t i o n of H a r m o n i c and H o l o m o r p h i c Maps Between Surfaces, Math. Ann 252 (1980), 2 7 52

[EL4]

Eells, J., and L. Lemaire, CBMS Regional Conf. 1981

[ES]

Eells, J., and J. H. Sampson, H a r m o n i c Mappings Manifolds, Am. J. Math. 86 (1964), 109 - 1 6 0

[EWI]

Eells, J., and J. C. Wood, Restrictions on Harmonic Maps of Surfaces, Top. 15 (1976), 263-266

[EW2]

Eells, J., and J. C. Wood, Harmonic ~ p s from Surfaces p l e x P r o j e c t i v e Spaces, Preprint, Univ. Warwick, 1981

[EW3]

Eells, J., and J. C. Wood, The Existence and C o n s t r u c t i o n of Certain Harmonic Maps, Ist. F r a n c e s c o Severi, Symp. Math. 26, (1982), 123 - 1 3 8

[Es]

Eliasson, H. I., A Priori Growth and H ~ i d e r E s t i m a t e s monic Mappings, Univ. Iceland, Preprint, 1981

IF]

Federer, H., Geometric Measure Theory, 153, New York, 1969

[Fe]

Fend%el, W., E l e m e n t a r e Beweise und A n w e n d u n g e n e i n i g e r Fixpunkts~tze, Mat. Tidsskr. (B) (1932), 6 6 - 8 7

[FT]

Fischer, E., and A. Tromba, On a Purely "Riemannian" P r o o f of the Structure and Dimension of the U n r a m i f i e d Moduli Space of a C o m p a c t Riemann Surface, SFB 72, P r e p r i n t 516, Bonn, 1982

[GR]

Gerstenhaber, M., and H. E. Rauch, On Extremal Q u a s i c o n f o r m a l Mappings I, II, Proc. Nat. Ac. Sc. 40 (1954), 808 - 8 1 2 and 991 - 9 9 4

[G]

Giaquinta, M., M u l t i p l e Integrals in the Calculus of Variations and Non L i n e a r E l l i p t i c Systems, SFB 72, V o r l e s u n g s r e i h e No. 6, Bonn, 1981

[GGI]

Giaquinta, M., and Giusti, E., On the R e g u l a r i t v of the M i n i m a of V a r i a t i o n a l Integrals, Acta Math. 148 (I 982) 31 - 46

[GG2]

Giaquinta, M., and Giusti, E., The S i n g u l a r Set of the M i n i m a of Certain Q u a d r a t i c Functionals, to appear in A n a l y s i s

[GH]

Giaquinta, M., and S. Hildebrandt, A Priori Estimates m o n i c Mappings, Journ. Reine Anaew. Math.

Durham

S e l e c t e d Topics in H a r m o n i c Maps,

Springer,

of R i e m a n n i a n

to Com-

for Har-

Grundlehren

for Har-

129

[GT]

Gilbarg, D., and N. S. Trudinger, E l l i p t i c Partial Differential E q u a t i o n s of Second Order, Springer, G r u n d l e h r e n 224, Berlin, Heidelberg, New York, 1977

[Go]

Gordon, W., Convex Functions and H a r m o n i c Mappings, M. S. 33 (1972), 433 - 4 3 7

[GKM]

Gromoll, D., W. Klingenberg, and W. Meyer, R i e m a n n s c h e Geometrie im GroBen, L.N.M. 55, Springer, Berlin, Heidelberg, New York, 21975

[~]

Gulliver, R., and J. Jost, H a r m o n i c Maps w h i c h Solve a Free B o u n d a r y Value Problem, in p r e p a r a t i o n

[Hp]

Halpern, N., A Proof of the Collar Lemma, Bull. London. Soc. 13 (1981), 141 - 1 4 4

Math.

[Hm]

Hamilton, R., Harmonic Maps of M a n i f o l d s w i t h Boundary, M. 471, Springer, Berlin, Heidelberg, New York, 1975

L.N.

[Ht]

Hartman, P., On H o m o t o p i c (1967) 673 - 6 8 7

[HtW]

Hart/nan, P., and A. Wintner, On the Local B e h a v i o r of Solutions of N o n p a r a b o l i c Partial D i f f e r e n t i a l Equations, Amer. J. Math. 75 (1953), 449 - 476

[Hzl]

Heinz, E., On Certain N o n l i n e a r E l l i p t i c Differential Equations and U n i v a l e n t Mappings, Journ. d'Anal. 5 (1956/57) 1 9 7 272

[Hz2]

Heinz, E., Neue a - p r i o r i A b s c h ~ t z u n g e n f~r den O r t s v e k t o r einer Fl~che p o s i t i v e r G a u B s c h e r Krthnmung durch ihr L i n i e n element, Math. Z. 74, (1960), 129 - 1 5 7

[Hz3]

Heinz, E., E x i s t e n c e Theorems for One - to - O n e Mappings Associated w i t h Elliptic Systems of Second Order, p a r t I, Journ. d'Anal. 15 (1965) 325 - 353

[Hz4]

Heinz, E., E x i s t e n c e Theorems for One - to - O n e Mappings Associated w i t h Elliptic Systems of S e c o n d Order, part II, Journ. d'Anal 17, (1967), 145 - 184

[Hz5]

Heinz, E., ~ b e r das N i c h t v e r s c h w i n d e n der F u n k t i o n a l d e t e r m i nante bei e i n e r Klasse e i n e i n d e u t i g e r Abbildungen, M. Z. 105, (1968), 8 7 - 8 9

[Hz6]

Heinz, E., Zur A b s c h i t z u n g der F u n k t i o n a l d e t e r m i n a n t e bei einer Klasse t o p o l o g i s c h e r Abbildungen, Nachr. Akad. Wiss. G6tt. (1968), 183 - 197

[Hz7]

Heinz, E., U b e r das R a n d v e r h a l t e n q u a s i l i n e a r e r e l l i p t i s c h e r Systeme mit isothermen Parametern, M. Z. 113, (1970), 99 - 1 0 5

[Hill

Hildebrandt, S., On the P l a t e a u P r o b l e m for Surfaces of Constant Mean Curvature, Comm. Pure Appl. Math. 23, (1970), 9 7 114

[Hi2]

Hildebrandt, S., N o n l i n e a r E l l i p t i c Systems and H a r m o n i c Mappings, Proj. B e i j i n g Symp. Diff. Geom. & Diff. Eq. 1980, Science Press, Beijing, 1982, also i n SFB 72, V o r l e s u n g s r e i h e No. 3, Bonn, 1980

H a r m o n i c Maps,

Proc.

Can. J. Math.

A.

19

130

[HJW]

Hildebrandt, S., J. Jost, and K ° - O . Widman, H a r m o n i c Mappings and Minimal Submanifolds, Inv. math. 62 (1980), 2 6 9 - 2 9 8

[HKWI]

Hildebrandt, S., H. Kaul, and K. -0. Widman, H a r m o n i c Mappings into Riemannian Manifolds with N o n - p o s i t i v e S e c t i o n a l Curvature, Math. Scand. 37 (1975), 257 - 263

[HKW2]

Hildebrandt, S., H. Kaul, and K. ,~O. Widman, Dirichlet's Boundary Value P r o b l e m for H a r m o n i c Mappings of R i e m a n n i a n Manifolds, M. Z. 147 (1976), 225 - 2 3 6

[HKW3]

Hildebrandt, S., H. Kaul and K. -O. Widman, An E x i s t e n c e T h e orem for H a r m o n i c Mappings of Riemannian M a n i f o l d s , A o t a Math. 138 (1977) I - 1 6

[HW~]

Hildebrandt, S . , a n d K. -0. Widman, Some R e g u l a r i t y Results for Q u a s i l i n e a r Elliptic Systems of S e c o n d Order, Math. Z. 142 (1975), 6 7 - 86

[HW2 ]

Hildebrandt, S., and K. - O. Widman, On the H61der C o n t i n u i t y of W e a k Solutions of Q u a s i l i n e a r E l l i p t i c Systems of Second Order, Ann. Sc. N. Sup. Pisa IV (1977), 1 4 5 - 1 7 8

[HOS ]

Hoffman, D. A., R. Osserman, and R. Schoen, On the Gauss Map of Complete Surfaces of C o n s t a n t Mean C u r v a t u r e in ~3 and ~4, Comm. Math. Helv. 57 (1982), 519 - 531

[Ho]

Hopf, H., Uber Fl~chen mit e i n e r R e l a t i o n zwischen den Hauptkr~mmungen, Math. Nachr. 4 (1950/51), 232 - 249

[Ih]

Ishihara, T., A M a p p i n g of Riemannian M a n i f o l d s w h i c h Preserves H a r m o n i c Functions, J. Math. Kyoto Univ. 19 (1979), 215 - 2 2 9

[JaK1]

J~ger, W., and H. Kaul, Uniqueness of H a r m o n i c M a p p i n g s and of Solutions of E l l i p t i c E q u a t i o n s on R i e m a n n i a n Manifolds, Math. Ann. 240 (1979), 231 - 250

[JaK2 ]

J~ger, W., and H. Kaul, Uniqueness and S t a b i l i t y of H a r m o n i c Maps and their Jacobi Fields, Man. Math. 28, (1979), 269 - 2 9 1

[J1 ]

Jost, J., E i n e i n d e a t i g k e i t h a r m o n i s c h e r Abbildungen, Diplomarbeit, Bonn, 1979, also Bonner Math. Schr. 129 (1981)

[J2]

Jost, J., Eine g e o m e t r i s c h e B e m e r k u n g zu S~tzen ~ b e r h a r m o n i sche Abbildungen, die ein D i r i c h l e t p r o b l e m l~sen, Man. math. 32, (1980) 51 - 5 7

[J3]

Jost, J., U n i v a l e n c y of H a r m o n i c M a p p i n g s b e t w e e n Surfaces, Journ. Reine Angew. Math. 324, (198]) 141 - 153

[J4]

Jost, J., Ein E x i s t e n z b e w e i s f~r h a r m o n i s c h e Abbildungen, die ein D i r i c h l e t p r o b l e m l~sen, mittels der M e t h o d e des W ~ r m e flusses, Man. math. 34 (1981), 1 7 - 2 5

[J5]

Jost, J., A M a x i m u m P r i n c i p l e for H a r m o n i c M a p p i n g s w h i c h solve a D i r i c h l e t Principle, man math. 38 (1982), 1 2 9 - 1 3 0

[J6]

Jost, J., E x i s t e n c e Proofs for H a r m o n i c M a p p i n g s with the Help of a M a x i m u m Principle, M. Z. 184 (1983), 489 - 496

131

[J7]

Jost, J., The D i r i c h l e t P r o b l e m for H a r m o n i c Maps from a Surface with B o u n d a r y o n t o a 2 - Sphere w i t h N o n - C o n s t a n t Boundary Values, P r e p r i n t

[JKI]

Jost, J., and H. Karcher, G e o m e t r i s c h e M e t h o d e n zur G e w i n n u n g von a - priori - Schranken fur h a r m o n i s c h e Abbildungen, man. math. 40 (1982), 27 - 77

[JK2 ]

Jost, J., and H. Karcher, A l m o s t L i n e a r F u n c t i o n s and a-priori Estimates for H a r m o n i c Maps, Proc. Durham. Conf. 1982

[JM]

Jost, J., and M. Meier, B o u n d a r y R e g u l a r i t y for M i n i m a of Certain Q u a d r a t i c F u n c t i o n a l s , Math. Ann. 262 (1983), 549 - 561

[JS ]

Jost, J., and R. Schoen, On the E x i s t e n c e of H a r m o n i c Diffeom o r p h i s m s Between Surfaces, Inv. math. 66 (1982), 3 5 3 - 359

[JY]

Jost, J., and S. T. Yau, H a r m o n i c M a p p i n g s and K ~ h l e r Manifolds, Math. Ann. 262 (1983), 145 - 1 6 6

[El]

Karcher, H., S c h n i t t o r t und konvexe M e n g e n in v o l l s t ~ n d i g e n R i e m a n n s c h e n M a n n i g f a l t i g k e i t e n , Math. Ann. 177 (1968), 105 121

[K2 ]

Karcher, H., R i e m a n n i a n C e n t e r of Mass and M o l l i f i e r Smoothing, CPAM 30, (1977) 509 - 5 4 1

[KW]

Karcher,H., and J.C. Wood, N o n - E x i s t e n c e Results and Growth Propezties for H a r m o n i c Maps and Forms, SFB-40 Preprint, Bonn, 1983

[K1 ]

Kaul, H., S c h r a n k e n fur die C h r i s t o f f e l s y m b o l e , (1976) 261 - 2 7 3

[Ke ]

Keen, L., Collars on Riemann Surfaces, Ann. Math. (1974), 263 - 2 6 8

[Knl ]

Kneser, H., L ~ s u n g der Aufgabe 41, Jber. (1926), 1 2 3 - 1 2 4

[Kn2 ]

Kneser, H., Die k l e i n s t e B e d e c k u n g s z a h l innerhalb e i n e r Klasse von F l ~ c h e n a b b i l d u n g e n , Math. Ann. 103 (1930), 347 - 358

ELU]

L a d y ~ e n s k a j a , O. A., and N. N. Ural'ceva, Equations aux d~riv~es p a r t i e l l e s de type elliptique, Dunod, Paris 1968

[Lv]

Lavrent'ev, M. A., Sur une classe des r e p r e s e n t a t i o n s continues, Mat. Sb. 42 (1935), 4 0 7 - 4 3 4

[Lz]

Leibniz, Hamburg,

ILl]

G. W., Die Theodizee, 21 968

Phil.

man. math.

Studies

19

79

Dtsch. Math. Ver.

Bibl.

35

71, Felix Meiner,

L e m a i r e , L., A p p l i c a t i o n s h a r m o n i q u e s de surfaces Riemanniennes, J. Diff. Geom. 13 (1978), 51 - 78

[L2]

Lemaire, L., B o u n d a r y Value Problems for H a r m o n i c a n d Minimal Maps of Surfaces into Manifolds, Ann. Sc. Norm. Sup. P i s a (4) 8 (1982), 91 - 1 0 3

[LS]

L e r a F, J., and J. Schauder, T o p o l o g i e et ~ q u a t i o n s nelles, Ann. Rcole Norm. Sup. 51 (1934), 4 5 - 7 8

fonction-

132

[Lw]

Lewy, H., On the E x i s t e n c e of a C l o s e d Surface Given Riemannian Metric, Proc., N.A.S. USA 24, 106

[Li]

L i c h t e n s t e i n , L., Z ur Theorie der k o n f o r m e n Abbildung. Konforme A b b i l d u n g n i c h t a n a l y t i s c h e r s i n g u l a r i t ~ t e n f r e i e r FI~c h e n s t ~ c k e auf ebene Gebiete, Bull. Acad. Sci. Cracovie, CI. Sci. Mat. Nat. A (1916), 192 - 2 1 7

[MS]

Meyers, N., 1055 - 1 0 5 6

[Si]

Misner, H a r m o n i c Maps Rev. D 18 (12) (1978)

[M1]

Morrey, Partial 166

[M2]

Morrey, C. B., The P r o b l e m of P l a t e a u on a R i e m a n n i a n fold, Ann. of Math. 49 (1948), 807 - 851

[M3]

Morrey, C. B., M u l t i p l e Integrals in the Calculus of Variations, Springer, Berlin, Heidelberg, New York, 1966

[Mu]

Mumford, D., A Remark on M a h l e r ' s A.M.S. 28 (1971), 289 - 2 9 4

[Na]

Nash, Math.

[0]

Olivier, R., Die E x i s t e n z g e s c h l o s s e n e r G e o d ~ t i s c h e r auf komp a k t e n M a n n i g f a l t i g k e i t e n , Comm. Math. Helv. 35 (1961), 146 152

[Rd]

Rad6,

[RV]

Rub, E. A., and J. Vilms, Trans. A.M.S. 149 (1970),

[SkU]

Sacks, J. a n d K. Uhlenbeck, The E x i s t e n c e of M i n i m a l sions of 2 - S p h e r e s , Ann. Math. 113 (1981), 1 - 2 4

[Sa]

Sampson, J. H., Some P r o p e r t i e s and A p p l i c a t i o n s of H a r m o n i c Mappings, Ann. Sc. Ec. Sup. 11 (1978), 211 - 228

[SUl]

Schoen, R., a n d K. Uhlenbeck, A r e g u l a r i t y theory nic maps, J. Diff. Geom. 17 (1982), 307 - 3 3 5

[su2]

Schoen, laneous

[SYI]

Schoen, R., and S. T. Yau, On U n i v a l e n t H a r m o n i c Maps Surfaces, Inv. math. 44 (1978), 265 - 2 7 8

[SY2]

Schoen, R., a n d S. T. Yau, E x i s t e n c e o f I n c o m p r e s s i b l e M i n i mal Surfaces and the T o p o l o g y of Three D i m e n s i o n a l M a n i f o l d s w i t h N o n - N e g a t i v e S c a l a r Curvature, Ann. Math. 110 (1979), 127 - 1 4 2

a n d J. Serrin,

H = W, Proc.

as Models

C. B., On the S o l u t i o n s D i f f e r e n t i a l Equations,

J., The E m b e d d i n g 63 (1956), 2 0 - 6 3

T.,

Aufgabe

Problem

41, Jber.

Nat.

for P h y s i c a l

Realizing 2 (1938),

Ac.

Sc.

51

Theories,

a 104 -

(1964)

Phys.

of Q u a s i - L i n e a r E l l i p t i c Trans.~M.S. 43 (1938), 126 -

Compactness

for R i e m a n n i a n

Dtsch.

Math.

The T e n s i o n 569 - 573

Ver°

Mani-

Theorem,

Proc.

Manifolds,

35

(1926),

Ann.

49

F i e l d of the Gauss Map,

Immer-

for h a r m o -

R., and K. Uhlenbeck, B o u n d a r y r e g u l a r i t y a n d m i s c e l results on h a r m o n i c maps, to a p p e a r in J. Diff. Geom. between

133

[SY3]

Schoen, R., and S. T. Yau, C o m p a c t Group Actions and the Topology of M a n i f o l d s w i t h N o n - P o s i t i v e Curvature, Top. 18, (1979), 361 - 380

[Sel ]

Sealey, H., Some P r o p e r t i e s of H a r m o n i c Mappings, Thesis, Univ. Warwick, 1980

[Se2]

Sealey, H., The Stress - E n e r g y H a r m o n i c Forms, P r e p r i n t

[Sh]

Shibata, K., On the Existence of a H a r m o n i c Mapping, O s a k a J. Math. 15 (1963), 173 - 2 1 1

[Si]

Siu, Y. T., The Complex A n a l y t i c i t y of H a r m o n i c Maps and the Strong Rigidity of Compact K ~ h l e r Manifolds, Ann. Math. 112 (1980), 7 3 - 1 1 1

[Sp]

Sperner, E., A priori G r a d i e n t E s t i m a t e s pings, S FB 72, P r e p r i n t 513, Bonn, 1982

[St]

Stampacchia, G., Le p r o b l ~ m e de D i r i c h l e t p o u r les ~quations e l l i p t i q u e s du s e c o n d ordre ~ c o e f f i c i e n t s discontinues, Ann. Inst. F o u r i e r (Grenoble) 15, 189 - 258 (1965)

[Tin]

TeichmHller, 0., E x t r e m a l e q u a s i k o n forme A b b i l d u n g e n und quadratische Differentiale, Abh. Preuss. Akad. Wiss. Math. - N a t . KI. 22 (1939)

[Td]

Tolksdorf, P., A Strong M a x i m u m P r i n c i p l e and R e g u l a r i t y for H a r m o n i c Mappings, P r e p r i n t

[Tr]

Tromba, A., A New P r o o f that T e i c h m ~ l l e r Space is a Cell, Preprint

[U]

[Wt]

Tensor and the V a n i s h i n g of L 2-

for H a r m o n i c Map-

Uhlenbeck, K., H a r m o n i c Maps: A D i r e c t M e t h o d in the Calculus of Variations, Bull. AMS 76 (1970), ]082 - 1 0 8 7 Wente, H., The D i f f e r e n t i a l E q u a t i o n s Ax = 2Hx u ~ x v w i t h V a n i s h i n g B o u n d a r y Values,

Proc. A.M°S.

50

(1975),

131 - 1 3 7

[Wy ]

Weyl, 1955

[Wi ]

Wiegner, M., A priori S c h r a n k e n f~r L ~ s u n g e n g e w i s s e r elliptischer Systeme, Man. math. 18 (1976), 279 - 2 9 7

[Wl ]

Wood, J. C., S i n g u l a r i t i e s of H a r m o n i c Maps and A p p l i c a t i o n s of the Gauss - B o n n e t Formula, Amer. J. Math. 99 (1977), 1329 1344

[W2 ]

Wood, J. C., N o n - e x i s t e n c e of Solutions Problems, Preprint, Leeds, 1981.

H., Die Idee der R i e m a n n s c h e n Fl~che, Teubner, Leipzig,

to Certain D i r i c h l e t