Harmonic Mappings and Minimal Immersion

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

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

Edited by E. Giusti

Springer-Verlag Berlin Heidelberg New York Tokyo

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

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

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

PREFACE

The

international course on Harmonic mappings and Minimal

Immersions was held

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

purpose

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

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

theory

recent years, surfaces, survey

of

and

topology,

of

all

harmonic

maps

has

received

several

important

contributions

in

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

recent

advances,

but

The course was not aimed at giving a complete rather

at

describing

in

detail

some

specific

topics. The lecturers,

S. Hildebrandt,

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

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

an

image

of

a

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

body of the present volume. During

the

course,

there

have

been

two

additional

lectures

by

J.

Sampson,

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

to

express

my

gratitude

to

the lecturers and to all the participants

for their contribution to the success of the course.

E. Giusti

C.I.M.E.

Session

on

"Harmonic

Mappings

and

Minimal

Immersiqns"

List of Participants

G. Arca,

Istituto Matematico

L. Boccardo, Dipartimento 00173 Roma R. Caddeo,

UniversitY,

di Matematica,

Istituto Matematico

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

Via Ospedale 72, 09100 Cagliari 2 ^ Universit~

UniversitY,

Department

Istituto Matematico

S. Hildebrandt, P.A. Ivert,

Mathematisches

Department

UniversitY,

Abteilung fur Mathematik,

T. Karlsson,

Department

Istituto Matematico

Istituto

Matematico

I.H.E.S.,

G.

Modica,

Istituto di Matematica 50139 Firenze

F.

Nicolosi,

K.

Ottarsson,

M.P.

Piu,

Y.S.

Poon,

35 Route

2 ^ Universit~

Ospedale

91440

Applicata,

72,

FacoltA

di

UniversitA,

Viale

A.

Institute,

University

des

Lumi6re,

Institute,

24-29

Sampson,

The Johns Hopkins Piazza dei Cavalieri

University 7, 56100

A.

Sanini,

Dipartimento Abruzzi 24,

M.

Seppala,

Department of Mathematics, SF-O0100 Helsinki

di Matematica, 10129 Torino

L. Sibner,

Polytechnic

Institute

R.

Mathematics

Department, N.Y.

11201,

N.Y.,

Doria

of Warwick, 68093 Giles,

Oxford

and Pisa

Seuola

Normale

OXI

of Helsinki,

College,

Brooklyn, City

95125

S. Maria

Catania CV4

7AL,

U.K.

3LB,

U.K.

Superiore,

University

St.,

Via

France

di Torino,

Jay

France

Coventry

Politecnico

333

Cagliari

6,

Mulhouse,

St.

Brooklyn USA

09100

Ingegneria,

Matematico,

Fr6res

S-581 83 Linkoping

Bures-sur-Yvette,

Trieste

4 rue

i0, Bonn

di Roma, Via O. Raimondo,

34127

Mathematical

S-881 83 Linkoping

Via Ospedale 72, 09100 Cagliari

Via

de Chartres,

i0, Bonn

D-4630 Bochum

UniversitY,

Mathematics

Brooklyn,

Fine Hall, Princeton,

Linkoping University,

UniversitA,

J.

Sibner,

Univ.,

di Matematica,

Seminario

I.S.E.A.,

72, 09100 Cagliari

Ruhr Univ. Bochum,

UniversitY,

Meeks,

Istituto

Princeton

di Matematica,

W.

R. Musina,

Via Ospedale

Linkoping University,

of Mathematics,

P. Marcellini, Dipartimento 00173 Roma P. Matzeu,

Av. 24 de Julho 194 3 ^ , 1300 Lisboa

Institut der Univ. Bonn, Wegelerstrasse

J. Kampmann,

I. Leurini,

72, 09100 Cagliari

Institut der Univ. Bonn, Wegelerstrasse

of Mathematics,

J. Jost, Mathematisches

Via Ospedale

of Mathematics,

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

di Roma, Via O. Raimondo,

Univ.

Corso

Duca

Hallituskatu

N.Y.

i1201,

N.Y.,

degli

15,

USA

3,

V

L. Simon,

Department of Mathematics, Canberra, ACT, Australia

Australian

P. Smith,

Department of Mathematics, Texas Y78~3, USA

Texas A. & M. University,

V. Vespri,

Scuola Normale Superiore,

G. Well, Polytechnic P. Wong,

Piazza dei Cavalieri

University,

P.O. Box 4,

College Station,

7, 56100 Pisa

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

Department of Mathematics, Indiana 46556, USA

R. Ye, Beringstrasse

National

University

of Notre Dame, Notre Dame,

4, D-53 Bonn

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

Lodz University,

ul. Stefana Banaeha 22,

TABLE

S.

HILDEBRANDT,

Harmonic Manifolds

J.

JOST,

J.

H.

L.

SIMON,

Lectures

SAMPSON,

OF

CONTENTS

Mappings

on H a r m o n i c

Harmonic

of

Riemannian

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

Maps

Maps

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

in K ~ h l e r

Isolated

Singularities

Geometric

Variational

of

Geometry

Extrema

Problems

..........

118

193

of

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

206

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

Stefan

I.

The D i r i c h l e t

2.

Harmonic

3.

Liouville

problem

mappings

4.

Estimates

5.

Riemann

6.

Existence

Removable

to m i n i m a l

for Jacobi

normal

7.

A priori

8.

Proof

9.

Miscellaneous

fields

maximum

harmonic

estimates

maps

Bibliography

2

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

11

in ~ n + m

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

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

44

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

62

principle,

and r e g u l a r i t y

69

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

theorem

maps

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

92

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

103

results "'"

10.

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

singularities.

for h a r m o n i c

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

mappings

submanifolds

coordinates

proof,

for w e a k l y

for h a r m o n i c

into s p h e r o i d s

theorems.

Applications

Hildebrandt

. . . . . . . .

"

. . . .

" ° ' "

. . . .

° ' ° ° ' ° ' - . . . - ° - I O 8

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

112

I. T h e

Let N

Dirichlet

X

and

problem

M

be complete

, respectively.

associate

for harmonic

mappings

Riemannian

To every mapping

manifolds

U : X ÷ M

of d i m e n s i o n

of c l a s s

C]

n

and

we can

energy

an

E(~) = fx ~Jd~!i Z' '2dv°l which

is o b t a i n e d

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

e(U)

of

U

with

respect

o f the m e t r i c respect

to

tensor

to the m e t r i c

functional

U :X ÷ ~

= int X

Here of

and

2e(U) M

. The

is s a i d

satisfies

is the t r a c e

under

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

the mapping

functional

E(U)

of

local

u N)

M

we can write

harmonic

to be

the E u l e r

In t e r m s on

X

]

~trx

U

taken with

is c a l l e d

energy

Dirichlet integral.

or

A mapping

X

of

=

= ~IldU ]i2

E(w~)

harmonic

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

for

mapping

there

is

of

all

w

class

no r e g u l a r

6 C

with

C 0 harmonic

14 map

within

We,

however,

Theorem

C

.

have

2.4.

If

the f o l l o w i n g

a ~

I

and

partial

n ~

The T h e o r e m s

2.3 and 2.4 imply a

NONEXISTENCE

RESULT

There

is

w(x)

=

no

regular

(x,O)

for

In o r d e r

7 ,

w : B ÷ S an

map

Theorem

= infcE

-

PROBLEM :

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

to e s t a b l i s h

E(w~)

then

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

result:

that

2. I-2.4,

with a ~

the

I

we shall

boundary

and

n ~

values

7 .

first derive

some a u x i l i -

ary results. Lemma

2.1.

For

L~(B,]R n+1)

any

tangential

along

the

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

Note

~I +w*

= ~I

- = i2 - ~