A survey of minimal surfaces

A Survey of

MINIMAL SURFACES

gyroid. An inflnitely connected periodic minimal surface containing no

straight lines, recently discovered and christened by A. H. Schoen. (NASA

Electronics Research Laboratory)

A Survey of

MINIMAL SURFACES by

ROBERT OSSERMAN Stanford University

DOVER PUBLICATIONS, INC. Mineola, New York

Copyright

Copyright © 1969, 1986 by Robert Osserman Ail rights reserved under Pan American and International Copyright Conventions. Published in Canada by General Publishing Company, Ltd., 895 Don Mills Road, 400-2 Park Centre, Toronto, Ontario M3C 1W3. Published in the United Kingdom by David & Charles, Brunei House, Forcle Close, Newton Abbot, Devon TQ12 4PU.

Bibliographical Note This Dover edition, first published in 1986 and republished in 2002, is a cor­ rected and enlarged republication of the work first published in book form by the Van Nostrand Reinhold Company, New York, 1969. (A fundamentally iden­ tical version had been translated into Russian and published in the journal Uspekhi Matematicheskikh Nauk in 1967.) A new Preface, an Appendix updating the original edition, and a supplementary bibliography have been specially pre­ pared for this edition by the author, who has also made severa! corrections.

Library of Congress Cataloging-in-Publication Data Osserman, Robert A survey of minimal surfaces. Bibliography: p. lncludes index. ISBN 0-486-49514-0 1. Surfaces, Minimal. l. Title. QA644.087 1986 516.3'62 85-1287 1

Manufactured in the United States of America Dover Publications, lnc., 31 East 2nd Street, Mineola, N.Y. l l501

PREFACE TO THE DOVER EDITION The flowering of minimal-surface themy referred to in the Introduc­ tion has contlnued unabated during the subsequent decade and a half. and has borne fruit of dazzling and unexpected varlety. In the past ten years sorne important conjectures in relativity and topology have been settled by surprising uses of minimal surfaces.

In

addition, many new

propertles of minimal surfaces themselves have been uncovered. We shall limit ourselves in this edition to a selected updating obtained by enlarging the bibliography (in "Additional References") and by the addition of "Appendix 3," where a few recent results are outlined. (The reader is directed to relevant sections of Appendix 3 by new footnotes added to the original text. l We shall concentrate on results most closely related to the subjects covered in the main text, with the addition of sorne partlcularly striking new directions or applications. Fortunately we can refer to a number of survey articles and books that have appeared in the interim, including the encyclopedie work of Nitsche !III and, for the approach to minimal surfaces via geometrie measure theory, the Proceedings of the AMS Symposlum-Allard and Almgren [1]; both books include an extensive bibliography. The section "Addltional References," following the original bibliography in the present work. starts with a list of those books and survey articles where many other aspects may be explored. In this Dover edition, a number of typographical errors have been corrected, and incomplete references in the original bibliography have been completed. Otherwise, with minor exceptions, the original text has been left unchanged.

NOTE: In references to the bibliography, Roman numerals refer to the list of books and survey articles in the Additional References (pp.

179-200), while Arabie numerals refer elther to the subsequent list of research papers or to papers ln the original References (pp. 167-178).

MSG and SMS

refer to

Minimal Submanifolds and Geodesies,

proceedings of a conference held ln Tokyo in 1 977, and to

Minimal Submanifolds,

the

Seminar on

a collection of papers presented during the

academie year 1979-1980 at the Institute for Advanced Study. listed as the first and second items in the Additional References. under books and survey articles.

ROBERT ÛSSERMAN

PREFACE TO THE FIRST EDITION

This account is an English version of an article (listed as Item 8 in the References) which appeared originally in Russian. ln the three years that have passed since the original writing there bas been a flurry of activity in this field. Sorne of the most strik­ ing new results have been added to the discussion in Appendix 2 and an attempt bas been made to bring the references up to date. A few modifications have been made in the text where it seemed desirable to amplify

or

clarify the original presentation. Apart from

these changes, the present version may be considered an e xact "translation" of the Russian original.

CONTENTS

Introduction

. . . .. • . . •

. . . . . . . . . . . . . . . , . •

§1.

Parametric surfaces: local theory .

§2.

Non·parametric surfaces

§3.

Surfaces that minimize area

§4.

Isothermal parameters

§5.

Bernstein's theorem

§6.

Parametric surfaces: global theory

. . . . . ,

. . •

• • . . ..

. . .

. . • .

. .. . . . . . . •

. . • . . . . . . .

.

• .

. . . . •

•

.

. .. .... 1

.

• . . . . .

.

.

.

. • . • .

.

.. . . . .

, . .. • . • . •

. • .. . . . . . . . • . . . • .

. . . . . • . . .

3 15 20 27 34

• . . . • . . . .

Generalized minimal surfaces. Complete surfaces 43 §7.

Minimal surfaces with boundary Plateau problem. Dirichlet problem

§8. §9.

. . . .

Parametric surfaces in E3• The Gauss map.

. . . •

• .

53

• • •

.. .. .

63

Surfaces in E3• Gauss curvature and total curvature 75 ..

§10. Non·parametric surfaces in E3

Removable sin gularities. Dirichlet problem

........

91

§11. Application of parametric methods to non·parametric

problems. Heinz' inequality. Exterior Dirichlet problem

105 n §12. Parametric surfaces in E : generalized Gauss map. 117 . .

.

. . .

.

.

• . . . .

Appendix 1. List of theorems .

.

.

.

.

.

.

. • . . .

. . . .

.

.

.

.

.

.

.

.

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

129

Appendix 2. Generalizations

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

135

AppendiX 3. Developments in minimal surfaces, 1970-1985

143

References

167

..

... . .

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

Addi Uonal References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Author Index Subject Index

. . . . . .

.

.

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

, . . . . . . . . . . . . . . . . . . . . . 201

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

The complete embedded minimal surface of Costa-Hoffman-Meeks (see Appendlx 3, Section 4). (Photograph copyright© 1985 by D. Hoffman and J. T. Hoffman.)

A Survey of

MINIMAL SURFACES

INTRODUCTION

The theory of min imal surfaces experienced a rapid develop­ ment throughout the whole of the nineteenth century. The major achievement s of this period are presented in deta il in the books of Darboux [1] and Bianchi [1]. During the first half of the pre sent century, attention was d irected almost exclusivel y to the solution of the Plateau problem. The bulk of result s obtained may be found in the papers of Douglas [1, 2], and in the book s of Rad6 [3] and Courant [2]. A major exception to this trend is the work of Bern ­ ste in [1, 2, 3] who con sidered m inimal surface s chiefly from the point of v iew of partial d ifferent iai equations. The last twenty years has seen an extraord inary flowering of the theory, partly in the d irection of general izations: to h igher dimen sion s, to Rieman­ nian spaces, to wider classe s of surfaces; and partly in the d irec t ion of many new results in the classical case. Our purpose in the present paper is to report on sorne of the major developments of the past twenty years. In order to give any sort of cohesive pre sentation it is necessar y to adopt sorne basic point of v iew. Our a im will be to present a s much a s possible of the theory for two -dimensional m in imal surfaces in a euc l idean space of arbitrary dimen sion, and then to restrict to three d imen­ sion s only in tho se case s where corresponding resu lt s do not seem to be available. For a more detailed account of recent results in the three-dimensional case, we refer to the expository article of 1

2

A SURVEY OF MINIMAL SURFACES

Nitsche [4], where one may also find an extensive bibliography and a list of open questions. The early history of minimal sur­ faces in higher dimensions is described in Struik [ 1]. Since it is impossible to achieve anything approaching com­ pleteness in a surve y of this kind, we bave selected a number of results which seem to be both interesting and representative, and whose proofs should provide a good picture of sorne of the methods which have proved most useful in this theory. For the convenience of the reader, a list of the theorems proved in the paper is given in Appendix 1.

For the most part the present paper will contain only an organ­ ized account of known results. There are a few places in which new results are given; in particular, we refer to the treatment of

non -parametric surfaces in E n in Sections 2-5, and the discus­

sion of the exterior Dirichlet problem for the minimal surface equa­ tion in Section 11.

ln Appendix 2 we try to give sorne idea of the various general ­ izations of this theory which have been obtained in recent years. One word concerning the presentation. ln most treatments of differentiai geometry one finds either the classical theory of sur­ faces in tbree-space, or else the modern theory of differentiable manifolds. Since the principal results of this paper do not require any knowledge of the latter, we bave decided to give a careful in­

troduction to the general theory of surfaces in En . For similar rea­

sons we have încluded a section on the simplest case of Plateau's problem in En , for a single Jordan curve. In this way we hope to

provide a route which may take a reader with no prevîous knowledge of the theory, directly to sorne of the problems and results of cur­ rent research.

3

PARAMETRIC SURFACES: LOCAL THEORY

§1.

Parametric surfaces: local theory We shaH denote by x = (x 1 , ..., xn)

E". Let D be a domain in the u-plane,

fine provisionally a tion

x(u)

surface in

a point in euclidean n-spac

u (u 1 , u2).

We shall de­

E" to be a differentiable transforma-
En. At

each point of

D

the following conditions are equivalent:

àx _ are independent, (1.3) the vectors .È.!!._, àu àu 2 1 (1.4) the ]acobian matrix M has rank 2,

( 1. 7) det G

>

0.

Proof: Formula (1.2), combined with elementary properties of the rank of a matrix , gives the equivalence. t DEFINITION. A surface S is regular at a point if the condi­ tions of Lem m a 1.1 hold at that point; S is regular if it is regular at every point of D. We shall write Sc

cr if x(u) c cr in

D;

i . e., each coordinate

xk is an r-times continuously differentiable function of ul' u2 in D.

5

PARA METRIC SURFA CES: L OCAL THEORY

We shall assume throughout that S ( cr, r :?_ L

Suppose that S is a surface x(u) f cr in D, and that u(ü) ( cr

is a diffeomorphism of a domain D onto D. We shall say that the �

�

surface S defined b y x(u(ü)) in Dis obtained from S by a change of parameter.

We say that a property of S is independent of param­

eters if it holds at corresponding points of ail surfaces S obtained

from S by a change of parameter. It is the object of differentiai geometry to study precisely those properties which are independent of parameters. Let us give sorne examples. We note first that if the J acobian m atrix of the transformation

u(ü) i s u then the fact that u(ii) i s a diffeomorphism implies that

a(ul' u2) a(ü1, a ) 2

det U f.

0 in

D .

Furthermore, by the chain rule, it follows from S ( cr and u(ü)

cr that s

(

cr, so that the property of belonging to cr is indepen­

dent of (Cr changes of) parameters. ln particular,

2

axi =

aük or

ax.1 au.

__l !. au. aük j= 1 1 -

M

MU,

whence

(1.8) and

(

-

G = UTGU

6

(

A SURVEY OF MINIMAL SURFACES

(1.9)

d et G

2 det G(det U)

=

=

det G

a(u , u ) 1 2

a(ü

1,

)

2

ü2)

An immediate consequence of this equation, in view of (1. 7) is that the property of S being regular at a point is indep end ent of param­ eters. Suppose now that L\ is a subdomain of D such that � where � is the closure of .!\. face x(u) to

u f

.!\.

C

D,

Let � b e the restric tion of the sur­

We define the area of � to be

A(�)

(1.10)

If u(ü) is a change of p arameter, and L\ maps onto ,!\, th en the cor­ responding surface � has area

using (1.9) and the rule for change of variabl e in a double integral. Thus the area of a surface is independent of parameters. We n e:xt note a special choice of parameters which is often use­ fui to consider. Let i and j denote any two fixed d istinct integers from 1 to

n,

and let D be a domain in the

tion s

x., 1

x. J

plan e. The equa-

(1.11) define a surface S in E".

A surface defined in this way will be

PARAMETRIC SURFACES: LOCAL THEORY

said to be given in nonpar ametric or explicit form. This is, of course, a special case of the surfaces we have b een c onsidering up to n ow, the parameters b eing chosen to be two of the coordi­ n ates in En. In other words, we may rewrite ( 1. 1 1) in the form (1. 12)

ul' xj= u2, xk=fk(u1 ,u2 ) , k�i, j.

xi

In the classical case n= 3 we have a sin gl e func tion f k, and the surface is d efined by expressing one of the coordinates as a fun c .

tion of th e other two. In ord er for a surface to b e expressible in n on-par ametric form, it is of c ourse nec essary {or the p rojection map (1. 13) when restric ted to the surface, to be one-to-on e. This is n ot true in general for the whole surface, but we h ave the following impor­ tan t lemma. LEM MA 1 .2. Let S be a surface x (u), and let u

a be a point

at which S is re�u lar. Then there exÎsts a nei�hborhood 1.\ of a,

\

such that the surface � obtained by restricting x (u) to 1.\ has a reparametrization � in non-par ametric form. Prpof: By cond ition ( 1 .5) for regularity, and using the inverse mapping theorem, we deduce that there exists a n eighborhood 1.\ of a in which the map (u 1, u2 ) more, if x (u)

t

_,

( xi, xi) is a d iffeomorphism. Further

cr, th e inverse map (xi, xj)

and the same is true of the c omposed map (1.14) which defines �. •

_,

(ul, u 2) is also cr,

1

l 1

8

A SURVEY OF' MINIMAL SURFACES

Thu s, when studying the local behavior of a surface, we may, whenever it is convenient, assume that the surface is in non-para­ metric form. Let us note also that the reparametriz ation (1.14) shows that in a neigh borhood of a reguler point the mapping x(u) is always one-to- one. In order to study more c losely the behavior of a surface near a given point, we con sider the totality of curve s passing through the point and lying on the surface. F irst of ali, by a curve C in E" we shall mean a continuously d ifferent iable map (1. 15) where [a,,B] is sorne interval on the real l ine. We shall also use the notation ( 1 . 1 6) The tangent vec tor to the curve at a point t0 is the vector (1.17) The curve is regular at t 0 if x '(t0) .;,. O. Suppose now that we have a surface S defined by

x

(u), u

and a curve C defined by (1.15). We shall say that C lies

on


( t0) C we h ave the formula

=

b. For the tangent vector to the curve

(1.18) where ax!aul and ax;au2

are e valuated at u

=

a.

LEMMA 1 . 3. A t a regular point of a surface S, if we consider

the set of ail curves which lie on S and pass through the point, then the totality of their tangent vectors at the point form a two·dimen· sional vector space. Proof: Since we can obviously find curves u (t) in D such that u ( t0)

=

a, and u ! ( t0), u2 ( t0) take on arbitrarily assigned values,

it follows from (1. 18) that the set of tangent vectors x' ( t0) consists of all l inear combin at ions of the two vectors ax;aul and ax;au2 . But b y condit ion (1.3) for regularity these vectors are independent and therefore span a two-dimensional space. • DEFINITION. The vector space described in Lemma 1 .3 is

called the tangent plane to the surface S at the point b

x (a),

and is denoted by n or n (a). Thus a surface S bas at every regular point a tangent plane, which by its defin it ion is independent of parameters. For the length of a tangent vector we have from (1. 1) and (1. 1 8):

10

A SURVEY OF MINIMAL SURFACES

(1. 19)

!x'(t0)!2

=

x'(t0) x'(t0) •

=

2 �

i, j=

1

d.;1 u;(t0)uj(t0).

Thus, the square of the length is expressed by a quadratic form in the corresponding tangent vector u'(t0), with matrix G ; this quad ­ ratic form is often referred to as the first fundamental form of the surface S. We have seen in (1.10) that the determinant of th1s form defi nes areas on the surface . Similarl y, the length of curves on the surface are obtained from (1.19), since the length of the curve x(t),

a $ t $ f3 in E n is given by L

(1.20)

J

13

1x'(t)1 dt

a

It is convenient t o associate with an arb itrary curve C of the form (1.16) the quantity

s(to)

(1.21)

=

r

to

lx'(t)ldt

'l

Since s'(t0)

1 x '(t0)1 � 0 for a $ t0 $ f3, we have a monotone

mapping

s(t ): [a ,/3]

(1.22)

-+

[O,L ] .

If furthermore, the curve C is regular, then

s '(t)

1 x '(t)1

>

0, and

the map (1.22) bas a differentia ble inverse t (s). The composed map (1.23)

-

�s): [0, L]

«s)

-

[a, {3]

�(0

En

defines a curve C which is called the parametrization of C with re­

spect to arclength. We h ave at each point the unit tangent vector

11

PARAMETRIC S URFACES: LOCAL THEORY

T

(1.24)

dx ds

lx'(t)l s'(t)

x'(t) s'(t)

1 .

We wish next t o stud y "sec ond order effects," and we shall as­ sume from now on that all curves cons idered are reguler c2 curves . We may then introduce the parametrization (1.23) w ith respect to arc length, and we define at each point the curvature vector

dT ds

(1.25)

as the derivative of the unit tangent with respect to arc length. We use the same notation as in the paragraph preceding Lemm� 1. 3, but we now add the assumption that the surface S f c2 , and we re­ strict the class of curves passing through the regul er point b

=

x (a)

on S to regul er c2 -curves l ying on S. We shall seek to describe the totality of curvature vectors to these curves evaluated at the point b

=

x (a). More precisely, if II is the tangent plane to S at

this point, let us denote by II 1 its orthogonal complement, an (n-2)­

dimensional space called the normal space to S at the point. Each

vector is determined b y its projections in II and II1. F or our pres­

ent purposes we shall examine the projection of the curvature vec­ tor into n1.

An arbitrary vector

N

f

II

1

is called a normal to S. Since such

a vector is in particular orthogonal to dx/du1, dx/du , we may

2

compute as follows:

dx ds (1.26)

�x ds2

-

·

� N = _..

b

. .

IJ

(N)

dui du. -

ds

1

_

ds

12

A SURVEY OF MINIMAL SURFACES

!

where we have introduced the notation

1

(1.27)

i

he vector a2 x!au1au being evaluated at the point i oting that

nd that

form

b

du./d s 1

=

N ds2 .

=

By

we may rewrite (1.26) in the

(du./dt)/(ds/dt), 1 �x

.28)

u =a.

l:.h;/N)u; ( t0)uj{t 0) l:.giiu ; (t o) u j (to)

_

The numerator on the right band side is a quadratic form in the tan­ gent vector

u '(t0)

whose matrix

surface and the normal of S with respect to

N.

N.

b. 1}

depends on the point of the

.(N)

It is called the second

fundamental form

We note that the entire right-hand side of

(1.28) depends on the particular curve C only to the extent of the

1tangent vector to C at the point. In fact, the homogeneity of the

i

right-hand side in the components of

u '(t0)

shows that it depends

ionly on the direction of the tangent vector: i.e., on the unit tangent 1 T. We may therefore rewrite (1.28) in the form i(l.29)

�x

N ds2 .

k ( N, T), N ll� (

Tt

ll,

where the nght-hand side is at f'ach point of Sa well-defined function of the normal

N

and the unit tangent

T, called the normal cur­

ature of S in the direction T with respect to the normal ix N, and let T vary, we obtain the quantities

N.

If we

13

PARAMETRIC SURFACES: L OCAL THEORY

(1.30)

k 1 (N)

=

max T

called the principal the normal

N.

k (N, T),

curvatures of

k 2( N)

min T

k (N, T)

S at the point, with respect to

Finally we introduce the average value

(1.31)

called the mean normal

N.

curvature of

S at the point, with respect to the

To obtain an explicit expression for

H(N) ,

we note that since

the right-hand side of (1. 28) is the quotient of quadratic forms, its maximum and minimum, which we have denoted by the roots of the equation

k 1 (N}, k 2( N},

are

0 .

(1.32)

Expanded, this equation takes the form

For the sum of the roots , we therefore have (1.33)

( }

HN

82 2 b1 1(N} gllb22 (N} - 2g1 2 b 1 2 (N} 2 det (gij)

+ = _;;;:_:_:...=------...:.. =-=--.=----=-=---= .: -

bI.J.(N) N for N ll�

It follows immediately from the definition (1.27) that the are linear in

N,

and from (1 .33) that

Thus there exists a unique vector

H(N} =

(1.34) H

H

·

N

H(N)

Hf

is linear in 1 n such that

for ali

N

f

f

111 .

thus defined is called the mean curvature vector of 1 s at the point. If e l, ... , e -2 is any orthonormal basis of n ' it n follows from ( 1 . 34) that the mean curvature vector H may be The vector

114

A SURVEY OF MINIMAL SURFACES

expressed as

p.35) ��

H =

n-2

!

k 1

[H(ek)] ek

•

DEFINITION. A surface S is a minimal surface if its mean

rurvature vector H

vanishes at every point.

The reason for this terminology will become apparent in Sec-

3. For the moment we note merely that by virtue of (1.34) and �1.35), H 0 if and only if H(N) 0 for aU N fi� Thus, using �1.33 ), minimal surfaces are characterized in terms of their first tion

=

=

e

and second fundamental forms by the equation

1

o.

15

NON·PARAMETRIC SURFACES

§2. Non-parametric surfaces. We consider in this section surfaces in the non-parametric form (1.11). By relabeling the coordinates in E n we may assume that the surface is defined by k

(2.1)

=

3, .

..

,n ;

or equivalently

Th en

and (2.4)

We note that the vectors ax !au1, ax!au2 are obviously indepen­ dent, so that every surface in non -parametric form is automatically regular.

We again denote by ll the tangent plane, and by lll the normal

space. 2.1. Let N3, ... , N

be arbitrary. Then there ex ist n ( Nl' ... , N ) is in nl. unique Nl' N2 such that the vector N n LEMMA

i

=

Proof: The vector N is in nl if and only if N . ax ;au.1 = 0,

�

1,2. By (2.3) we have N1 = -� =3Nk (a!J!au1), i

=

1,2. +.

16

A SURVEY O F MINIMAL SURFACES

We give an application of this lemma to arbitrary surfaces.

Let x (u) define a surface S Cr, let b x (a) be a regular point of S, and let N be a normal to S at this point. Then there exists a neighborhood A of a, and N(u) cr-l in A such that N(u) IT1(u) and N(a) N. Proof: By Lemma 1 . 2 we may find a neighborhood A of a in LEMMA 2.2.

f

=

f

f

=

which S has a reparametrization in the form (2. 1) (assuming suit­ able labeling of the coordinates in E n ). Let

set

N (N 1, .. . , Nn ), ==

and

1, 2. Then

N(u)

has the desired properties. t

For the remainder of this section we consider surfaces S

f

c2•

From (2.3) we deduce (2.5)

2

a x --a ia i

u u

Thus, for an arbitrary normal (2. 6)

b1}.. (N)

N (Nl' . . . , Nn ), =

we have

a2Jk I Nk _ k 3 a u i a uj n

=

__

The equation (1. 36) for a m inimal surface therefore takes the form

0,

17

N O N- PARAMETRIC SURFACES

for all normal vectors ponents

N 3, . .. , N

n

Since, as we observed earlier, the com ­

may be chosen arbitrarily, it follows that each

of the coefficients of tain th at

N. Nk

must vanish, for

k

=

3,

... , n.

We thus ob­

n - 2 equations for the n- 2 functions 13, ln. If we recall u 1 x l' u2 x 2 , and if we introduce the vector notation • • •,

=

=

(2.7)

then these equations may be written as a single vector equation

This is the minimal

surface equation for non-parametric minimal

surfaces in En . Every reguler minimal surface provides lot;;: al so­

lutions of this equation, by Lemma 1.2. We shall use this fact later on to aid in the local study of minimal surfaces. For the present, let us observe that Equation (2.8) allows us to find explicitly a number of specifie exemples of minimal surfaces. Sorne of these surfaces turn out to be very useful in the general theory because of certain extrema! properties which they possess.

3. Then

f(x 1 , x 2 ) f s<x1, x2 ) and (2. 8) reduces to a single equation for the scalar function f(x l' x2 ), First let us consider the case

n

We have the following classical surfaces.

The helicoid: (2. 9)

=

18

A SV RVEY OF MINIMAL SVRFACES

One can show that this is the only solution of (2.8) which is also a harmonie function, and that the helicoid is the only ruled minimal surface. (For a more detailed discussion of all these surfaces we re fer the reader to Darboux [ 11.)

The catenoid: cosh-1

(2. 10)

r,

r

)x/ + xi

or (2. 1 1)

This is the only minimal surface which is also a surface of rotation.

Scherk's su face : r

cos x 2 log--=­ cos x 1

(2. 12)

This is the only minimal surface of translation; that is (2. 1 2) is jhe only solution of (2.8) of the form

f(x l' x 2)

g (x 1 ) + h (x2 ).

There are two remarks we should make about these solutions. First of ali, the image of a minimal surface under a similarity trans­ formation is again a minimal surface, so that one can obtain other solutions of (2. 8) trivially by this method. Second of all, we have not specified the domain of definition of the above solutions, but we may note that none of them is defined for ali

x l' x2 .

This turns

out not to be an accident, s ince the theorem of Bernstein which we shall prove later on (Section 5) states that for

n

=

3 there are no

non-trivial solutions of Equation (2.8) valid in the whole plane. We turn now to the case of arbitrary each

fk

is linear in

x l' x2,

n.

xl' x2-

We note first that if

then (2 .8) is satisfied trivially. In this

NON•PA RAMETRIC SURFACES

S

case the surface

19

is a plane.

n i s even we have the following special solutions. :j:.,et z x 1 + ix 2, and let g1(z) , , g m (z) be complex analytic functions of z, where n 2m + 2. Then settin g If

=

. .•

k 2p + 1 k 2p-+ 2 for

p

==

l,

.

. , m, .

we obtain by direct verification a solution of (2.8).

the graph of a complex·analyti curve, considered as a surface in real euclidean space, is always a minimal surface. For the case of a single function g 1(z) , the We may word this result as follows:

corresponding minimal surfaces in E4 were studied in great detail

by Kommerell [1].

A SURVEY OF MINIMAL SURFACES

§3. Surfaces that minimize area. We shall now discuss briefly the problem which led historically 1to the theory of minimal surfaces. Namely, that of characterizing those surfaces which have !east area among all surfaces with the 1 same boundary. Specifically, we shall consider the following situ•ation. Let S be a regular surface defined by x (u)

l

c2 in a domain D.

Let r be a closed curve in D which bounds a subdomain �. and

let :E be the surface defined by x ( u) restricted to �. Suppose th at •the area of I is less than or equa l to the area of every surface I ldefined by

x(u)

u on x (u) ?

in� such t hat for

does this imply about the surface

r,

x(u)

==

x ( u)

.

What

We shall apply the standard methods of the calculus of variations in two different forms. First we s hall make normal variations of the surface, and later we sh aH consider non -p arametric surfaces and .variations perpendicular to the x l' x 2 -plane. To start with, let us suppose that N(u) l c1 in D, such that

W(u)

is normal to S at x (u). That is

N(u)

(3.1)

·

ax a ui

_

0,

Î

1, 2.

Differentiating this equation yields (3. 2)

[

We now consider an arbitrary function h (u) •ach real number À we form the surface SÀ : e find

R

(u)

==

x (u)

+

l

Àh (u) N(u),

c2 in D, and for

u

l

D.

SURFACES THAT MJNJMIZE AREA

ax. au i

=

Èl!:_ au i

+

A rh aaN l u

i

+

l J

È1!_ N au i

21

,

and usin g (3.1), (3.2)

g . l] .

=

ax . ax aÛ. au. l 1

=

g

l]

. .

- 2Ahb

l]

. .

(N)

+

2 A c

IJ .

.

where c l].. is a continuous function of u in D. It follows that

(3.3) where

and a2 is a continuous function of ul' u2, À for u in D. As a first consequence of this formula, using the fact that S is regular we deduce that a0 bas a positive m inimum on "3., and since a 1 and a are continuous in D, there exists g > 0 such that 2 < €., the 0 < €. and u € �. In other words, for for det

�ii>

lAI

lAI

surfaces I,À defined by restricting x(u) to � are all regular sur ­ faces. To compute their area A(A) A(l: À), we note that in view of (3.3), we have =

(3.5) for u in �. where M is a positive constant. By (1.10) and (3.4) we have

and integratin g (3.5)

22

A SU RVEY OF MINIMAL SURFACES

or

A( À) - A( O) À

1 Letting

a0, al'

À

-f�i�a0 du1 du2 1 2 ·vr-::-

< M1

À

•

tend to zero, substituting in the expressions (3.4) for

and recalling the formula (1. 33) for the mean curvature, we

arrive at the expression

A' ( O)

(3 .6)

-2

// H(N) h (uh/det�?ijdu1 du2 �

for the rate of change of area as a function of À.

f ( u) is an arbitrary continuous u in L\ we may define the integral of f with respect to surface area on I, as We may note in passing that if

function of

J1 f (u)dA Jf f (uh/det gijdu 1 du2 �

(3.7)

k

With this notation, if we choose our family of surfaces ting

h ( u)

=

l, th en formula (3. 6) reduces to

A '(0)

-2

•

SÀ

by set­

1! H(N)dA k

which provides an interesting interpretation of the quantity

H(N).

We now return to our original problem, and we make the follow ­

in order for I. to minimize area, its mean curvature must be identically zero. This follows immed iately from (3. 6) using ing a ssertion:

the standard argument of the calculus of variations. Namely, if the

23

SURFACES THAT MINIMIZE AREA

mean curvature did not vani sh identically, then there would be a point

u "' a

in 6. and a n ormal

H(N) > O. a, and N(u) C 1

may assume

N "' N(a)

H(N) fo

such that

O. We

By Lemma 2 . 2 we can find a neighborhood

V1 of in V 1, such that N(u) is normal to S at x(u). Then we will have H(N) > 0 throughout a neighborhood V2 , a ( v2 c vl ' and if we choose the function h (u) so that h(a) > 0, h(u) 2: 0 for all u, and h (u) 0 for u' v2, the i ntegral on the ri ght-hand side of (3.6) will be strictly positi ve. However, if V2 x(u) on r, so that i s sm all enough so that v2 c 6., then i(u) l

=

=

�À w ill be a surface with the same bound ary as�. The assump­ tion that � minimizes area implies that A(À) 2: A(O) for all À,

whence A '(0)

O. Thus we would obtain a contradiction to (3.6),

and the assertion is proved. Thus minimal surfaces arose originally in connection with mini­ mizing area, and it is from this connection that they derived their name. However, as we shall see, they also arise naturally in a num­ ber of other connections, and many of their most important propertîes are totally unrelated to questions of area. Before leav ing the sub­ ject we shall use the property of mini mizing area to derive severa! other forms of the minimal surface equation. We start from a surface in non-parametric form:

3, . , n ,

k

..

and introduce the vector notation (3 .8)

q

r

24

A SURVEY OF' MINIMAL SUR FACES

!Then the minimal surface equation (2.8) may be written as

1!(3.9)

0,

or as

(3. 10)

IThe equations (2.4) take the form (3. 1 1) whence

(3. 12) One often u se s the notation W

(3. 13)

ydet g ii

or non -parametric surfaces. Suppose now that we make a variat ion in our surface, setting k

=

3,

,

. . .

l where

n,

À is a real number, and hk c c1 in the domain of defin ition 'v of the fk" In vector notation, setting h (h3, , hn) we have • • •

Ê

=

f + Àh,

p

W2 where

+

2ÀX

+

À2 Y ,

25

SURFACES THAT MINIMIZE AREA

and Y is cont inuons in xl' x2• It follows that

where

z

is again continuo ns.

We now consider a closed curve ï in the domain o f definition

of f(x1, x2), and let ô. be the region bounded by ï. I f the surface xk f(x1, x2) over ô. min im izes the area among all surfaces with �

the same boundary, then for every choice o f h such that

ï, we have

h

0 on

whic� is only possib le if

Substituting in the above expression for X , integrating by parts , and using the fact th at h

0 on ï, we find

B y the s ame reasoning as above, it follows that the equation (3. 14)

0

must hold everywhere. Once we have found this equat ion, it is easy to verify that it is a consequence o f the minimal surface equation (3.1 0). In fact the le ft -hand side of (3. 14) may be written as the sum o f three terms:

A SURVEY OF MINIMAL SURFACES

The first term vanishes by (3. 9). If we expand out the coefficient of

p

in the second term, we find the exp ression

(3. 15)

=

_!_

w3

[(p q)q -( 1 + 1 ql 2 )p] ·

•

[(l + 1 ql

2 )r - p q)s 1 2 )t 2( + ( + IPI ]

which vanishes by (3. 10). Interchanging see that the coefficient of

q

•

p

and

q, x 1

and

x2 ,

we

in the th ird term vanishes a lso, thus

roving (3. 14). ln the process we have also shown that the two equations

b . 16) are satisfied by every solution of the minimal surface equation

•

3 . 10). These equat ions have long been known in the case

n

=

3,

and the fact that they are in divergence form al lows one to derive many consequences which are not immediate from (3 . 10). (See, for example, Rad6 [3].) We shall see tha t Equations (3. 1 6) have equal ­ y important consequences in the case o f arbitrary

n.

27

ISOTHERMAL PARAMETERS

§4. lsothermal parameters. When studying properties of a surface which are independent of choice of parameters, it is convenient to choose parameters in such a way that geometrie properties of the surface are reflected in the parameter plane. As an ex ample one can ask that the mapping of the parameter plane onto the surface be conformai, so that angles between curves on the surface are equal to the angles between the corresponding curves in the parameter plane. Analytically, thi s con­ dit ion is expressed in terms of the first fundamental form ( 1 . 1) by

0

(4. 1) or

À(u)

(4.2 ) Parameters

u1 , u2

>

0 .

satisfying these conditions are called

isothermal

parameters. Many of the basic quantities considered in su rface theory s impl i f y considerabl y when referred to isot herm al parameters. For exam­ ple, from

(4.3)

(4. 2) we have clet g 1]

. .

=

.\4

and the formula (1.33) for mean curvature becomes

(4.4) We also have the following useful formula for the Laplac ian of the coordinate vector of an arbitrary surface.

LEMMA 4.1. Let a regular surface .S be defined by (u) C2 where u l' u 2 are isothermal parameters. Then x


k ( /;) : a) cf>k( /;)

is analytic ir. 1;


xk

is harmonie in

u 1 , u2 ;

30

A SURVEY OF MINIMAL SURFACES

b)

ul' u2

are isothermal parameters

(4.9) c)

if

u1, u2

are isothermal parameters, then S is regular
gii À oii < > (À/À)U an orthogonal matrix < > u(ü) is either conformai or anti-

(�.8),

=

conformai. •

34

A SURVEY OF MINIMAL SURFACES

§S. Bernstein' s Theorem. In this section we shall prove several results related to Bern­ stein's Theorem. Although this theorem is of a glo bal, rather than a local nature, we include it here before our general discussion of surfaces in the large, for two reasons. First, because the proof requires a purely elementary argument, and second, because Bern­ stein's Theorem provides the motivation for a number of the results that we discuss later on. We begin with sorne element ary lemmas.

5.1. Let E(x l, x 2) ( c 2 in a convex domain D, and suppose that the H es sian matri x LEMMA

( a.�;;J)

is positive definite. Define a mapping

(5.1) Then if x and y are two distinct points of

D,

and if u and v are

their respective image points under the map (5.1), the vectors y -x and v-u satisfy the equation (v-u).(y-x) > 0 .

(5. 2)

Proof:

Let

G(t) = E(ty + (1-t )x), 0 .::; t.::; l.

G '(tl =

and

1

�

1

[/� x

(ty + 0- t)x)

}

Then

y1-x1) ,

35

BERNSTEIN'S THEOREM

2

�

G "(t)

i,j = l >

[ a2E

� cr -

ux.ox. 1

1

(ty+ 0- t)x)l (y.-x .)(y.-x.) J 1 1 J J

0 for 0 < t < 1 .

Hence G'(l)>G'(O), or �v.(y.-x.)>�u.(y.-x.), which is 1 1 1 1 1 1 (5.2).

•

LEMMA 5. 2. (Lewy [ 1]). Under the hypotheses of Lemma 5.1, if we define the map

1

(5.3)

where u / x1, x2) is defined by (5.1), then for any two distinct points

x and y of D, their images

1;

and 11 satisfy

(5.4) Proof: Since 11

from (5. 2).

1; =

(y- x)+ (v-u), this follows immediately

+

COROLLARY. Under the same hypotheses, we have

\17-/;1

(5.5)

>

Ir- xl

.

Proof: By the Cauchy-Schwarz inequality,

which applied to (5.4) yields (5.5).

+

LEMMA 5.3. ln the notation of the previous lemmas, if the disk xf + x

f

2 < R ,

D zs

then the map (�.3) is a diffeomorphism of

D onto a domain � which includes a disk of radius

R about

/;(0).

36

A SURVEY OF MINIMAL SURFA CES

Proof: The m ap (5 .3) is continuously differentiable, since

E( x 1 , x 2 ) < C 2 . If x ( t ) is any differentiable curve in D, and ÇU ) i ts image, then it follows from (5. 5) that I Ç'( t ) l

>

l x '( t ) l , and

bence the Jacobien is everywhere greater than 1. Th us the m ap is a local diffeomorphism, and by (5 .5) it is one-to-on e, bence a glo­ b al diffeomorphism onto a domain !'!. We must show that !'! in­ eludes all points Ç such th at 1 Ç- Ç( 0) 1




1 x ( k ) l by (5 . 5), it

follows that 1 Ç- Ç{O)I 2: R, which proves the lemma.

+

/{ x 1 , x 2) be a solution of the minimal sur­ face equation (3 . 1 0) for x + x < R 2 . Th en using the notation (3.8), (4. 12), th e map {x ' x 2 ) ... (Ç1 , Ç2 ) delined by (4. 13) is a dif­ l LEM M A 5 . 4 . Let

f

;

feomotphism onto a domain !'! which includes a disk of radius

R

about the point Ç(O).

Proof: It follows from e quations (4. 1 2) that there exists a func­ tion

E{x l ' x 2) in x

+

x

i

àE àx 1

( 5 . 6) Then

f

E(x 1 , x 2 )

iJ 2 E à x 12

1




0'

det

a 2E àx1 à x2

à(F, G) à( x 1 , x 2)

1

37

BER NSTEI N•$ THEOREM

by (3 . 12), (3 . 13) and (4. 12). Thus the function E(x 1, x ) b as a po­ 2 sitive-definite Hessi an m atrix and we may apply Lemmas 5 . 1-5. 3 to it. But by ( 5 . 6), the map (4. 1 3) is just the map (S . 3) applied to this function. Th us Lem m a S . 4 follows immediately from Lemma •

S .3 .

LEMM A s . s . L et f{x r x 2)

t

c 1 in a domain D, where f is

real-valued. N ecessary and sufficient that the surface S: x3 f{x 1 , x 2 ) lie on a plane is that there exist a nonsingular linear

=

transformation { u 1 , u 2 ) parameters on S.

""*

{x 1 , x 2) such tha t u 1 , u 2 are isothermal

Proof: Suppose such parameters u 1, u 2 exist. functions cp (() by (4 .6), k 1, 2, 3, we see that k constant sin ce x and x 2 are linear functions of 1 ( 4. 9), cp 3 must also be constant. Thi s mean s th at

Introducing the

c/> 1 and cp 2 are u 1 , u 2 . But by x3 bas const ant'

gradient wîth respect to u 1 , u , bence also with respect to x 1 , x . 2 2 Thus H x 1 , x ) Ax C. Conversely, if f i s of this form + B x + 1 2

2

=

1

it is easy to write down an explicit l inear transformation yielding isothermal coordin ates; for example, x 1 À Au 1 + Bu , x 2 2 /..B u 1 - Au2 , where t..2 1/0 + A 2 + 8 2 ) . t

=

=

TH EOREM S . l . (Osserman

1

j

[7}). Let f{ x 1, x 2) be a solution o 1

the minimal surface equation (2.8) in the whole x l ' x 2 -plane. Then , i there exists a nonsingular linear transformation (S.7)

xl = u l x2

=

au 1 + bu 2 ,

b

>

0,

such th at (u 1 , u 2 ) are (global) isothermal parameters for the sur­ face S defined by k

=

3, . . . , n .

38

A SURVEY OF MINIMAL SURFACES

COROL LARY 1. (Bernstein [ 3]). ln the case n = 3, the only

solution of the minimal surface equation in the whole x l ' x 2 -plane is the trivial solution, f a linear fun ction of x 1 , x2 . COROLL ARY 2. A bounded solution of equation (2.8) in the

whole plane must be constant (for arbitrary n) . COROL LARY 3. Let f( x ' x ) be a solution of (2.8) in the

} 2 whole x l ' x 2 -plane, and let S be the surface defined by k = 3, . . , n

(S.8)

.

obtained by referring the surface S to the isothermal parameters given by (S .7). Then the functions (S .9)

k = 3, . . . , n

are analytic functions of u 1 satisfy

n

(S. 10)

I

k= 3

+

iu 2 in the whole u l ' u 2 -plane and

cf>; '-" - l - c 2,

c = a- i b .

Converse/y, gi ven any complex constant c = a - ib wi th b

>

0,

and given any entire functions (f> 3, . . , if; of u 1 + iu2 satisfying n ( S . 1 0) , equations (S. 9) may be used to defi ne harmonie fun etions .

l (u 1 , u 2 ) and substituting u 1 , u as functions of x , x from 2 1 2 k (S .7) into equations (5. 8) yields a solution of the minimal surface

equation (2 .8) valid in the whole x l ' x2 -plan e. Proof of Coro11ary 1: This is an i mmediate consequence of Theorem S . l and Le mma S . S .

t

39

BERNSTEIN'$ THEOREM

Proof of Corollary 2. B y Lemma 4 . 2, each xk , for k

3,

. . . , n,

will be a bounded harmonie function of (u 1 , u2) in the whole u 1 , u2J pl ane, hence constant.

+

Proof of Corollary 3: This is an immediate consequence of L emm a 4. 3 , using the fact th at in view of (5 . 7),

•

- 1

Proof of the theorem: We introduce the map (4. 13) , which i s now defined i n the entire x l ' x 2 -plane. It follows from Lemm a 5 . 4 that this m ap i s a diffeomorphis m 0 f the x 1 , x 2 -plane onto the en­ tire � 1 . �2 -plane . We know from (4 . 14) that (�1 ' �2) are isothermal p arameters on the surface S defined by xk By Lem m a 4.3, the functions

fk( x l ' x2), k

k

=

3, . . . , n .

1, . . , n .

are analytic functions of (. We note the identity a (x l ' x2) a< � l ' �2> and sin ce the J acobian on the right is alw ays positive, we deduce first th at cp 1 -J. 0,

cp2 f. 0 everywhere, and further that

Thus the function cf>/cf> 1 is analytic in the whole (-plane, has negative im agin ary p art, and must therefore be a con stant:

40

A SURVBY O F MINIMAL SURFACB S

Ç> 2

(5. 1 1)

cÇ> 1 ;

c

a- ib, b > O .

=

T aking the real and i m agin ary parts of equ ation

(5 . 12)

�x 2

a ç2

=

(5. 1 1),

we find

b �� a���+

açt

aç2

If w e now introduce the t r ansfo r m ation

(5. 7),

equations

(5. 12) t ake

the form

1

a u2 a ç2

__

:;:;:;;

-

au l a ç2

--

that is to s ay, the Cauchy-Riemann e qu ations, expressing the con­ : dition that

u 1 + iu2

is a complex- analytic function of

But this m e ans, by Lemm a 4 . 5 , that

( u 1, u2)

· p arameters, which proves the theorem. We m ay note th at in the cas e n function E( x 1 ( s ee Jôrgens

\ (5.3) !

,

x2)

[ 11 ,

defined by

p.

133),

(5 .6)

Ç1 + iÇ2 .

are also i sothermal

+

3 the introduction of the w as suggested by E . H einz

whereas the applic ation of Lewy ' s m ap

to this function and the resulting elementary p roof of Bern ­

s tein' s Theorem i s due to N i t s che

[ 1 , 2].

1 The point o f Coroll ary 3 is that it provides a kind of represen� ation theorem for all sol u tions of the minimal surf ace equation :(2.8) over

the whol e

detail the case n 1 s ection

2,

=

x l'

x -pl ane. It i s interesting to examine in 2

4. Then, as we h ave rem arked at the end of

one h as in addition to the tri v i al solutions lk linear,

the solutions

(5 . 13)

13

+ il4

g ( z),

z

x 1 + ix2 ,

41

B ERNSTEIN ' S TH EOREM

where g i s an analytic function of z. For an arb itrary entire func­ tion g ( z), equ ation

(5. 13) defines

a solution of the m i ni m al surface

e qu ation in the whole x ' x -pl ane. The s ame is true if one sets l 2

(5. 14) Now by Theorem

5. 1,

to every global solution f3 ( x 1 , x ) , f ( x , x ), 4 1 2 2

there corresponds a transform ation

�3 (w), �4 (w),

(5.7)

and entire functions

where w = u 1 + iu , sati sfying 2 -2 -2

=

Thu s to each non-orient able sur -

E" corresponds an orientable surface x

(p ) :

M -->

E",

x ( g ( p )), and m any properties of the former can be

re ad off immedi ate! y from corresponding properties of the l atter. In p articular, only such p roperties of non-orientable minimal surfaces will be derived in this s urvey .

2 D E F I N I TION . A regular C -surface S in E" is a minimal sur­ fac e if its mean curvature vector v ani shes at each point. L EM M A 6 . 1 . L et S be a regular minimal surface in E" defined

by a map x ( p) :

M.

M --> E".

Then S induces a conformai structure on

Proof: We are assuming by our convention th at S i s ori ent abl e. L et A be an ori ented atlas of

M. Ra

Let A be the collection of ali

(Ra' Oa, �� such that is a plane dom ain, Oa i s an open 1 set on M, Fa is a homeomorphism of Ra onto Oa, Ffi 0 � pre­ serves orien t ation wherever defined , and x o Fa : Ra --> E" defines ...

-

...

-

tri ples

-

a local surface in i sotherm al p arameters. By Lem m a 4 . 4 the union

Oa equals M, so that A i s an atlas for M, and by Lemma 4 . 5 e ach Fa - 1 o F{3 i s conformai wherever defined, so th at À defines a conformai structure on M. t -

o f the

-

46

A SUR VEY O F MINIMAL SURFACES

Let us note that the introd uction of a conformai structure in thi s way m ay be carried out in gre at generality, since low-order d ifferentiability conditions on S guarantee the existence of local i sothermal p arameters; however, the proof of their exi stence is far more difficult in the general case. We no w discuss sorne b asic notions connected with conforma! s tructure. We note first that if M has a conformai structure, then we can define all concepts which are inv ariant under conformai m apping. In particul ar, we c an s peak of harmonie and subharmonic functions on M , and (complex) analytic m aps of one such m anifold M into another M. A meromorphic function on M is a complex­ analytic map of M into the Riemann sphere. The l atter m ay be de­ fi ned

as

the unit sphere: 1 xl = l in E 3, with the conformai struc­

t ure defined by a pair of m aps (6. 1)

F1 : x =

(-

2u

(

2ü 1

1 wl

/+-l

'

l wl

1 wl 2 - l ' l 1 wl 2 + l

)

'

1 - 1 wl 2 ' � 1 w1 + l + l

)

'

+

w

and (6 . 2)

l w-1 2

+

1' l w-1

The m ap F1 is c alle d stereographie projection from the point

(0 , 0 , 1 ), the i m age being the whole sphere minus this point. The m ap F1 -

1

is given explicitly by

x 1 + ix 1 2 F1 - .. w = l - x3

(6.3)

1 F1 - o F is simply w "" l/w , a conformai m ap of 0 < l w! < "" 2 onto 0 < 1 w 1 < "" and

.

P ARAMBTRI C S URFACBS: GLOBAL THBORY

47

E" is a n non-constant map x{p) : M ... E , where M i s a 2 -manifold with a D E F I N ITIO N ·

A generalized minimal surface S in

conform ai structure defined by an atlas A =

!(Ra> 0a' Fa)l, such

that each coordinate function x (p) is harmonie on M, and further­ k more

n

I. cp k2 ( () k= 1

(6.4)

where we set for an arbitrat:y

=

0 .

a,

i:) h

k aç 1

-

. dh

- 1

k aç2

--

Let us make the following comments concerning this definition . First of ali, i f

S

is a regular minimal surface, then using the con­

formai structure defined in Lemma 6. 1, it follows from Lemm a 4 . 3 th at

S

is also a generalized minimal surface. Thus the theory of

generalized minimal surfaces includes that of regular minimal sur­ f aces. On the other band, if

S

is a generalized minimal surface,

then since the map x ( p) is non-constant, at least one of the func­ tions x (p) is non-constant, which implies that the corresponding k analytic function cp (() c an have at most i solated zeroes. Thus k the equ ation

(6.5)

n

I.

k= l

l cf> k2 ( {;) 1 = 0

can hold at most at i solated points. Then again by Lemma 4 . 3 , if we delete these i sol ated points from

S,

the remainder of the sur­

face is a regular minimal surface. TQe points where equ ation (6.5) h olds are called branch points of the surface. If we allow the

48

A SURVEY OF M INIMAL SURFACES

case e ither l{ ()

.

f '(()

n

=

2 in the defi nition of a generalized surface, we find that

x 1 ix 2 +

or

x 1 - ix 2

is a non-constant analytic function

The poin ts at which (6.5) holds are simply those where

0, corresponding to branch points, in the classical sense,

of the i nverse

rn

epping. In the case of arbitrary

n, the difference

between regul er and generalized mi nimal surfaces consists in al ­ lowing the possibility of i solated branch points. There are both p ositive and negati ve aspects to enlargin g the c lass of surfaces to be studied in thi s way. On the one hand, there are certain theo­ rems one would like to prove fo r reguler minimal surfaces, but which have, up to now, been settled only fo r generalized minimal s urfaces. The classical Plateau problem i s a prime example! On the other h and, there are many theorems where the possible exis­ ltence of branch points has no effect, and one may as well prove them for the wider class of generali zed minimal surfaces. Let us ,

give an example. LEMM A 6 . 2 . A generalized

Proof: map x(p) : onic on

minimal surface cannot be compact.

S b e a generalized minimal surface defined by a M En . Then each coordinate function xk(p) i s har­ M, and if M were compact x k(p) would attain its maxi­ Let

....

mum, hence it would be constant , cont radicting the assumption hat the map

x(p)

is non-constant.

+

Finally, concerning the study of generali zed minima l sur­ faces , let us note that precisely properties of the branch points 1themselves may be an object of investigation. See, for exemple,

�ers [ 2] and Chen [ 1].

For the sake o f brevity we make the fol lowing convention. We

$hall suppress the adj ecti ves "generalized" and "reguler, " and i•see

Appendtx 3, Section 1.

49

P ARAMETRIC SURFACES: GLOBAL TH EORY

we shall refer simply to "minimal surfaces" except in tho se cases where either the statement would not be true without suitably qual­ i fying it, or else where we wish to emphasize the fact that the sur­ face s in question are "re gular" or " generalized. " We next give a brief discussion of Riemannian manifolds. DEFIN ITION . Let M be an n-manifold with a C'-structure de­ fined by an atlas a

A =

! (Ra ' 0 a ' Fa) l.

A

Riemannian structure on M,or

Cq-Riemarmian metric is a col lection of matrices

Ga, where the 0 $ q S r - l,

e lements of the matrix Ga are Cq·functions on Oa� and at each point the matrix Ga is positive definite, whi le for any

a , f3

such that the map

u ( u) Fa- 1 o Ff3

is defined, the relation

(6.6) m ust hold, where

Fa- l o F (3 •

U

is the J acobian matrix of the transformation

differentiable curve on M i s a differentiable map p(t) of an i nterval [a, b] of the real line into M. The 1ength of the curve p( t), a :S t S b, with respect to a given A

Riemannian metric is defined to be

f \ (t)dt '

(6.7)

a

where for each

p( to) oa, (

(6.8)

t 0, a S t 0 S b,

and we set

h ( t)

=

(

Î 1 é.;/p(t)) u;(t) u/( t � Y:< ,

i,j

Ga

(gii) ,

t0, where u 1 , u 2 are coordinates in Ra . By the definition of h ( t ) is independent of the choice of oa .

for t sufficiently near (6.6),

we choo se an 0a such that

50

A SURVEYO F MINIMAL SURFACES

divergent path on M

A

i s a continuous map

p(t), t ?: , 0,

of the

non-negative reals into M, such that for every compact subset Q o f M, there exists

t0

p( t) i

such th at

Q for

t > t 0.

If a div ergent path i s differentiable, we define its length to be

!0

(6.9) where

h( t )

00

h ( t) dt

i s again defined by (6 .8).

DEFIN I TION. A manifold M is

complete with respect to a

given Riemannian metric if the integral (6.9) diverges for every d ifferentiable divergent path on M. The first investigation of complete Riemannian manifolds was made by Hopf and Rinow [ l] in 193 1 . Since that time thi s subject has been studied extensively, and it is generally accepted that the notion of completeness is the most useful one for the global study o f mani fold s with a Riemannian metric. One of our aims in this survey wi ll be to discuss in detail the s tructure of complete mi ni ­ may surfaces. Fi rst let us make the following observations.

En

Let a cr-s urface S i n

be defined by a map

x(p) :

M

_.

En .

Then thi s map induces a Riemannian structure on M, where for each

a

we set

x( u)

=

x( Fa (u)),

whose elements are (6. 10)

g 1. 1 = a x .

and we define Ga to be the matrix

-

ax.1

·

au.J

Then e quation (6.6) is a consequence of ( 1 .8), and the matrix Ga will be positive definite at each point where S is regular. Thus to each regular surface S in

En

corresponds a Riemannian

51

PARAMETRIC SURFACES: GLOBAL THEORY

2-manifold M. We say that S is complete if M i s complete with r espect to the Riemannian metri c defined by (6. 10). If S is a generalized minimal surface, then there will be iso1 ated

points at which the ma tri x defined by (6. 10) will not be posi­

tive definite. However, the function

h ( t)

defined by (6.5) i s sti ll

a non-negative function, independent of the choice of

a,

and we

may still define S to be complete i f the integral (6. 9) diverges for every divergent path. We conclude this section by recal ling sorne basic facts from the theory of 2-m anifolds.

universal covering sur­ lace which consists of a si mply-connected 2-manifold M and a map 11 : M M, with the property that each point of M has a neighbor­ First of ail , each 2-manifold M has a

...

hood V su ch th at the restriction of

11

to each component of

i s a homeomorphism onto V. In parti cular, the map

11

11- 1 ( V)

is a local

homeomorphism, and it follows that any structure on M : c r, con­

M. I t is not hard to show that M is complete with respect to a given Riemannian metric if and only il M is complete with respect to the induced Riemannian metric. formai, Riemannian, etc. induces a corresponding structure on

Suppose now that S is a mini mal surface defined by a map

x( p) :

M

...

E".

We then have an associated simply-connected mini­

rn al surface S, called the by the composed map 1 ar,

universal covering surface of S, d efined x(TT(p)) M E". It follows that S is regu:

...

if and only if S is regular, and S is complete if and only if S

is complete. Thus, many questions concerning mini mal surfaces may be settled by considering only simply-connected minimal sur­ faces. In that case we have the following important simplification .

52

A SUR VEY OF MINIMAL SUR FACES

L EM M A 6 . 3 . Every simply-connected minimal surface S has a

reparam etri zation in th e form x( () unit disk,

1 (1


0, so that log \ w - w 0 \ < h(w) . The function log [ ( l + \ w\ 2 )/\ w - w \ 2] is con­ 0 tinuous on th e comp act set E, hence h as a fi nite m aximum M. Thus i f w 1 is any bound ary point of D, we have lim [ log O + \ w\ 2 ) W -> wl

-

2h(w)] S l im

w -> wl

[ log ( l + \ w\ 2 )

-

2 1 og \ w- w 0 \]

< M. 2

ut log( l + \ w\ ) i s subh armoni c in D, and by the maximum prin­ iple we have log( l + \ w\ 2 ) :S 2h( w) + M throughout D.

+

3 P ARAMETRIC SURFACES IN E :

TH EOR E M

8. 2." Let

E 3 . Then either age of

S

S

71

TH E GAUSS MAP

be a complete regular minimal surface in

S

is a plane, or else the set E omitted by the im­

under the Gauss map has capacity zero.

Prao!: If

S

is not a plane , then the i m age of S under the Gauss

m ap is an open connected set on the sphere, and the complement of the i m age is therefore a compact set• E. If E is empty there i s nothing to prove. Otherwi se we m ay assume that the set E includes the point ( 0, 0, l), after a preli minary rotation of coordinates. Again we m ay pass to the universal covering surface

S

of S, whose G auss

m ap omits the same set E. The surface S is given by a map

x ( Ç} : D -> E 3 , where D is either the plane or the uni t disk. In the former c ase we know that the set E can contain at most two points and hence certainly has capacity zero. Let us examine the case where D is the disk

1 (1
:ero. Then in the image

D 1 of D under the m ap w

=

g ( () , there would be a h armonie func­

tion h ( w) m ajori zing log( l + l wl 2 ) . Then h (g ( () ) i s harmonic in

D, and is the real part of an afl alyti c function · G( () in D. Finally, F(() eG ( ( ) is analytic in D and never zero. For an arbitrary path =

C in D, we h ave the length

But the function

/(() F ( ()

never vanishes in D, and by Lemma

8 .5

there would be a divergent path C for whi ch th e integral on the right converges, and the surface S would not be compl ete. Thus the set

E must in fact h ave capacity zero , and the theorem is proved. •see Appendlx 3, Section 4.

+

72

A SURVEY O F MINIMAL SURFACES

TH EO REM 8 . 3 . Let E be

an

arbi trary set of k poin ts on the

unit sphere, where k ::; 4. Th en there exists a complete regular minimal surface in E 3 whose image under the Gauss map omits preci se/y the set E. Proof: By a rotation we m ay assume that the set E contains t h e point

(0, 0, 1).

If t h i s is only point , then Enneper' s surface

d e fined e arl ier (by s etti n g f( ()

=

1, g ( ()

()

solves the problem.

O t herwi se l et t h e other points of E correspond to the points wm ,

m

=

1, . . . , k - 1 , under stereographie proj ec tion. If we set g ( ()

f ( ()

'

'

( (- wm )

n m

=

1

and use t h e repre sentation (8.2), (8 .6) in the whol e ( - p l ane minus the points wm

,

we obtain a m inim al surface whose normals omit

precisely the points of E, by (8 .8), and whi ch is complete, because a divergent p ath C must tend ei ther to

oo

or t o one of the points

wm , and in e ith er c ase, we have

We m ay note t h at the integrais (8 . 6) may not be single-valued, but b y p assing to t h e unive rsal coverin g surface we get a single-valued m ap d efining a surface h avin g the same properti es.

t

Let us r eview briefly the historical development of the above heorems. Theorem 8 . 1 , with t h e additional assumption th at S be simply connected w as conj ectured by Ni renberg as a n atural gener­ alization of Bern stei n ' s theorem, and it was proved in O sserm an[ l]. T h e p resentation given h ere fol lows th at of O sserman

[ 3] ,

where it

73

3 P ARAMETRlC SURFACES IN E : THE GAUSS MAP

i s observed th at simple connecti vity is irrelevant and where the formulas

(8 .2), (8 .6), (8.8)

are used to reduce geometrie statements

of this kind to purely analytic o n es . Lemm a the c ase f ( z)

,f, 0, and Theorem

8.2

8.5

i s given there fo r

is st ated as a conjecture. At

the end of the p aper there is given a proof of Theore m

8.2

due to

Ahlfors, who obs erved that u si n g the above-mentioned reduction to an alytic functions the result followed from a theorem of Nevanlinna. Ahlfors also s uggested the reasoning wh ich allows one to include in Lemma

8.5

functions / ( z) with a fini te n u mber o f zeros . Thi s

allows u s to m ake the followin g geometrie conclusio n : Theorems

8. 1 and 8. 2 remain valid lor generalized minimal surfaces, provided that they are simply connected and have only a finite number of branch poin ts. F urthermore, Ahlfors observed that Lemma

8.5

con­

t inues to hold if f( z) h as an i nfi nite number of zero s, provided that their Bl ashke product is convergent.

Thus Theorems

8.1

and

8.2

continue to hold fo r a certain class o f general i zed minimal surfaces which h ave an i n finite number of branch points, but it is not clear how to characterize this cla s s geometrically. On the other band, let u s note the followi n g. There exist complete generalized minimal

s urfaces, not lying in a plane, whose Gauss map lies in an arbi tra­ ri/y small neighborhood on the sphere. In fact, we need only choose D to be the unit disk

1 ��

g( ()

< l,

=

E ( for any gi ven E >

f ( (> a fu nction an alytic i n D such th at fe l / ( () 1 I d(! d i vergent p ath C.

= oo

0, and

for every

Such functio n s / (() may be constructed in a va­

r iety of ways . Retur n in g to Theorem

8 . 2,

the proof given here, which does not

depend on Nevanl i n n a ' s theory of functions of bounded c h aracteris­ t i c , is t aken from Osserman

[4}.

Theorem

8. 3

is due to Voss

[ ].] .

A SUR VEY OF MINIMAL SURFACES

i

An example of a complete surface whose normal s omit 4 directions had been given earlier i n Osserman [3], and it was l ater observed i n Osserman [6], that the classical minimal surface of Scherk pro­ v ides still another example. The obvious question which arises when comparing Theorem s 8 .2 and 8.3 is the exact size of the set E omitted by the normals.* Specifically the following: Problems 1 . Do there exist complete regular minimal surfaces

1 whose i m age under the Gauss map covers ali of the sphere except : for any arbitrary finite set of points E given in advance? 2. Does th ere exist a complete regular minimal surface whose i m age under the G auss m ap is the complement of an infinite set E !o f capacity zero?

.•see Appendix 3, Section 4.

!

3 SURFACES IN E . GAUSS CURVATURE AND TOTAL CURVATURE

75

§9. Surfaces in E3 . Gauss Curvature and Total Curvature. We continue our study of minimal surfaces in E 3 , using the represent ation (8. 2). The second fundamental fonn was defined in (1. 27) with re­ spect to an arbitrary normal vector N . Since i t is line ar in N, and

since in E3 the nonnal space at each point is one-dimensional, it i s sufficient to define

b I J.( N) .

for a single normal N which is usu al-

ly chosen to be that in (8 . 8). By a calculation one finds the following expression for the second fundamental form for a minimal surface in the representation (8. 2) :

Since by (8 . 7) we h ave I g

. .

lJ

dÇi �Çi dt

dt

=

[ i t i 0 + 1 Ai 2 ) ] 2 i !i.'- ! 2 , 2

dt

it fo llows by (1. 28) that the normal curvature i s given by

The maximum and minimum of this expression , as to 2rr, were defined

in

a

varies from 0

( 1 . 30) to be the principal curvatures. They

are obviously (9.2) Now for an arbitrary regular c 2 -surface in E 3 one defines the Gauss curvature K at a point as the

v atures:

product of the principal cur­

76

A SURVEY OF MINIMAL SURFA CES

(9. 3) LEMM A 9 . 1 . Let x ( () : D -+ E 3 define a minimal surface. Then

using the representation (8. 2), the Gauss curvature at each point is given by (9.4)

Proof: (9. 2) and (9 . 3).

•

CO RO L L ARY. The Gauss curvature of a minimal surface is non-positive and unless the surface is a plane i t can have only iso1 ated

zeros.

Proof: By (8.8), S is a plane .. oii . Consider now, for an arbitrary minim al surface in E 3, the follow­ =

ing sequence of mappings, where l denotes the unit sphere: (9.6)

x( ()

Gauss m ap

s tereographie proje ction

D --� s ---� l ------ w-plane.

he composed map, as we have seen in the proof of Lemma 8. 3, is

3 SURFACES IN E . GAUSS CURVATURE AND TO TAL CURVA TURE

77

g ( () : D -+ w-plane.

Con si der an arbitrary differenti able curve (( t) in D, and its image under each of the maps in (9 .6). If s ( t ) is the arclength of the im­ age on S, then as we have seen: (9 . 7)

The arclength of the image in the w-plane i s s imply (9.8)

1 dw 1 dt

=

1 g '( () 1 1 �K 1 dt

If we let a ( t ) be arclength on the sphere, then it follows by the formula (6. 1) for stereographie projection that (9.9)

da = dt 1

+

2 1 dw 1 l wl 2 dt

Combining these formul as with (9 .4), we find (9. 10)

Thus we have an i nterpretation of the Gauss curvature in terms of the Gauss map. If we now let 11 be a domain whose closure is in

D , the surface defined by the restriction of x ( () to 11 has total curvature gi ven by

which is the negative of the area of the image of 11 under the Gauss map. (This is in fact the original deqnition used by Gauss to defin

the Gauss curvature of an arbitrary surface.) Of course if the image

78

A SURVEY O F MINIMAL SURFACES

i s multiply-covered the total area is that of ali the sheets. We may note that (9. 1 1) may be regarded cquivalently as the spherical area of the image of � under g ( ,) .

Throughout this discussion we have been assuming that the surface S was defined in a plane domain D. However, the normal N a nd the value of the function g are independent of the choice of

parameter. In fact, if we introduce new isothermal parameters by a d' conformai map ,( ,) , then we have .. i dzi

0 .5: t


2k- 2m

nd

Jfs

K dA

=

-

4rrm S: 2rr(2- 2G- 2k)

REMARK . The inequality

1

J J5 K dA S:

2rrx

2rr(x- k) .

•

was shown by Cohn-

!Vos sen to hold for an arbitrary complete Riemannian 2-manifold [With fini te total curvature and finite X . It follows in particular rom (9. 22) that in the case of minimal surfaces, equality can never old in Cohn-Voss en ' s inequality. This question is discussed in a recent paper of Finn [ 6] , who introduces at each boundary compo­ ent a geometrie quantity which acts as a compensating factor beween the two sides of the Cohn-Vos sen inequality. One obtains in this way a geometrie interpretation of the order lat '

p1

. •

v.

1

of the pole of

f

87

3 SURFACES IN E : GAUSS C U R VA TURE AND TOTAL CURVA TURE

There are only two complete regular minimal surfaces whose total curvature is - 4rr. These are the catenoid and Enneper' s surface. Proof: This is the case m l in Theorem 9 . 2 . This means that the function g is meromorphic of order 1, hence maps M con­ formally onto the Riemann sphere L. Thus M has genus G 0, and inequality (9 . 2 2) reduces to k :5 2. We may therefore choose M to be L minus either one or two points, in which case we have g ( �) �' and by Lemma 9 . 6 f( �) is a rational function. Taking TH EO REM 9.4.

=

into account the completeness and the fact th at the functions are s ingle-valued, one easily finds that the only choices of

xk (Q

/ ( �)

compatible with these conditions yield precisely the two su rfaces named.

t

Enneper' s surface and the catenoid are the onl two complete regular minimal surfaces whose Gauss map is one-toone. Proof: If the Gauss map is one-to-one , then the total curvature COROLL ARY .

being the negative of the area of the image, satisfies

By Theorem 9 . 2, equality holds on the left, and the result fol lows from Theorem 9 .4 .

t

The methods used to prove the above theorems on the total curvature can also be used to study more precisely the behavior of the Gauss map, supplementing and sharpening the results obtained in the previous section . Let us give aq example.

88

A SUR VEY O F MINIMAL SURFACES

THEOREM 9.5. Let S be a regular minimal surface, and sup­ pose that ail paths on the surface which tend to some isolated boundary component of S have infinite length. Then either the normals to S tend to a single limit at th at boundary component, or else in each neighborhood of i t the normals take on ali directions except for at mos t a set of capacity zero. Proof: By a neighborhood of an isolated boundary component

we mean a doubly-connected reg1on whose relative boundary is a J ordan curve

y.

This region on the surface may be represented by

E3, where D is an annular domain 1 < I Ç'I < r2 :S ""• the curve y corresponding to 1 '1 = 1, and we may introduce the

x ( �') : D

...

representation (8 .2) in D . Suppose now th at in sorne neighborhood of this boundary component the normals omit a set of positive ca­ pacity. This means that for sorne r 1 � 1, the tunction w g ( Ç') omits a set of positive capacity in the domain D ': r 1 < 1 ' 1 < r2 . By Lemma 8 .6 the re exists a harmonie function h ( w) i n the image of D ' un der g ( Ç') , such th at log( l + 1 wl 2) ::; h( w) . Since the met­ rie on S is given by À = � 1 �( 1 + l s l 2), we have log À( Ç) .S log

�

+

log h ( �( Ç))

Since the right-hand side is a harmonie function in D ', we may ap­ p ly Lemma 9.3 and deduce that r2 ""· But then g(Ç') could not have an essential singularity at infinity by Picard's theorem. Thus =

g ( Ç) tends to a limit, finite or infinite, as Ç tends to infinity, and the same is true of the surface normal . t Without giving the details , let us note that further analysis a long the above lines leads to the following result:

SURFACES IN E 3 : GAUSS CURVATURE AND TOTAL CURVATURE

89

3

Let S be a complete minimal surface in E • Then S has infinite total curvature
3. There,

e ven bounded solutions of the minimal surface equation may have i solated n on-remo vable singu l arities (Osserman

[ 10] ). t

We tum next to another application of Lemma

10.2.

We wish to

inves tigate the sol vability of the Dirichlet problem for the minimal s urface equation. For this purpose we introduce the following no­ tation.

DEFIN ITION .

Let D be a plane dom ain and P a boundary

point of D. We say that P is a point of concavity of D if there exists a c i rc le C through P and sorne neighborhood of P whose inters ection with the exterior of C lies in D. The circle C i s c alled a circle o f inner con tact at P.

L EM M A 10.3.

Let D be a boun ded plane domain, P a poin t of

concavity on the boundary, and C a circle of inner con tact at P. li F is a s olution of the minimal surface equation in D, th en the •see Appendix 3, Section 5. tSee Appendlx 3, Section 5.

99

N ON -PARAMETRIC MINIMAL SURFACES IN E 3

boundary values of F at the point P are limi ted by the boundary values o f F on the part of the boundary exterior to C. P ro of: We may assume that the circle C is centered at the o ri gin and h as radi us r 1 . The entire domai n D is contained in

s orne circ le r < r2 • Suppose th at lirn F � M for all boundary points of D exterior to C. Then by Lernrna 10. 2 applied to the intersec­

tion of D with the exterior of C, F � G(r ; r 1) - G( r2 ; r 1 ) + M for

r 1 < r < r2 , and therefore

•

( 10.1 4)

L EM M A 1 0 . 4 . Let D be an arbi trary plan e domain. Th en D i s

c onvex i f a n d only if th ere does not exist a point o f concavity on th e boundary of D.

P roof: Suppose first that P is a point of concavity of D, and l et C be a circle of inner contact at P. Then a segment of the tangent line to C at P wi ll have its endpoints in D, but the seg­

m ent its e lf contains the point P not in D. Hence D is not convex C onvers ely , if D is not convex there exist two points Q 1 , Q 2 in

D, such that the segment joining them is not in D. Let Q( t ) ,

be a curve i n D j oining Q 1 to Q • T he n there i s a 2 s rnallest v alue t of t for which the segment L from Q 1 to Q( t ) 0 0 is not entirely in D. We h ave t > 0, and for a l l srnaller values of 0 t sufficiently near t the li ne segments from Q 1 to Q( t ) lie in D 0 a nd a re on the sarne side of L. There is therefore an open subset 0 ::; t ::;

l

!'! of D consisting of all points on one side of L and sufficiently n ear to L to gether with all points in a disk around each of the endpoints . By choosing a sufficiently flat circular arc lying in !'!

lOO

A SURVEY OF MINIMAL SURFACES

join ing the endpoints of

L

and tran slating it until it first contracts

the boundary of D, one finds a point of concavity.

+

Let D be a bounded domain in the plane. N ecessary and sufficient that there exist a solution of the minimal surface equation (10.1) in D taking on arbitrarily assigned contin� uous values on the boundary is that D be convex. Proof: Suppose first that D is convex. Then by Theorem 7.2 T H EOREM 10.3.

o ne can solve the boundary value problem for arbitrary continuous boundary values. If, on th e other hand, D is not convex, then by L emma 1 0.4 there exists a point of concavity

P

on the boundary

of D. If we choose boundary values which are sufficiently large at

P

and small outside a neighborhood of

P,

then no solution can

+

exis t by v irtue of the inequality ( 10. 14).

T his theorem and Lemma 10.3 are both due to Finn [5}. We may n ote that one can easi ly const ruct special cases of non·solvabi lity d irectly from Lemma 10. 2 . For example, if D is the part of the annulus

r 1 < r < r2

lying in the first quadrant, and if we assign

c ontinuous boundary values whi ch are equal to the circle

r r1 , =

G(r ; r 1 )

outside

and positive somewhere on this circ le, then no

solution can exist by (10.5). The question of non-existence of so­ lutions in non-convex domains had been considered earlier by a h umber of authors starting with Bernstein

[2], but in

each case the

arguments h ad been valid only for special domains . For a detailed d iscussion of this question, see Nitsche

[6].

W e consider next a more general si tuation in which solutions m ay have infinite boundary values . We prove first the fol lowing re­ s uit, also contained in the paper of Finn {5}.

101

NON·PARAME TRIC MINIMAL SURFACES IN E3

Let D be an arbitrary domain having a point of concavity P on its boundary. Then a solution of the minimal surface equation in D cannot tend to infini ty at P. Proof: Let C be a ci rcle of inner contact at the point P. By choosing a s li ghtly larger circle tangent to C at P, if necessary, we m ay assume that there is an arc y of C which contains P but T H EOR EM 10.4.

no other boundary points. If C' is a larger circle concentric with C and sufficiently close, then the region D ' bounded by radial segments, and an arc

y

'

y,

two

of C ' wil l be entirely in D. Any ,

solution F of ( 10 . 1) in D will have a finite upper bound

M

on the

part of the boundary of D ' exterior to C. We may then apply Lem ­

m a 1 0. 3 to the domain D ', and by ( 10 . 14) F cannat tend to infinity at the point

P.

+

The interest of the above theorem is that solutions of the mini ­ mal surface equation can indeed take on infinite boundary values, even a long an en tire arc of the bound ary , as in the case of Scherk's s urface

F(x 1 , x 2)

=

log

cos x 2 cos x 1

--­

w hich is d efined in the square 1 x 1 1 < to e ither

+oo

or

-oo

77/2,

l x 2\
M for 0 < / '1 < r1 • L et M 1 = max / x 1 + ix2/ for / (/ = r1 • We assert that every point

l i n the exterior of the circle xi

1

(once in

0

1 choose r2

<


0,

a nd given a ny entire functions 1;> , . . , if>n of u1 + iu2 satisfying 3 the above e quation, these may be used to define a solution of the .

m inimal surface equation (2. 8) in the whole x1 , x2 �plane.

TH E OREM 7.1. Let r be an arbitrary Jordan curve in E". T hen there exis ts a simply�connected generalized minimal surface bounded by r.

T H E OREM 7.2. Let D be a bounded. convex domain in th e x1, x 2 �plane, and let C b e its boundary. Le t gk(x1 , x2 ) b e arbi­ trary continuous functions

on

C, k

3, . . , n. Then there exists a .

s olution /(x1 , x2) = {1 (x 1 , x2 ), . . . , / (x 1 , x2)) of the minimal 3 n s urfa ce e quation (2. 8) in D, such that f (x 1 , x 2 ) takes on the k boundary values g (x 1 , x 2) . k

TH E OREM 8.1. Let S be a complete regular minimal surface

3 in E • Then either S is a plane, or else the normals to S are every where dense.

TH E OREM 8.2. Let S be a complete regular minimal surface

in E 3• Then ei ther S is a plane, or else the set E omitted by the image of S under the Gauss map has logarithmic capaci ty z ero.

T H E OREM 8.3. Le t E be an arbi trary set of k points on th e s ph ere, where k .::; 4. Then th ere exists a complete regular minimal surface in E 3 whose image under the Gauss map omi ts precisely the set

E.

131

LIST OF THEOREMS TH EO R EM 9. 1 .

Let M be a complete Riemaimian 2-manifold

whose Gauss curvature satisfies K .$ 0, JJM I Ki dA < Then there exis ts a compact 2-manifold M, a fini te number of points p l ' ' " ' pk oo.

on M, and an isometry between M and M- { p 1 , T H E O R EM 9

.

2. Let S

Then either ff;< dA

=

be

- oo ,

. • .

, pk l.

a complete minimal surface in E 3 •

or else JJ5 K dA

=

- 4rrm, m

=

0, l,

2, . . . .

Let S be a complete regular minimal surface 3 in E • Then JJ5 K dA .$ 2rr ( x - k}, where x is the Euler charac­ teristic of S, and k is the number of boundary components of S. THEOREM 9.3.

There are only two complete regular minimal 3 surlaçes in E whose total curvature is - 4rr. These are the cate­ T H E O R E M 9. 4.

noid and Enneper' s surface. Enneper' s surface and the catenoi d are the only two complete regular minimal surfaces in E 3 whose GausS� map is C O RO L L A R Y .

one-to-one. THEOREM 9.5.

Let S be a regular minimal surface in E 3 • and

suppose th at ali paths on the surface which tend to sorne isolated boundary component of S have infinite length. Then either the n ormals to S tend to a single limi t at that boundary componen t, or e lse in ea•::h neighborhood of it the normals take on ali the direc­ tions except for at mos t a set of capacity zero. Let F( x 1 , x 2 ) be a solution of the minimal s urface equation (10. 1) in a bounded domain D. Suppose that for T H E O R E M 10. 1 .

every boun dary point ( a 1 , a 2) of D, wi th the possible exception o f a finite n umber of points, the relation s

132

A SUR VEY O F MINIMAL SURFACES

Hm

(x1, x 2) ...,. (a

1,

a�

F(x 1 , x 2 ) .:S M ,

lim

F(x 1 , x 2)

(xl, x2} ...,. (a l, 82)

are known to hold. Then m .:S F( x 1, x 2) .5 M throughout

.2

m

D.

T H EO REM 1 O. 2. A

solution of the minimal surface equation cannat have an isolated singularity. Let D be a bounded domain in the plane. N ecessary and sufficient th at there exist a solution of the minimal surface equation in D taking on arbitrarily assigned continuous values on the boundary is that D be convex. TH EOREM 10. 3 .

Let D be an arbitrary domain having a point of concavity P on its boundary. Then a solution of the minimal surface equation in D cannat tend to infinity at P. THEOREM 10.4.

Let D be a plane domain, y an arc of the l boundary of D, L the line segment joining the endpoints of y. [f y and L bound a subdomain D ' of D, th no solution of the minimal surface equation in D can tend to infinity at each point of y. T H EO REM 10.5.

en

Let x3 F( x 1 , x 2 ) define a minimal surface in the disk D: xi � < R 2 • Let P be the point on the surface !lying over the origin, let K be the Gauss curvature of the surface at P, and let d be the distance along the surface from P to the 1 boundary. Then the inequality I KI .:s c/� w� holds, where c is an absolute constant, and w� is the value at the origin of w 2 l + (aF /ax 1 ) 2 (aF /ax2) 2• CO ROL LARY . Under the same hypotheses, there exists an 2 � bsolute constant c0 such that l KI .:S c0/R . THEOREM 1 1 . 1 .

=

+

=

+

UST OF THEOREMS

1 33

Let x3 F( x 1 , x2) define a minimal surface in the region x� x� > R 2 • Th en the gradient of F tends to a limit at infinity. THEOREM 1 1 .2 .

=

+

If the exterior Dirichlet problem has a bound· ed solution, then it is unique. TH EOR EM 1 1 .3 .

There exist continuous functions on the cir­ cle � + x� l which may be chosen to be arbitrarily smooth, such that no bounded solution of the minimal surface equation in the exterior of this circle takes on these values on the boundary. THEO REM 1 1 .4 . =

T H EOREM 12 . 1 . Let S be a complete regular minimal surface E n . Then either S ia a plane, or else the image of S under the

zn Gauss map approaches arbitrarily closely every hyperplane in

p n- l(C )

.

CO ROLLARY· The normals to a complete regular minimal face S in En are everywhere dense rmless S is a plane.

sur·

APPENDIX 2 GENERALIZATIONS One of the important roles played histori cally by the theory of m inim al surfaces was that of a s pur to obtain more general results. I n the followin g pages we shall try to give a brief idea of sorne of the e xtensions of the theory whi ch we have d iscussed . We shaH group the results in severa! cate gories . I . Wider c lasses of surfaces in E3•

A . Surfaces of constant mean curvature. F rom a geometrie point of vi ew, the most natural generali zation1 of surfaces in E3 satis fying H

H

=

0 consists of surfaces s atisfying

c . Certain properties e xtend to this class, whereas others are

quite d if fe rent accord ing as c

0, or c f,

U,

Let us start w ith the P lateau proble m . Theorem 7 . 1 w as e x­ tended first by Heinz

[ 2],

wh o pointed out that in general , one

w ould e x pect a solution only if the mean curvature is not too l arge c ompared to the size of the curve r. (See a lso the di scussion be­ l o w for the n on-parametric case .) The results of Heinz were later i m pro ved on by Werner

[ 1] ,

who obtained, in particular, the fol low­

i n g result: Let r be a Jordan curve whi ch lies in the unit sphere.

1 cl

Then lor any value of c, s urface satisfying H

=

< Y. , there exists a simply-connected

c, bounded by r.

C om pl ete surfaces of constant m e an curvature are studied i n K lotz a n d Osserman

[ 1]

using method � s i mi lar t o those of section

8 . The follo wing result is proved : L e t S be a complete surface o f cons tant m ean curvature.

Il K

>

135

0 on S, th en S is a plane,

APPENDIX 2 GENERALIZA TIONS One of the i m po rtant roles played historic ally by the theory of m inimal surfaces was that of a s pur to obtain more general results. ln the following pages we shall try to give a brief idea of sorne of the e xtensions of the theory whi ch we have d iscussed . We shall group the re sults in several cate gories . I . Wider classes of surfaces in E3•

A. Surfaces of constant mean curvature.

F rom a geometrie point o f vi ew, the most natural generalizatio�

o f surfaces in E3 satis fying H

H

=

=

0 consists of surfaces satisfying

c . Certain properties e xtend to this class, whereas others are

quite d if fe rent according as c "" 0, or c

�

u,

Let us start w ith the P lateau problem . Theorem 7 . 1 w as ex­ tended f irst by Heinz

[ 2],

wh o pointed out that i n gen eral , one

w ould e x pect a solution only if the mean curvature is not too l arge c ompared to the s ize of the curve r. (See a lso the di scussion be­ lo w for the n on-parametric case .) The results of Heinz were later i m proved on by Werner

[ 1] ,

who obtained, in particular, the fol low­

i n g result: Let r be a Jordan curve whi ch lies in the unit sphere.

1 cl

Then for any value of c, s urface satisfying H

=




0 in a disk

define a surface

xi x� +


O. the graph oj thejunction w •

czz

represents a minimal suiface in E4 with total curvature - 2'lf. Every complete minimal suiface in En with total curvature

- 2'lf lies ln a

jour-dtmensional affine subspace and is congruent to one qf the above suifaces. This theorem is a somewhat sharpened form of a theorem of C. C. Chen [3). For this and further related results we refer to Hoffman and Osserman i 1 ]. The above results characterizing complete minimal surfaces with total curvature - 211' and - 411' follow from a general characterlzatlon of complete minimal surfaces of genus zero and given total curvature (Hoffman and Osserman [ 1 ), Proposition 6 . 5 ) . In particular, one has a general construction for genus zero minimal surfaces. As mentioned on p. 89, above, it would be interesting to have more examples ln the higher genus case. A paper of Gackstatter and Kunert

I l l shows that

given any compact Riemann suiface M, there exist points p1,

•

•

•

Pk>

and a complete minimal suiface S ojjtnite curvature ln E3 dejined by a map x(p): M-E3, where M =M - {p1,

•

•

•

pk}. ln fact, their method

152

A SURVEY OF MINIMAL SURFACES

gives many such surfaces for any given

M, including a continuum of

conformally distinct types . What still remains to be studied is to what degree one can prescribe the points p1

•

•

•

.,

Pk· (See also Chen and

Gackstatter ( l l. and, for non-orientable surfaces. Oliveira ( l J, ) With n o assumption o n finite total curvature. we have very recent extensions of Xavier's Theorem to E4 by Chen 1 4 1 and to En by Fuj imoto

I l l. Fuj imoto shows that if S is a complete minimal surface in

En with

non-degenerate Gauss map g. then g(S) cannotfail to tntersect more than n2 hyperplanes in general position. In a second paper. Fujimoto 121 gives a remarkable generalizatlon of ali these Picard-type theorems in

the form of Nevanlinna-type theorems for the Gauss map.

5. Minimal graphs Many results on minimal graphs in E3 extend only partially. or not at ali. to higher codimension. Theorem 7.2 asserts the existence of a solution to the minimal surface equation in a convex plane domain corresponding to an arbi­ trary set of contlnuous boundary functions. For the case of a single boundary function, it follows from Lemma 10. 1 that the solution is unique. It turns out that in the general case uniqueness does not hold. In fact. even when D is the unit disk, one can show that there exists a

pair of real analyticfunctions on the boundary of D to which corre­ spond three distinct solutions in D of the minimal-surface equation (Lawson and Osserman

[ I l).

Another result that falls when going from E3 to En is Bers' Theorem (Theorem 10.2) asserttng that a solution of the minimal-surface equa­ tion cannot have an isolated singularity. There are simple counterexam­ ples as soon as n = 4, such as the complex functlon w = 1/z, considered as a pair of functlons of two real variables. However, one does have the

DEYELOPMENTS 1970-1985

153

following result (Osserman [ 12 11: letj(x1,x2) be a vector solution Q/ the

2 2 minimal s urface equation (2.8) in O<x + x 0. S uppose 1 2 th at all components ojj with at most one exception extend continuously to the ortgin. Thenj extends to the origtn, is smooth there, and sattsjies (2.8). This result contatns as special cases both Bers' Theorem and the followtng result proved tndependently by Harvey and Lawson

2

ijj(x1,x� is continuous i n x + x 2 < 11. 2 and is a solution Q/ (2.8) in l 2 2 0 < x + x2 < 11. 2 , thenj is a solution in thejull disk. l

Ill:

2

Finally we note that Simon [ 2 1 has generaltzed the removable stngularity result of Nitsche and of de Giorgi and Stampacchia (see p. 98, above) by eltminating the hypothests that the excepttonal set E be a compact subset of D. 6. Generalizations We follow here the order adopted in Appendix 2, and list just a few of the subsequent results most pertinent here. 1. Wider classes oj surfaces in E3

A. Surfaces qf constant mean curvature A whole new approach to Plateau's p roblem for surfaces of constant mean curvature was devised by Wente

I l l and elaborated in later papers

of Steffen 1 1. 2 1 and Wente [ 2 ]. (See the last of these for details and further references . ) Wente's method involves minimizing area subject to a volume constratnt, in contrast to the earlier method of Heinz 1 2 1 based on a variational problem wlth the mean curvature H p rescribed in advance. An analog of Theorem 8. 1 has been proved by Hoffman, Osserman,

and Schoen [ 1 ] . They show that if S is a complete surface oj constant

mean curvature in E3 whose Gauss map lies in a closed hemisphere, , then S is either a plane or a right circular cylinder. Examples of

154

A SURVEY OF MINIMAL SURFACES

complete surfaces of revolution of constant mean curvature show that the Gauss map can lie in an arbitrarily narrow band about an equator. An important observation due to Ruh

I l l is that a surface has

constant mean curvature If and only If its Gauss map ls a harmonie

map. For details on this and the general theory of harmonie maps see Eells and Lemaire 1 1 .2 1 . (See also the comments on harmonie maps ln VI below. ) Kenmotsu

I l l derived a representation theorem for surfaces of con­

stant mean curvature, similar to the Weierstrass representation theorem of Lemmas 8. 1 and 8.2. He proved that if g ts a harmonie map Qf a

simply-connected plane domain D into the unit sphere I. then there exists a surface S oj constant mean eurvature in E3 dtiftned as a branehed immersion

x:

D

_..,

E3 where the coordinates in D are iso­

thermal parameters on S and the map g is the eomposttton Q/ the immersion map D

_..,

S with the Gauss map S

_..,

I. More generally. for

arbitrary surfaces ln E3 of variable mean curvature H. Kenmotsu derives an lntegrability condition relating H with the Gauss map. and obtains a generalized Weierstrass representation theorem. For further results along these lines, see Hoffman and Osserman [4). Although somewhat further afield, the most strlklng recent result on surfaces of constant mean curvature is the answer by Wente [31 to an old problem of Heinz Hopf: Are the re any compact immersed surfaces of constant mean curvature ln E3 other than the standard sphere ? Wente showed that there is such a surface in the form of an lmmersed torus. Severa! notions of stabillty are possible for surfaces of constant mean curvature. For a discussion and some recent results, see Barbosa and do Carmo [ 5 ), Palmer

I l l. and da Silveira [ 1 ].

B. Quastmtnimal surfaces The question ralsed on p. 137, above, has been settled by S imon

155

DEVELOPMENTS 1970-1985

(14], Theorem 4 . 1 ) who showed that Bernstein's Theorem holds for arbitrary quaslminimal surfaces. The paper of Schoen and Simon [21 referred t o earlier, where Bernstein's Theorem i s generalized b y impos­ ing a bound on area growth rather than a non•parametrtc representa­ tion, is actually valid for all quasiminimal surfaces. Another paper of Simon ! 5 1 gives generalizatlons of Heinz' inequaltty

( I L 7) and of Bers'

Theorems, Theorem 1 0 . 2 and 1 1 . 2, for quasimlnimal surfaces that are defined via solutions to equations of "mean curvature type. " This class of equations is considerably broader than similar ones con�idered earlier by Finn [2,4) and Jenkins and Serrin [ 1.2). C. Complete surfaces Q{flnite total curvature A recent paper of White [3] shows that many of the results of Chapter 9 conœrning the Gauss map and total curvature of a complete surface hold in great generality. Without assuming minimality or any other local condition, White gives new proofs and generalizations of Theorems 9. 1 and 9. 2. and Lemma 9. 5. He assumes only that S is a complete surface in E3 satsfying the condition f5(2H2

- KJdA

�J .

where p = min{m , n - m} and K = 1 if

m + 1. lf [ det(gulP'2

component!J Q/f is a linearfu nctton Q/ x 1 ,

j30 everywhere, then each

:!S;

•

•

•

, Xm.

DEVELOPMENTS 1970-1985

159

Thus, the only entire solutions come from affine subspaces, pro­ vided a suitable gradient bound holds. In the hypersurface case n=

m + 1, the condition reduces to a uniform gradient bound on the

defining function, and the theorem reduces to that of Moser [ 1 ] . One problem in dealing with minimal submanifolds o f high dimen­ sion and codimension is the pauc i ty of examples . In that respect, recent work of Harvey and Lawson [ 1 . 3 1 is of special interest. They show that a certain class of closed differentiai forms can be used to single out submanifolds of euclidean spaces that are area-minimizing in their homology class. A special case of their construction is the family of Kahler submanifolds of 4>. where Cn is identified with E2n. They give other explicit examples, along wtth the partial differentiai equations that must be satisfted by non-parametric submanifolds in each class. A a special case they recover the minimal cone of Lawson and Osserman referred to above, which is thereby not only a solution of the minimal surface equation, but absolutely area-minimizing with respect to its boundary. For absolutely area-minimizing submanifolds (and more generally, integral currents) , Almgren [ 4 ] has recently proved the long-sought sharp isoperimetric inequality, wi th the same constant as obtained for an open subset of euclidean space of the same dimension. Concerning complete m inimal submanifolds, there are sorne strik­ ing recent results of Anderson [3 ]. He shows that the main theorems concerning the structure and the Gauss map for complete minimal surfaces of ftntte total curvature, proved in Chapter 9 for surfaces in E3 and extended by Chern and Osserman [ I l to surfaces in En, have extensions to minimal submanifolds of arbitrary dimension and co­ dimension.

160

A SURVEY OF MINIMAL SURFACES

IV. Minimal subvartettes of a Riemanntan manifold The move from euclidean to more general Riemannian spaces as the amblent manifold represents undoubtedly the area of greatest actlvlty ln recent years. One of the main differences is that tt allows minimal submanifolds to be compact. From the vast amount of work that has been done, we select just a few results.

A. Existence theorems Ustng a combinatlon of the methods of geometrie measure theory, in particular as developed by Almgren and Yau

I l l.

[ 11.

and those of Schoen, Simon,

Pltts III has obtalned the most striking general existence

theorems, including the followlng: let M be a compact n-dtmenstonal Riemannian manifold of class Ck, where 3 � n � 6 and 5 � k �

oo,

Then there extsts a non-empty compact embedded minimal hypersur­ face of class Ck- l ln M. Using a somewhat different approach, Schoen

and Simon

Ill

were able to extend Pitts' results to n � 7, as well as to

prove an important regularity theorem for stable minimal hypersurfaces in arbitrary dimension. In both papers stability plays a fundamental role, as tt does in the paper of Schoen, Simon, and Yau

(1],

rouch of

which applies to stable minimal hypersurfaces in an arbitrary Riemann­ lan manifold. There are a number of more special but very important existence theorems. Among them we note the theorem of Lawson

{6),

that there

exist compact minimal surfaces of every genus in the 3-sphere, and a kind of dual result of Sacks and Uhlenbeck

I l l.

proving the existence of

generalized minimal surfaces of the type of the 2-sphere in a broad class of Riemannian manifolds. A later paper of Sacks and Uhlenbeck 1 2 1 roves existence of minimal immersions o f higher-genus compact sur­ aces. Results of a similar nature were also proved by Schoen and Yau

DEVELOPMENTS 1970-1985

l l l.

161

and applied to the study of three-dlmensional manifolds (see V.A.B.

below). B. Minimal surfaces in constant curoature manifolds Minimal surfaces in spheres continue to be studied extenstvely. For recent results and references to earlier ones, see the papers of Barbosa

[ 1 ] and Fischer-Colbrie I l l. There has also been sorne work on minimal surfaces ln hyperbolic space and ln compact flat manifolds. For the latter, see Meeks [ 1 .2], Mlcallef 121. and Nagano and Smyth { 1 ), and further references there. Among the topics covered, we may mention : a) an analog of Theorems 8. 1 and 8.2 stating that a compact minimal submanifold in the sphere must be a lower-dimensional great sphere if the normals omit a large enough set. The first theorem of that type is in Simons [ 1 ) ; for the best results to date and for earlier references see Fischer-Colbrie ( 1 ]. b) stability: again the first results are due to Simons ( 1 ), and then later, Lawson and Simons

[ 1].

We note in particular the fact that there

does not extst any stable compact minimal submanifold on a stan­ dard sphere. Extensions of the Barbosa-do Carmo Theorem [ l i have been made by a number of authors, in particular Barbosa and do Carmo

[2 ,31. Mort { l i. and Hoffman and Osserman 121. For a more detailed survey of stability, see do Carmo [1]. In V. below, we give a number of applications of stability results. c) the "spherlcal Bernstein problem": Hsiang [41 has shown that for

n = 4,5,6, there exist minimal hyperspheres embedded in sn, different from the great hyperspheres. C. Foliations with mintmal leaves Minimal surfaces have turned up in a somewhat surprising manner

162

A SURVEY OF MINIMAL SURFACES

as leaves of a foliation. In particular, a number of recent papers treat the question of characterizing those foliations such that there exists a Riemannian metrlc for which the leaves of the gtven foliation are ali minimal submanifolds (see Rummler ( 1 ), Sullivan [ 1 ], Haeflinger [ 1 ], and Harvey and Lawson 141 1. V.

Applications Q{ minimal surfaces ln recent years the theory of minimal surfaces has been applied to

the solution of a number of important problems in other parts of mathematics. We gtve here several examples. A.

Topology

ln a series of papers, Meeks and Yau [ 1-5 1 have shown how adept use of solutions to the Plateau problem in Riemannlan manifolds can lead to important consequences of a purely topologtcal nature. The most striking example was their part in a string of results whlch when combined led to the solution of a long-standing problem ln topology known as the Smith Conjecture (see Meeks and Yau [5J). Similar methods are used to obtain other purely topological results in the theory of 3-manifolds ln a recent paper of Meeks, Simon, and Yau

I l l.

In a somewhat different direction, Schoen and Yau ( 1 , 2] use the existence of certain area-minimizing surfaces to obtain topological obstructions to the existence of metrics with positive scalar curvature on a given manifold. The stability of the minlmizing surface plays a key role in the argument. More recently, Gromov and Lawson

I l l have made

a somewhat different use of stable minimal surfaces to study existence of metrlcs with varlous conditions on the scalar curvature. Those results are in turn used to derive topological properties of stable mini­ mal hypersurfaces in manifolds with lower bounds on the scalar cur­ vature. The existence of stable minimal 2-spheres is used to derive a

DEVELOPMENTS 1970-1985 homotopy result by J. D. Moore due to Lawson and Simons

[ 1 ]. Other results of a related nature are

[ 1 ) and Aminov I l l. Still other applications of

minimal surfaces to topology have been given by Hass Nakauchi

163

1 1 .3 1. and

[ 1 ].

B. Relattvtty By much more delicate arguments. but fundamentally an extension of those used in the papers referred to above. Schoen and Yau

( 3.41 were

able to prove a well-known conjecture ln general relativity . the "positive mass conjecture . .. Using results in these papers, they later obtained a mathematical proof of the existence of a black hole (Schoen and Yau

[5]). Other applications to relattvity are gtven by Frankel and Galloway

[ 1 ]. C. Geometrie inequalittes Let C be a Jordan curve ln En and let B be a closed set such that B and C are linked. (When n

=

3 . B would typically be another closed

curve . ) Gehrlng posed the problem of showing that tf the distance between B and C is

r.

then the length L of C satisftes L � 27Tr. Several

proofs of this inequaUty were given, including one ( Osserman

[ 1 3)) that

used the solution of Plateau's problem for C and the lsoperimetric lnequality on the resultlng minimal surface. lt was pointed out ln that paper that the same argument would yield

an

analogous result ln ali

dimensions. provided one has a parametrlc solution to Plateau's prob­ lem and a sharp isoperimetrlc inequality for the resulting surface. Neither of those results was available at the Ume, but they have slnce been proved by White I l l and Almgren

(4], respectively. ln the meantime

a somewhat different proof of Gehring's lnequality was obtained by Bombierl and Simon

1 1], also using minimal surfaces, and a strengthen-

164

A SURVEY OF MINIMAL SURFACES

ing of the result was glven by Gage [ 1 ). A generalizatlon of Gehring's lnequallty was subsequently obtained by Gromov

I ll.

p. 106, as part of a

major new approach to whole classes of geometrie inequal!ties. Solu­ tions to generaUzed Plateau problems are basic to Gromov's arguments.

VI. Harmonie maps The class of harmonie m appings has proved to be of increasing Importance in recent years. Harmonie maps have many ties to minimal surfaces. First of ali they represent a direct generalization. in that,

if a

mapf: M -"' N ls an isometrtc immersion qf one Riemanntan manifold into another, thenf ls harmonie if and only iff(MJ ls a minimal submanifold of N. When M ls two-dlmenslonal, the same result holds If is assumed to be conformai, rather than an lsometry. Sl ightly more generally, one has the following { Hoffman and Osserman [21):

letf:

M -"' N be a conformai map where the coriformalfactor p ls a smooth non-negattvefunction wtth p > 0 except on a set qf measure zero. Then if dim M = 2 , f ts harmonie if and only iff(MJ ls a generalized minimal submanifold of N; t.e . j ls an immersion almost everywhere .

with mean curvature zero; if dim M > 2. thenf ls harmonie if and only iff ls homothetie (p ls constant on each connected component of MJ

andf(MJ ls a minimal submanifold qf N. Given a mapf:

1

M -"' N, we may assume that N is embedded

isometrlcall y in sorne En. If dim parameters

u�> u2 on M, and thus get a representation off in the form

x(u1,u�. where x = (x1, (ilx,/ilu1) 1

M = 2, we may choose local isothermal

•

•

•

,

xnJ. We may form the functlons (f)�rW =

t{ ilx�rlilu2), as in (4.6), and we define q>{U

I t turns out that

iff ls a harmonie map, then

=

Ï q>�(tJ.

k�l

q> ls a holomorphie

function. {See Chern and Goldberg [ 1 ), §5. Sacks and Uhlenbeck [ 1 ),

i Prop. 1 .5 ,

and T. K. Milnor

1 li.J

Furthermore, under change of isother-

: mal parameters, q> behaves like the coefficient of a quadrauc differentiai

165

DEVELOPMENTS 19 70-1985

N ts harmonte if and only if the graph ofj is

mtntmal ln M

x

N

(Eells [ 1]).

We note also that if a foliation is deflned by a Riemannian submer­ sion]: M

--�>

N,

thenj is harmonie If and only If the leaves are minimal

ln M. ( Eells and Sampson Tondeur 1 1 J, )

Ill. For related results, see Kamber and

Finally, we note the basic theorem of Ruh and Vilms

I ll: let M be a

submanifold oj En. Then the generalized Gauss map g qf M lnto the

Grassmann! an ts a harmonie map if and only if M has parallel mean eurvature. In part!cular, g ts harmonie if M ts mtntmal. For further basic facts and references on harmonie maps. see Eells and Sampson

I l l and Eells and Lemaire 1 li.

Among the Important recent applications of harmonie maps that have been made, we mention:

A. Hildebrandt, Jost, and Widman [ 1] proved a Liouv!Ue-type the­ orem for harmonie maps, and by applytng lt to the Gauss map via Ruh and Vilms, they obtalned the Bernstein-type theorem for arbltracy dimension and codimension cited in

III

above.

B. Sacks and Uhlenbeck ( 1 ,2) have proved existence theorems for harmonie maps together wlth arguments concerning conformai struc­ ture to get a conformai harmonie map w hich ls thereby a minimal surface. C. Harmonie maps have recently been studied by physictsts {see

166

A SURVEY OF MINIMAL SURFACES

Misner [ 1 ] for a discussion of their relevance as models for physical theories}. In particular, a number of physicists have studied the ques­ tion of characterizing ali harmonie maps of the standard 2-sphere 52 into complex projective space cpn (Din and Zakrzewski [ 1 , 2 1. Glaser and Stora [ 1 ]). By vlrtue of the result mentioned above. that ali such maps are conformai, the question is equivalent to that of finding ali minimal 2-spheres in CPn. Inspired in part by the work of the phys iclsts, Eells and Wood [ l i gave a complete classification of harmonie maps of a class of compact Riemann surfaces, including the sphere, into CPn. A dif­ ferent approach to the same problem was given by Chem and Wolfson [ 1].

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