A survey of minimal surfaces

(S . 15)

E

- d,

d

± i, then equation

W e consider two cases. F i rst, if c

(5. 10)

re ­

d u ces to a=

(5. 16) Thus the t r ansfo rm ation

(5. 7) is

or a reflection: w = z, or w = z,

± i<jJ 3 whi ch i s equiv alent to 13

.±

z. Th us the case

c =

lutions

(5. 14).

(5. 13)

and

0, b =

+

l.

either the ident i ty transformation and equation

(5. 16)

i mplies

+ if

4 an analytic function of z or

i corresponds precise! y to the spec i al so ­

In the second case, c

,P .±

i, w e m ay write e qu ation

(5 . 15) i n

the form

d ,p o .

(5. 17)

Thus, each of the factors on the l eft i s di fferent from zero. In par­ ticul ar, the function <jJ 3 - i

� 3 - i� 4

=

eH(w)

for sorne entire function H( w). By

(5. 17)

we have

A SURVEY OF MINIMAL S URFACES

- de-H ( w )

J

1and combining these two e quations yields

t•

5 .18)

�3

=

} ( eH ( w) _ de-H ( w ))

'

�4

4

= ( eH ( w ) + de- H ( w 'l)

We can thus describe explicitly in the case

n

4 al i solutions

.of the mini m al surface equation which are valid in the whole x 1 , x 2 jp lane. They are either of the speci al form (5 . 13) or (5. 14), or el se

1 they m ay be obtained by m aking an arbitr ary linear transformation of the form (5 .7), and inserting the constant d

=

1 + (a - i b) 2 in

(5.18) together with an arbitrary entire function H( w) . Let us give a simple ill u stration. We choose a

=-

0, b

=

2 in

1(5.7), and H( w) = w. Then

� 3 � ( e w - 3e- w), �4 = }< e w + 3e-W),

x 1 = u l ' x2 = 2u2 , x3 x4

==

J �3 dw = % cos u2( eu 1 - 3 e- u 1) Re J �4 dw = % sin u2( eu 1 - 3e- u 1 )

Re

=

-

•

W e thus obtain the surface i n non-parametric form: 3 e- x l)

which, as one may verify by a direct computation, provides a glob 1 al solution o f the mini m al surface equ ation for

n

=

4. This surface

[ will be a useful example in connection with our general di scus sion later on. • (See p . 124 . )

*See Appendlx 3 , Section 2 .

P ARAMETRIC SURFA CES:

43

GLOBAL THEORY

§6. Parametric Surfaces: Global Th eory In the previous sect ion we were able to obtain global result s because o f the speci al circumst ance th at we had a global p aramet­ rizati on in terms of two of the coordinates x l ' x 2 . In the general case we have a surface covered by neighborhoods in each o f which a p arametrization of the form considered in section 1 i s given. In o rder to study the whole surface, we have to first give precise defi­ nitions. We begin by recalling sorne facts about differentiable mani­ folds. DEFINITION S. An n-manifold i s a Hausdorff sp ace, each point

of which h as a n eighborhood homeomorphic to a domain in En .

An atlas A for an n-manifold M i s a collection of triples (Ra ,

Oa , F ), where R is a domain in En , O is an open set on M, a a a F is a homeomorphism of R o nto O ' and the union of ali the a a a O equ als M. E ach triple is called a map. a A m anifold M is ori en table if it possesses an atlas for which e ach transformation F - 1 o F{3 preserves orientation wherever i t a i s defined. A n orientation o f M is the choice o f such an atlas. A cr-s tru cture on M is an atlas for which F - 1o F {3 f cr a wherever it i s defined. A conforma! s tructure o n M i s an at las for which F;- 1 o F {3 is a confo rmai map wherever it is defined. REM ARK . By "conformai, " we mean "strict!y conform ai . "

Th us a conform ai structure o n M automatically provides an orien­

t ation of M.

-

Let M be an n-manifold with cr-structure A, and M an m-mani-

fold with a cr-structure A . A map t : M map,

denoted f f CP, for

wherever it i s defined .

p

-

-7

M will be called a CP-

� r, if ea�h map

F{3- l

a

foF a

f

CP,

44

A SURVEY OF MINIMAL SURFA CES

Let us note, in p arti cul ar, that for ali =

E",

r,

E"

h as a canoni c at c r-structure

defined by !etting A consist of the single triple Ra

=

0a

Fa the identity m ap.

D E FI N I TIO N . A Cr-surface S in E" is a 2 -manifold M with a

c r-structure, to gether with a c r-m ap x ( p) of M into

E" .

Let S be a C r -surf ace in E", A the c r -s tructure on the asso­ c i ated 2-m anifold M , Ra a dom ain in the u-pl ane, and Rf3 a domain in the Ü- plane. Then the composition of Fa with the m ap x ( p) is a

m ap x ( u) of Ra into E" whi ch defines a local surface in the sense of section 1. The corresponding m ap x ( Ü ) of R f3 into gn defines a loc al s urface obtained from x ( u) by the change of parameters

u(Ù)

=

Fa- l o F {3 . Th us ali local properties of surfaces which are

i ndependent o f parameters are weil defined on a global surface S given by the above definition. In particular, by a point of S we

shall m ean the p air (p 0 , x ( p 0 )) where p0 is a point of M, and we may speak of S being regular at a po int, or of the tangent plane and the mean curvature vector of S at a point, etc .

The global properties o f S will be defined simply to be those

of M. Thus S will be called orien table if M is ori entable , and an

orient ation of S is an orient ation of M. Simil arly for topological

p roperti es of S : S compact, connected, simply conn ected, etc. We sh all now

rn

ake a convention which we shall adhere to

throughout this paper. Ali surfaces considered will be connected and orien table. Obviously, if a surface is not connected one can

consider sep arately eac h connected component, which is also a surface. As for non-o ri ent able surfaces, they are certainly of inter­ e st, and in p arti cul ar in the case of mini m al su rfaces, since they ari se both analyti c ally as element ary surfaces given by e xpli cit

45

P ARAMETRIC SURFACES: GLOBAL THEORY

formulas, such as He nneberg' s surface (see for exam ple D arboux

[ 1] , §226), and physically as so ap-fi lm surfaces of the type of the Mo bi us strip, bounded by a simple closed curve (see for ex ample Courant [2] , Ch apter IV) . However it is an element ary topological f act that to e ach non-orientable c r- m anifold

M -

an o rientable cr -rn anifold

M

and a cr -map g :

there corresponds

M M, -

__,

su ch th at g

i s a local diffeomorphi sm, and the i nverse i m age of each point of

M

M. -

consi sts of two points of

face x ( p) : where x ( p )

M --> =

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

<

l,

:

D

�

E", wh ere D i s ei th er th e

or th e entire ( -p lane.

Proof: Let S be defined by x ( p) : M

�

E" .

By L em m a 6 . 2 , M

is not co mpact , and by the Koebe uniform i z ation theorem ( see , for examp le, Ahlfors and S ario

[ l] ,

Ill , 1 1 , G) M i s confo rm ally e quiva­

lent to either the unit disk or the plane. The composed m ap

D

�

M

�

E" gives the result.

t

Fin ally, let u s recal l sorne terminology wh i ch distinguishes the t wo c ases in Lemm a 6 . 3 . A 2-m anifold M with a confo rm ai stru cture i s c al l ed hyperboli c i f there exists a non-const ant negative subh armon ic fonction on M, and parabolic otherwise. The fonction R e l (

-

unit disk

1 (1

<

l

ll

sho w s that the

is hyp erbo lic, and it is not h ard to show that the

entire p l an is p arabolic. There are m any interrel ations bet ween the conform ai structu re, the topolo gical structu re , and the Rie manni en structure of a surface. F urthermore, in the c ase of minim al surfaces, we shall see th at e ach of these stru ctu res p l ay s a role in studyin g the geometry o f t h e s u rface.

_

§7.

Minimal Surfaces wi th Boundary. In th i s s ection we shal l investigate sorne properties of minimal

s u rfaces with bound ary. We do not go into det ail on Pl at e au ' s prob­ l em fo r the reasons mentioned in the introdu ction , but we i nclude a brief discu ssion i n the context of the present p aper.

DE FINITION . A sequence of points pk on a m anifold M is divergent if it h as no points of accumu l at ion on M. I f S is a m i nim a l su rface defined by a m ap x ( p) : M -+ E", the boun dary values of S are the set of points of the form li rn x ( p ) , k for all divergent se qu ences pk o n M. REM A RK . If M is a bounded domain in the p l ane , then a se­ quence p

k

in M is div ergent i f and only if it tend s to the bound ary

I f x( p) extends to a continuous m ap of the do sure

M,

th en the

b oundary values of S are the i m age of the bound ary of M . LEM M A

7 . 1.

Ev ery minimal surface li es i n the con vex hull o f

i ts boundary values. Proof: Let S be a mini m al surf ace defined by x ( p) : M -+ E". Suppose t h at the bound ary v alues of S lie in a h al f sp ace n

L ( x) =

� k=

The function h ( p)

=

a x - b :::; o . k k 1

L( x ( p) ) i s h armonie on M, and by the maxi ­

mum pri nci p le, h ( p) 5: 0 on M.

Namely, if s u p h ( p) = m, we m ay

c hoose points p k such t h at h ( p k) -+ m.

lf the pk h ave a po i nt of

accum u l ation in M , then h( p) wo uld assume its maxim u m at t h i s point, h ence be constant. B ut choo sing an arbitrary divergent seq u en ce qk we wo uld h ave h ( q ) k

=

m for

�11 k,

) � 0,

and l i m h( q

54

A SURVEY O F MINIMAL S U RFACES

m < O. On the other hand, i f pk i s divergent, then again limh ( pk) :5 O. Thus L(x(p)) :5 0 on M, so that S lies i n the h alf-space L(x) :::; O. But the convex hull of the boundary values bence

m

is the intersection of ali the half-spaces wh ich contain them, and

S

lies in thi s i ntersection.

t

Let x(u) be a minimal surface in isothermal pa­ rameters defined in a plane domain D. Then x(u) cannot be con­ stant along any fine segment in D. Proof: Since the map x(u) i s one-to-one in the nei ghborhood LEMM A 7 . 2 .

of a regular point, and since the branch points, i f any, are i solated, the result follows im mediately.

t

Let x(u) be a minimal sur­ face in isoth ermal parameters defined in a semi-disk D : 1 ul < E , u 2 > O. Suppose there exists a line L in space such thal x(u) .... L when u 2 .... O. Then x(u) can be extended to a generalized minimal surface defined in the full disk 1 ul < E . Furthermore this extended surface is symmetric in L. Proof: By a rotation in En we may suppo se that L is given by the equations xk 0, k 1, , n - l. Then the functions xk(u), for k 1, . . . , n- 1, m a y be extended by setting xk(u 1 , 0) 0, xk (u 1, u 2) = - xk (u 1, - u 2). These extended functions are harmonie LEM MA 7 .3. {Reflection principle).

=

=

. . .

=

=

in the full disk, by the r eflection principle for harmonie functions. T hus the functio n s

k=

l,

. . .

,n

-

1

are analytic in the full disk and are pure i maginary on the real axis.

MINIMAL

SURFACES

WITH

BOUNDARY

55

By the equation

4>; we see that

4>;

n- 1 = -

I

k=

1

4>;

extends continuously to the real axis and has non-

negative real values th ere. lt follows that

4> n

extends continuous­

1

xn a xn ;au2 0

ly to the real axis and has real values there. By i ntegration, extends continuously to the real axis, and satisfies there. If we then set

xn (u 1, u 2) xn (u 1, - u 2 ) =

d i sk we obtain the desired result.

=

in the lower h alf­

t

Let x(u) be a minimal surface in isotherme! pa­ rameters defined in a disk D. Then x(u) cannot tend to a single point along any boundary arc of D. Prool: If it did, we could apply Lemma 7.3, after a preliminary LEMM A 7.4.

conformai map of D onto the upper half-plane, and extend the sur­

face over a segment of the real axis. lt would then be constant on this segment, contradi cting Le mm a 7. 2.

t

We now state the fundamental existence theorem concerning th exi stence of a minimal surface with prescribed boundary. TH EO R EM 7 . 1 (Douglas

[1] ). Let 1 be an arbitrary jordan curve in E". Then there exists a simply-connected generalized* minimal surface bounded by r. W e content ourselves here with an outline of the proof, based on i m portant modifications due to Courant [ 1,

2].

We refer also to

the versions given in the books of Rade [3] , Lewy

hedi an [ 1].

Fi rst of

ali,

*See Appendlx 3.

[21 , and Gara­

let us give a precise statement of the concl usion.

Section 1.

1

56

A SURVEY O F MINIMAL SURFACES

Let us use the following notation:

D is the unit disk: u f + uff < 1 D i s the closure : uf + uff :5 1

C i s the boundary :

uf u] +

=

l.

The result is th at there exists a map

x( u) x( u) x(u)

i) i i) i ii)

x(u) :

D

....

En, such th at

is continuous on D , restricted to D is a minimal surface, restricted to c is a homeomorphism onto r .

This mapping i s obtained by the method o f minimizi ng the Di richlet i ntegral. To each m ap x(u) < C 1 in D, we denote by

i ts Dirichlet integral. In order to single out a suitable class of m appings in which the Dirichlet integral will attain a minimum, we consider monotone maps of C onto r ; that is, maps such th at if c is traversed once in the positive direction, then r is traversed ont:e also in

&

given direction, although we allow arcs of C to map

onto single points of r.

To a given Jordan curve r we associ ate the class H of m aps

lx(u) :

a) b) c) d)

D

...

En h aving the following properties:

x ( u) x(u) �( x) x(u)

is continuous in D ; 1 in D ; f. C <

oo

restricted to c is a monotone map onto r.

t may well be that H i s empty. However, we m ake the following assertions. 1.

If r i s piecewise differentiable (in particular, a polygon),

57

MINIMAL SUR FACES WITH BOUNDARY

o r i f r is even rectifiable, then H is not empty. 2. Whenever H i s not empty there exists a map y ( u) that :i) ( y) :S. � ( x) for all x ( u) f H .

<

H such

3. This map y ( u) s atis fies the conditions i), ii), iii) above fo

a solution of our problem. We now outline the proof o f these assertions. 1. If we l et s be the parameter of arc length on r, L its total

2rr s/L, then the m ap x ( ei t ) : C .... r c an be extend­ ed by the Poisson integral applied to each coordinate xk( e i t ) to a

length, and t

m ap o f i5

d v

-+

=

En which will be in H.

2. Let d inf :D ( x) for x < H, and let x"'(u) < H sueh th at :D ( xv) .... d. For e ach , define xv(u), by !etting x� ( u) be v

=

the h armonie function in D h aving the same boundary values as x�(u). Then x Y(u)

:î) ( X Il) .:5 :D ( x v), by the pro perty o f harmonie funetions to minimize the D irichlet integral <

H, and bence d .:5

with given boundary values. Now fix three points on C and three points on r' and let H be the subset of H for which x ( u) maps the former onto the l atter in a given order. By a linear fraetional transformation o f D, we can obtain from e aeh i Y( u) an i V(u) < Îl, and � ( � v)

:i) ( XY), One then proves the fundamental lemma that

boundary m aps x v : C

....

r must be equicontinuous. We ean there­

fore find a subsequenee of the iV whieh converge uni formly on C, bence in the closed disk D, and the limit will be a m ap y ( u) with :D (y) d.

<

Îl,

3 . The m ap y ( u), being the uni form limit o f harmonie functions

in D is also h armonie in D. One shows by a variation & argument that such a map must define a generalized minimal sur face in D. Since the bound ary eorrespondence is monotone. the only way it

58

A SURVEYOFMINIMAL SURFACES

c ould fail to be one-to-one i s if a whole boundary a rc m aps onto a point, but by Lem ma 7 . 4 this i s impossible. Th us the map y ( u) s ati sfies conditions i), ii), iii) and solves the problem. For the final st ep in the proof of Theorem 7 . 1 , Dougl as shows t h at i f r i s an arbitr ary Jord an curve, it can be approxi m ated by a sequ ence of polygon s , and the corresponding sequence of mini mal surfaces will tend to a surface which again s at i sfies i ) , ii), iii). +

REM A R K S · 1 . The re st riction that the surfac e S be si mply­ connected is neither necessary, nor from a certain point of view, n atural. ln fact, i f one proceeds to construct minimal surfaces physically, using a piece of wire and soap solution, one finds i n many cases surfaces o f higher genus and connectivity, including, among the si mplest, the Mobius stri p. For further di scussion of this theory we refer to the fundamental paper of Dougl as [ 2] , and the book of Courant [ 2] . We l imit ourselves here to mentioning two of the most striking f acts concerning the order of compl ications t h at m ay arise. First of ali there exist rectifi abl e J ordan curves r which bound a non-denumerable number of d istinct mini m al surfaces 1(Lév y [ l] ) ; s econd, there e xi st rect ifi able J ordan curves r for which a surface of m inimum area bounded by r must have i nfinite connectivity (Fleming [ 1 ] ). 2. In the past few years an entirely new appro ach to Plateau ' s ,problem b as been instit uted, i n which one seeks a minimum of

l area among a very general c l ass o f objects,

instead of rest ricting

�he competition to surface s , and then shows that there is a mini­ b izing object whi ch is in fact

a surface. In a pioneering work o f

R eifenberg [ lL the class o f objects cons idered are comp act subets h aving a given bound ary in a certain sense, and the 2-dimen-

59

MINIMAL SURFACES WITH BOUND ARY

s i on al H ausdorff m e asure is minimi zed. Two m ajor advantages of this method is t h at it c an be applied to higher di mensional sub­ v arieties, in which case one minimizes the m-dimensional Haus­ dorff m easure, and that in the 2-dimensional case one obt ai ns not o nly a generalized m i n i m al surface, but in f act a regul ar one. The l atter follows from the combi ned results of Reifenberg

[ 2] ,

2,

3] ,

an

4, p. 70 of [ 1] together with [ 3]. Flem­

in particular, from Theorem i ng

[ 1,

b asing on the methods of Federer and F leming

[ 1] ,

shows

the exi stence of a regular oriented m i n i m al surface bounded by an oriented rectifi ab le J ord an curve

'

See also t he discussion in

[ 3] .

Almgren

3.

r.

One of the major open problems concerning Theorem 7. 1 i s

whether there, in fact, e xi sts a regular simply-connected mini m al surface bounded by an arbitrary J ordan curve

r.• Th is w as actual­

l y the way P lateau 's p roblem was ori ginal ly envis ageâ , and in thi s form it is still unsol ved. In the following lemm as we giv e sorne in­ formation concerning c ases in which the surface constructed i n Theorem 7 . 1 must, in fact, b e regul ar. L EM M A 7 . 5 (Rado

[31 ,

h armonie in the uni t disk

III . 7 ) . Let h ( u 1 , u ) he non-constant 2

D,

continuous in the closure

that the gradien t of h vanishes at a poin t (a , a ) in 1 2 h(a , a ) 1 2 of

D

>

D.

Suppose

Th en if

b, there are at !east four distinct points on the boundarY1

where h takes on the value b.

Proof: L et h

fi.

tl be a component of the set of points in ii where

b. Th en at bound ary points of

S ince h

/=

b, e ach component

bound ary of

D with

h > b.

tl interior to D we h ave

h

=

b.

tl m u st have bou n dary points on the

Simil arly for each compon ent of the set

where h < b. In the nei ghborhood of the ·poi nt ( a , a ) there are at 1 2 •see Appendtx 3, Section

1.

60

A SUR VEY OF MINIMAL SURFA CES

l e ast two components of the set h > b, and two of the set h < b. If the former were part of a single component � in which h > b, then we could find an arc j oining the� which could be completed to a Jordan curve in D through (a 1 , a 2) , and 'thi s curve would com­ p letely enclose a domain in wh ich h < b, contradicting our previous conclusion that such a dom ain must go to the boundary of D. T hus

there are at l e ast two distinct components where h > b and two

where h < b, and e ach of these intersect s the bound ary at points where h

1= b.

But if there were at most three bound ary point s where

h "" b, there would be at most three complement ary arcs on e ach of which h > b or h < b, bence at most three components of sets

h

>

b and h

<

b, which contradicts what we h ave shown.

•

L EM M A 7 . 6 . L et r be a Jordan curve in En, and let x ( u) be

a simply-connected generalized minima! surface bounded by r in the sense of Theorem 7. 1 . Then ei ther x ( u) is in !act a regular

minimal surface or else r has the property that for sorne point in

En, every hyperplane through this point intersects r in at /east four dzstinct poin ts.

Proof: We u se the not ation of the proof of Theo rem 7 . 1 . Thus x ( u) is continuous in

D,

and defines a generali zed mini m al surface

i n D. Suppose that at sorne point ( a 1 , a2) of D, x ( u) fails to be regul ar. Then at this point all the functions

must vanish. If L( x)

=

� ak x k

+

c·"" 0 is the equ ation of an arbi­

t r ary hyperpl ane through the point x ( a 1 , a 2) , then h ( u 1 , u2 )

=

( x ( u 1 , u 2)) is a function sati sfying the hypotheses of Lemm a 7 . 5 ,

61

MINIMAL SURFACES WITH BOUNDARY

with b on

=

O. It follows that there are at le ast four distinct points

r where L ( x)

0, which proves the lemm a.

=

This lem m a shows that unless

•

r is fairly complicated, the

s urface that it bounds has no branch points. In many specifie c ases we m ay assert the existence of regul ar sim ply-connected s urfaces boun ded by

r, thus giving in those c ase s a complete so­

l ution of P l ateau ' s problem. TH EO R EM

7 . 2 .• Let D 1 be a bounded convex domain in the

( x 1, x ) - plane, and let C 1 be its boundary. Let gk( x 1 , x ), 2 2 3,

. .

k

=

. , n, be arbi trary continuous fun etions on C 1 . Th en there ex­

ists a so lution

of th e minimal surface equation

(2 . 8) in D 1 , such that fk( x1 , x2 )

takes on the boundary values gk( x l' x ) . 2 Proof: The boundary C 1 of D 1 is a J o rdan curve( even recti ­ f i able) and the functions gk( x 1 , x ) define a Jordan curve 2 which proj ects onto C 1 . ous rn ap x( u)

:

D

->

By Theorem

r in E "

7 . 1 , there exists a continu-

E " of the unit disk

1 u 1 ::;

l whic h de fines a

m i nim al surface in the interior D, and t akes the boundary C homeo morphi cally onto

r. It follows that the m ap (u l' u 2 ) ... ( x l , x ) i s 2

a continuous m ap of

D

whi ch is h armonie i n t h e interior an d m aps

C homeomorphically onto C 1 . By Lemm a

7 . 1 , the im age of D must

lie in D 1 . Furthermore , the J acob ian ne ver vanishes in D. N ame ly , if at sorne point (a 1, a ) in D the J acobi an were to vanish, th en 2 the rows of the J acob ian m atrix would be linearly dependent at this point, i . e . , for suit able constants ,\ 1 , •see Appendtx 3. Section 5.

À::!•

we would have

62

A SUR VEY O F MINIMAL SURFACES

o .

0,

Th en the function h ( u l ' u ) "" A 1 x 1 ( u 1 , u 2 ) 2

+

A2 x 2 (u 1 , u ) wou l d

s at i sfy t h e hypotheses o f Lemma 7 . 5 and it would fol lo w th at

A 1x1

+

A2 x 2 would take on the s ame value at fou r d i s t inct points

o f C 1, which is impossible by the convexity of D . We therefore

1

conc lude that the m ap ( u 1 , u ) ... ( x 1 , x 2) is a local di ffeomorphism

2

i n D , and since it maps C h omeomorphi cally onto C , i t i s a g l o ­ 1 b al di ffeomorphism o f D onto D . 1

xk , k

3,

.

.

We m ay therefore express the

. , n a s funct ions of x l ' x2, and these functions will

s at i s fy the concl usio n of our theorem .

t

Let us note in conclu sion th at except for the proof of Theorem

. 1 , ali the results in thi s section h ave been adapted from th e treat ­ ment f o r the c as e

n

=

3

i n t h e book o f Rado

[3].

3 PARAMETRIC SURFACES lN E . THE GAUSS MAP

§8. Parametric Surfaces in

E 3 . Th e Gauss Map.

In everything we have clone up to now, there is e i ther no d i f­ ference at a l l , or e l s e very little difference, between the c lass i cal c ase of three dim ens ions and the case of arbitrary n.

We shal l

now discuss severa! re sults whi ch e i ther have not been extended to arbitrary n , or else require a much more e l aborate d i scussion to d o so. W e s t art with the import ant observation that for the c as e n =

3

w e are able to describe explic itly ali so lutions of the equation

(8 . 1) L EM M A 8 . 1 . Let

D be a domain in the cornplex (,-plane, g ( (,)

an arbi trary meromorphic lune tion in D and f( (,) an analytic lun e

l

1

tion in D having the property that at each point wh ere g ( (,) has él pole of order m, f( (,) has a zero of order at !east 2m. Then the functions cf>

(8 . 2)

3 = fg

will be analytic in D and s atisfy (8 . 1) . Conversely, every triple o f analytic functions in D satisfying (8. 1 ) may be represented in the form ( 8 . 2), except for

c/> 1

i cf> , 2

c/> 3 � 0.

Proof: Th at the functions (8. 2) satisfy (8. 1) is a di rect cal cu - ' l ation. Conversely , given any solution of (8 . 1 ), we sf't

(8.3)

If we w r i t e (8. 1) i n the form

(8.4)

64

A SU R VEY OF MINIMAL SURFACES

we find

(8.5)

2 = -

.

= -

[� 2

.

Combining (8 .3) and (8 .5) yie lds (8. 2). The condition relating the zeros of

1

and the poles of

�

must obviously hold , since otherwise

by equation (8 .5),

=

0 , which is the exception al cas e mentioned.

Every simply-connected minimal surface in E 3 can be represented in the form LEM M A 8 . 2.

k

(8 .6)

=

1, 2, 3

where the E 3 where D is either the disk or the plane, the coordin ates xk being h armonie in ' If we set ·

:

·

then these functions will be analytic and (8.6) will hold (the inte-

P ARAME TRIC SURFACES

IN

E 3 . THE GAUSS MAP

65

tegral being independent of path). F or a generalized minimal sur­ face the equation (8. 1) must hold and by Lem ma 8 . 1 we have the r epresentation (8 . 2). The surface will fail to be regul ar if and only if all the when pole.

f

==


v anish simultaneously, which happens precisely

0 where

•

g

i s regular or when

fg 2 0 ==

where

g

has a

Let us note that representations of the form (8 . 2), (8.6) were first given by Enneper and Weierstrass, and have played a major r ole in the theory of minimal surfaces in

E 3.

For one thing, they

allow us to construct a great variety of specifie surfaces having i nteresting properties. For example, the most obvious choice :

f 1, g ( () =

=

(, le ad s to the surface known as Enneper' s

surface.

M ore important, this representation allows u s to obtain general theorems about minimal surfaces by translating the statements in­ to corresponding statements about analytic functions. ln order to do th is we must first e xpress the basic geometrie quantities asso­ ciated with the surface in t erms of the functions

f, g .

F ir st o f ali, the tangent plane i s generated by the vectors

It follows that (8.7) where

F urthermore

66

A SURVEY OF MI NIMAL SURFACES

and substituting in (8.2) we arrive at the expression

From thi s it fol lows that

and

(

(8.8)

! gl + 1

---,

2I m l g l

2 R el g l '

! gl 2 + 1

is the unit norm al to the surf ace with the standard orientation. Now given an arbitrary regular surface x ( u) in E 3, one de­ fines the Gauss map to be the map

(8.9 )

x ( u)

-.

ax aul

N(u) 1

é1..?!.

au l

x

x

é1..?!.

au 2 ax 1 au2

of the surf ace i nto the unit sphere.

LEMM A 8 . 3 . If x( ( ) : D -. E 3 defines a regular minimal sur­

face in isothermal coordinates, then the Gauss map N( () de/ines a complex analytic map of D into the uni t sphere considered as the Riemann sphere. Proof: Formul a (8 .8) compared with formul a (6. 1) for stereo­ -> N( () followed

graphie proj ection shows that the G auss map x ( ()

3 P ARAMETRIC SURFACES lN E : THE GAUSS MAP

67

by stereographie projection from the point ( 0, 0 , 1) yields the mero·

•

morphic function g( �).

Let us note th at in general the G aus s map cannot be defined i f a surface is not regul ar. However, for a generalized mini m al sur· face the G auss map extends continuously, and even an alytical ly, t o the branch poin ts, the normal N being given by the right·hand side of (8.8). LEM MA 8.4. L et x( �) : D -+ E 3 define a generali zed minimal

surface S, where D is the entire �-plane. Then either x ( �) lies on a plane, or else the normals to S take on ali directions with at most two exceptions. Proof: To the surface S we associ ate the function g ( �} which { ails to be defined only if 4> 1 = i 2 , 4> 3 O. But in thi s c as e x3 i s constant and the surface lies in a plane. Otherwi se g ( �) is m eromorphic in the entire �-p lane, and by Pic ard's theorem it either t akes on ali v alues with at most two exceptions, or else is constant. But by (8 .8) the same alternative applies to the norm al N, and in the l atter case S lies on a plane.

•

LE M M A 8 . 5 . Let f( z) be an analytic function in the unit disk

D which has at most a finite number of zeros. Then there exists a

divergent path C in D such that (8. 10)

f \ f( z) \ \ dz \ c

< ""' .

Proof: Suppose first that f( z) /. 0 i n D. Define z w F(z) f( �) d� . =

=

1 0

68

A SUR VEY O F MINIMAL SURF ACES

Th en F ( z) maps 1 zl < 1 onto a Riemann surface which h as no bran ch points. If we let z funct ion satisfy ing G(O)

G( w) be that branch o f the inverse

0, then since I G( w) l < 1, there is a 1 argest d isk 1 wl < R < "" in which G( w) is defined. There must =

then be a point w0 with l w0 1 "' R such that G( w) cannot be ex­ tended to a neighborhood of w0 . Let L be the line segment w t w0 , 0 :S t < 1, and let C be the image of L under G( w). Then C must be a divergent path, since otherwise there would be a se­ quence t

...

1 such that the corresponding sequence o f points z n on C would converge to a point z0 in D. But then F( z0) w0, and since F '( z0) f( z0) 1= 0, the function G( w) would be extend­ n

=

=

able to a neighborhood o f w0 . Thus the path C is divergent, and we h ave

/_ l f{ z) l l dz l = j0

1

c

l f( z) l l �� ! dt = dt

10

1

1 q_'!:. l dt = R < dt

e.o .

This proves the lemma if f( z) h as no zeros. But if it has a finite n umber o f zeros, s ay o f order v k at the points zk , then the func­ tion f( z) n

(� - )11k 1

z - zk

never vanishes, and by the above argument there exists a divergent path C such that fe l f 1 ( z) l l dzl < oo. But l f( z) i < l f1 ( z) l through­ out D, and (8 . 1 0) follows. t THEO R EM 8. 1. L et 3

in E . Then either where dense.

S

S

be a complete reguiar minimal surface

is a plane or else the normals to

S

are every­

3 P ARAMETRIC SURFACES IN E : THE GAUSS MAP

Proof:

69

Suppose that the normals to S are not every where dense.

Then there exists an open set on the unit sphere wh ich is not inter­ sected by the i m age of S under the G auss m ap . By a rot at ion in sp ace we m ay assume that the point (0, 0, Then the unit normals

N

�

(N 1, N 2, N 3 )

1)

is in thi s open set.

s at i sfy

s arn e i s true of the uni versai covering surfac e b e represented in the form x ( () : t he unit disk. But

N 3 :S Tf < 1 n ot vanish. But

and by Lemma

D

D

...

E

3

N 3 :S:'I < 1.

S

where

o f S, which m ay

D

i s the plane or

(8.8), f(() can-

cannot be the unit disk, be c ause by

1 g( () 1 s M < oo, and since S i s regu l ar by (8.7) the length of any p at h C would

8 .5

The

there would exist a divergent path

C

be

for which

this integral converges, and the surf ace would not be complete.

D

Thus

is the entire p l ane , and since the normals om i t more than

t wo points, it follows from Le m m a

8.4

that

S

must lie on a plane.

The s ame is then true of S, and since i t is complete, S must be the whole pl ane.

+

Let us note that Theorem

8. 1

h as as an i m me di ate consequence

the theorem of Bernstein. In fact, a non-pararnetric m i n i m al surface in

E

3

defined over the who le x 1, x 2 -plane i s a complete regular

surface whose normal s are contained in a hemi sphere, hence it must be a plane. On the other h and, Theorem

8. 1

l eads naturally to the qu estion,

given a complete m i n i m al surface S , not a pl ane, wh at can be sai d about the s i ze of the set of points on the sphere omitted by the G auss m ap of S? Theorem

8. 1

tell s u s that, at le ast if S i s regular,

70

A SURVEY OF MINIMAL SURFACES

the omitted set c annot contain a neighborhood of any point. A much stronger result can be obtained by introducing th e notion of sets of

Iogarithmic capacity zero. We may define these to be closed sets on the sphere whose complement is parabolic in th e sense of con­ form ai structure. (See the discussion at the end of section

6. )

lt

is weil known that p arabolicity i s e quival ent to the non-existence of a Green' s function. ( See, for example, the book of Ahlfors and Sario [ 1] , IV L EM M A

6

and IV

22.)

8 . 6. Let D be a domain in the complex w-plane. The

complement E of D on the Riemann sphere has Iogarithmic capac­ i ty zero if and on !y if the fun ction log ( l + \ w\ 2) has no harmonie majorant in D. Prao!: Suppose first th at there exist s a harmoni e function h(w) in D such th at log( l + \ w\ 2 ) S h( w) every where. Th en - h( w) is a negative h armonie function in D, and hence E h as positive log­ arithmic capaci ty. Conversely, if E has positive logarithmic cap aci ty, th en for any point w0 in D the re exi sts a G reen ' s function G( w, w 0 ) with pole at w0 . By defini tion , G( w, w 0) + log \ w- w 0 \

h(w), where h ( w) i s h armoni e in D, and G( w, w 0) > 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 <

l. We have the associ ated function g ( ()

which i s analytic in D , and by the regularity of

S� /(() f. 0

in D.

Suppose n o w that E did n o t h ave capacity >: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 <� N constant � g con­

stant � g '

=

O. But g ' is an alytic and either has i solated ze-

ros or is identically zero.

•

Let us note the import ant formula (9.5)

which may be verified by a direct computation from (8. 7) and (9. 4). Actually it is proved in differentiai geometry that formul a (9.5) holds for an arbitrary surface in terms of isothermal parameters, g 1. ]. 2 >.. 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
and by its definition (8.3) the value of g is unchanged. Thus, if we are given an arbitrary minimal surface we may replace (9.6) by (9 . 12)

x(p) MS

s tereographie

G au s s m ap

� _......,!;P..:;ro�i..;;.e.;.ct;;;;.io;;;.;n:.;.__ w-plane ....,

where the composed map (9. 13)

g ( p) : M

..

w-plane

is a meromorphic function on M . The total curvature of S is still

the negative of the area of the image on 2:. ln order to study the

nature of this map for complete minimal surfaces, we prove a series of lemmas, of which the first is a generalization, and in a certain respect, a clarification of Lemma 8 .5. LEMMA 9 .2. Let D be a plane domain and gi j A2oi j a Rie­ mannian metric in D, where À = À ( z) f c2, such that D is complete wi th respect to this metric. If there exists a harmonie function h(z) in D such that

(9. 14)

log À( z) < h ( z)

3

SURFACES IN E

79

: GAUSS CURVATURE AND TO TAL CURVATURE

throu&hout D, then D Îs eith er the entire plane, or else the plane with one point removed. Prool: Introduce a second metric �1 } = À2o 1} , where À . .

-

. .

eh .

Then À � À throughout D, and if C is any divergent path in D, we have

J À \ dz\ J À \ dz\ ?

c

c

so that D is complete with respect to this metric also, and the same is true for the universal covering surface 6 o f D with re­ spect to the induced metric. But in a neighborhood of any point of

6 we may introduce an analytic function l( z) whose real part is h ( z), and the mapping w =

J e f( z) dz

which satisfies (9 .15)

\ 4� 1 dz

=

\ e f( z ) \

=

eh ( z ) = À

Thus the length of any curve on 6 with respect to the metric

� I.J.

is equal to the euclidean length of its image in the w-plane. By the simple connectivity of D there exists a global map of 6 into the w-plane satisfying (9 . 15), and by the completeness of D, this map must be a one-to-one map of fi onto the entire w-plane. Thus the

unive rsal covering s urface of D is conformally equivalent to the plane, and it follows that D itself is of the type asserted in the lemma. LEMM A 9.3. Let D be the domain 0 <

r1

< ! z\

< r 2 � oo, and

80

A SURVEY OF MINIMAL SURFACES

let g1. J.

=

>.. 2S 1. J. be a Riemannian metric in D satisfying (9. 14).

Suppose that fe >.. i dzi

0 .5: t

<

for every path C of the form z ( t ),

= oo

1, such that lim ! z( t ) l t ... 1

Proof: Suppose r2

<

oo.

=

r2 . Then r2

oo ,

Then we may assume that �

We introduce in � the metric �1- J·

=

1z:

r2

<

1 zl

< r2 1

À S . . , where À (z) 1}

•

À (z) >.. ( l/z) .

Then one verifies easily that this metric satisfies the hypotheses of Lemma 9.2 in �. but the conclusion is not valid. Thus the as­ sumption r2

<

oo

leads to a contradiction.

t

LEMMA 9.4. Let D be a hyperbolic domain and gI. J.

=

>..2S1. ]. a

metric with respect to which D is complete. Suppose that

� log À ;:: 0

(9. 16)

/l i � log >..! dx dy

(9. 17)

<

oo ,

z = x + iy .

Then there exists a harmonie function h ( z) in D satisfying (9 . 14). Proof: Since D is hyperbolic, for each point ( in D there ex­

i sts a Green' s function g ( z, () which is harmonie and positive for z -! (, and such that g ( z, () + log 1 z- t;;l = H(z, () , a harmonie function of z throughout D. (See, for example, Ahlfors and Sario [ 1] ' IV 6.) Set u ( t;}

2�

f�

G( z, () � log À( z) dx dy .

Thi s integral exists, by (9 .17) and the basic properties of the

S URFA CES lN E 3 : GAUSS CURYA TURE AND TOTAL CURYATURE

81

Green's function, while u ( C:::') 2: 0 by (9. 16). But by Poisson's for­ - � log À, so that h ( z) "' u

mula, we have �u

monie in D. But sin ce u 2: 0, li z) 2: log À.

+

+

log À is har­

REM ARK . The formula (9.5) , which holds for the Gauss curva­ t ure of an arbitrary surface in E3, can also be used to defi ne the Gauss curvature of an arbitrary Riemannian metric of the form g1. J. =

À 28 lJ. . . One can verify directly th at it is invariant unde r conformai

changes of parameters . Then (9. 16) is equivalent to

K _:s O

(9. 18) and (9. 17) is equivalent to (9. 19)

THEOR EM 9. 1 . Let M be a complete Riemannian 2-manifold whose Gauss curvature satisfi es (9. 18) and (9. 19). Then there ex­ ists a compact 2-manifold M, a fini te number of points on M, and an isometry between M and M - l p 1 , . . . , pk } .

Pp

. . . , pk

Proof: By a theorem of A. Huber, the condition (9. 19) on a com­

plete 2-manifold M i mplies that M is finitely connected (A. Huber [ 1] , p. 61). Thi s means that there exists a relatively compact re­ gion M 0 on M bounded by a finite number of analytic Jordan curves y l' . . . , yk , such that each component Mi of M - M0 is doubly con­ nected. (See Ahlfors and Sa rio [ 1] , 1 44 D and II 3 B. ) Th en each M j can be mapped conformally onto an annulus D i : 1 < l zl <.. ri .:5 oo , where the curve yJ. corresponds to 1 zl 1. The metric on MJ. takes =

the form g l. J.

À281. J in DJ. , and the conditions (9. 18) and (9. 19) re.

duce to (9. 16) and (9. 17) respectively. The region DJ. is obviously

82

A SURVEY OF MINIMAL SURFACES

hyperbo lic, since the function Re l !_ z

l!

is a negative harmonie

function. By Lemma 9.4 there exists a harmonie function

h(z) sat-

isfying (9. 14) and by Lemma 9.3 (and the completeness of M) i f follows that

r1.

=

oo ,

Let D. be the extension of D . t o a disk on 1

1

the Riemann s phere obtained by adding the poi nt at

oo .

-

Let M be

the compact surface obtained by " we !ding" the disks D to M 0 i along y . ( see Ah lfors and Sario [ 1) , Il 3 C). Then M is conformally J

-

is the point D . - D . , and J 1 by carrying over the metric on M thi s correspondence becomes an

equi valent to M -

!p 1 , . .. , pk!'

where

-

p. 1

t

isometry.

Let x ( p) : M -+ E 3 define a complete regular mini­ mal surface S. li the total curvature of S is finite, then the con­ clusion of Theorem 9. 1 is valid, and the function g (p) in (9 . 13) ex­ tends to a meromorphic function on M. Proof: We know that on a minimal su rface K S 0, and therefore Jf I KI dA, so that finite total curvature i s equivalent 1 ff K dA I LEMM A 9.5.

=

to (9. 19) . Thus Theorem 9 . 1 may be applied, and we may consider

lp l ' ... , pk!. If any of the points p. were an essential singularity of g, then by Picard's 1 theorem, g would assume every value infinitely often, with at most

g (p)

to be a meromorphic fun ction on M-

two exceptions. But this wou ld imply that the spherical area of the image is infinite, and hence also the tot al curvature, contrary to assurnption. Thus

g

has at most a pole at each •

fore merornorph i c on all of M.

t

p1. ,

and is the re-

Let S be a complete minimal surface in E3 . Then the total curvature of S can only take the values -477m, m a non-negative integer, or TH EOR EM 9 . 2 .

-oo ,

SURFACES lN E 3 :

Proof:

83

GAUSS CURVATURE AND TOTAL CURVATURE

Since K ::: 0, either the integral ff K dA over the whole

surface diverges to

--oo ,

or else the total curvature is finite. In the

latter case we apply Lemma 9 . 5 and find that the total curvature is the negative of the spherical area of the image under

l p l ' . . . , pk !.

But since

g

constant, in whi ch case K

'=

fixed number of times, say

m.

M

then

t

41Tm.

LEM M A 9 . 6.

r 1 < 1 zl < infinity, we have

0

<

oo.

g

of

i s meromorphic on M, it i s either 0, or else it takes on each value a Th e spherical area of the image is

Let f(z) be analytic and different from zero for Suppose that for every path C which diverges to

f

(9.20)

c

l f( z) l l dzl

""'

=

Then f( z) has at most a pole at infinity. Prool: Since f( z) ! 0, log l f( z) l is harmonie,

and we have the

Laurent expansion at infinity log where

h ( z)

l l( z)l

= a

log l zl

h ( z) H(z) , +

is harmonie and bounded for

is harmonie in the fini te plane 1 zl ger great er th an (9.21)

+

a,

<

oo.

r 1 < r2 < ! zl

<

""•

and

H(z)

If N is any positive inte­

it follows th at for 1 zl

>

r2 ,

1 f( z) l

where M i s a suitable constant, and whose real part equals

w

=

H(z). F(z)

G(z)

i s a n entire function

We introduce the entire function

J�e� ( z)dz ,

F(O)

=

0,

84

A SURVEY OF MINIMAL SURFACES

and note that there exists a single-valued branch

�( z} i n a neighborhood of z

=

=

[F(z)] 1

/N+ 1

0, satisfying �'( 0) f:. O. We therefore have

an inverse z ( �) in a neighborhood of �

0, and either this inverse

' extends to the whole �-plane, or else there is a largest circle

1 '1


to whi ch it can be e xtended. But the latter cannot occur,

for then there would be a point �0 satisfying 1 �01

R

over which

the inverse could not be extended . If we consider the curve �( t )

, t �0 , 0 ::; t

<

l , its inverse image i n the z-plane will be a path C

such that

If the path C were to diverge to infinity, then from sorne point on it would be in the region l zl

>

r2 ,

and because of (9 .21) thi s would

contradict (9 .20). Thus C cannot diverge to infinity, and there must exist a sequence of points zn on C which have a finite point l of accumulation z0 , such that the images of these points in the K,'-plane tend to �0 . But since F '( z 0) f:. 0, we could ex tend the in! verse mapping over � . We th us conclude that the inverse z ( �) � s defined in the whole �-plane. Furthermore,

rN + 1

'=> 2

•

Thus each value is taken on at most N + l times , and z ( �) must

be a polynomial. But z( �)

=

0

for sorne integer k. Finally, z '(O) IThus F( z}

( z/A) N + 1 , and

�

=

0, and bence z ( Q

Ak� k - 1

f- 0

k=

A �k l, A f:. O. =

G( z) must be constant. The same is

SURFACES IN E 3: GAUSS CUR VATURE AND TOTAL CURVA TURE

then true of

H(z},

so that l l( z) l

has at most a pole at infinity. THEOREM 9 . 3 .• Let

in E 3 .

<; M 1 1 zi N

85

near infinity, and

t

f(z}

S be a complete regular minimal surface

Then

Jjs K dA

(9 .22)

:5

2rr( x- k) .

where X is the Euler characteristic of S, and k is the number of boundary components. Proof: If the integral on the left of (9 . 22) diverges to the result holds tri vi ally. Otherwise S h as finite total curvature, and - oo ,

we may apply Theorem 9 . 1 and Lemma 9 .5. By virtue of the rela­ tion (8.8) between the function

g

and the normals to the surface

we may assume (after a preliminary rotation in s pace) that oo

at the points

p ., J

and tha t the po les o f

g(p) -f. O,

g (p) are ali simple po les. ln a neighborhood of any point of M M - l p l ' . .. , pk l we have the representation (8.2) in terms of isothermal parameters (. As we have noted earlier, under a conformai change of parameters ((Ô, the corresponding functions �k( Ô satisfy �k( Ô cf>k(()d(/d( , and similarly for f((). This implies that the existence of a zero of cf>k or f at a point, as we il as its order, is independent of the choice of loca l parameters . By Lemma 8 . 2, f must have a double zero at each of the simple poles of g, and f has no other zeros. Thus, if g is of order m on M, f has exactly 2m zeros on M. At each point p. we may introduce local coordinates ( so that ( 0 J corresponds to p . . For 0 < 1 '1 < E we have [ gl < M, bence f -f. O. J Furthermore, i f C is a curve on M tending to p., we have by the J completeness of S that -

=

=

•see Appendlx 3, Section 4.

186

H

A SURVEY OF MINIMAL SURFACES

f

follows from Lemma 9 .6 that

xk((,)

rom the fact that the functions n 0

< 1 (,1

< E

,

must have a pole at the origin. in (8 . 6) are si ngle-valued

and because of the assu mption that

follows easily th at the order

]

f( (,)d(, îs

quai to 2 . Thus

f

of the pole of

v.

at

g ( p.) f 0, it

p.J

]

is at least

a meromorphic differentiai on

M, and

ne has the Riemann relation that the number of poles minus the number of zeros equals 2 - 2G, where

lfors and Sario [ l] ,

V

G

i s the genus of M (Ahl­

27 A). Furthermore , the Euler characteristic

X of M is equal to 2 - 2G-

2- 2G

k.

We have, therefore

k =

j

l:

v.

1 ]

2m > 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 <

the normals to S take on ali

directions infinite/y olten wi th the exception of at most a set of logarithmic capacity zero;

S has finite n on-zero total curvature <

> the normals to

S

take

on ail directions a fini te number of times, omitting at most three directions; S

has zero total curva ture

S i s a plane.

This result, and the other theorems in this section are contain· ed in the two papers Osserman [4, 5]. The presentation given here is somewhat different. Lemma 9.4 is based on an argument of A . Huber [ 1] . which he used to obtain the fol lowing result: i f S is a complete surface, and ff5 Ir"dA converges, where x-

= max { - K, Ol, then S is parabolie.

(This result was obtained earlier by Blanc and Fi al a in the case th at S is s imply connected and the metric real analytic.) Lemma 9 .6 was announced (without proof) by MacLane [ 1] and Voss [ 1] . The proof given here is taken from Finn [6] who obtains b y the s ame reasoning a much more general result, not needed for our purposes. The corollary to Theorem 9.4 was observed independent­ ly by Osserman [6] and Voss [ 1] . In conclusion , let us note that i t would be interesting i n con­ nection with the above results to have more examples of complete minimal surfaces whose genus is different from zero: It was shawn by K lotz and Sario ( 1] that the re exist complete minimal surfaces of a rbitrary genus and connectivity. Furthermore , it can be shawn (Osserman [6]) that the classical surface of Scherk has infinite genus. There remains the following o pen question. •see Appendtx 3, Section 4.

90

A SURVEY OF MINIMAL SURFACES

Problem: Does there exist a complete minimal surface of finite

total curvature who se normals omi t three directions? If so, the value "three" is the precise bound. If not, the maxi­ mum is "two , " and is attai ned by the catenoi d.

NON·PARAMETRIC MINIMAL SURFA CES IN E3

91

§ 1 0. Non-parametric Minimal Surfaces i n E 3 • The study of minima l surfaces in non-parametric form consists of the s tudy of solutions of the mi ni ma l surface equation ( 1 0.1 ) and as such it may be considered as a chapter in the theory of non· linear elliptic partial differentiai equations . Just as we have seen in previous s ections that the theory of functions of a complex va­ riable and of Riemann surfaces led to many geometrie results about minimal surfaces, we shall now show that many properties of minimal surfaces can be attributed to the form of equation ( 10. 1). In the next section we shall reverse the process and use the para­ m etric the ory to derive properties of solutions of this equation. The underlying idea in the present section is the comparison of an unknown solution of (10. 1) wi th a fixed solution whos e prop­ erties are weil known. This method is possible by virtue of the following basic lemma . L E M M A 1 0. 1 . Let F and G b e two solutions of equation (10.

in a bounded domain

Suppose th at Hm (F - G) ,::;; M for an arbi·

D.

tracy s equence of points approachin/5 the boundary of F - G 5: M throuf5hout

D.

Then

D.

Proof: This lemma holds for a wide class of equations of whic� (10.1) is a special case. We refer to the discussion in CourantHilbert [l], p. 323.

+

An interesting feature of the study of equation (10. 1) is that its s olutions behave in sorne ways precisely as one would expect from the general theory of elliptic equations, as in the case of the above l emma , and in other ways they reveal completely unexpected

92

1! properties.

A SUR VEY OF MINIMA L SURFA CES

The non-existence of any solution in the whole plane

other than the trivial linear ones (Bernstein 's theorem) is an exam­ 'ple of the latter. We shall now give sorne further examples. We first show that for certain domains D, the con clusion of Lemma

10.1

may be valid even though the hypotheses are known to hold

on only part of the boundary. Let us i ntroduce the following nota­ tion. Let

(10.2) The equation

1(10.3)

r

=

Jxi + x�

,

k:l efines th e lower half of the catenoid, and is a solution of the min­ imal surface equation in the entire exterior of the circle

�,

xî + x� =

taking the boundary values zero on this circle.

10.2. Let

D

be a domain which lies in the annulus in 1 < r < r2 , and let F(x 1 , x 2) be an arbitrary solution of . If the relation L EMMA

(10.1)

�10.4)

lis known to hold for an arbitrary sequence of points which tends to a boundary point of D not on the circle r r 1 , then

i

=

i

10.5)

throughout D. R EM ARK. Perhaps the most striking i l lustration of this lemma ·

s the case where D coin cid es with the annulus

r 1 < r < r2 •

Then

NON·PARAMETRIC MINIMAL SURFACES IN E

3

93

the fa ct th at (10.4) holds on the outside circle r = r 2 i mplies that ( 1 0.5) holds throu ghout D, and bence lim F ( x 1, x 2) ::; M on the inne c ircle r

r 1 • Geometrically, one can describe this situation as

follows : given an arbitrary minima l surface lying over an annulus, if one places a catenoid al " cap" over the annulus in such a posi· tion that the surface lies below it alon g the outer circle, then the s urface lies below it throughout the annulus. This behavior is i n s triking contrast with that of harmoni e functions where on can pre· s cribe arbitrarily large values on the inner circle, independently of the v alues on the outer circle, and find a harmonie function in the a nnulus taking on these boundary values.

Proof:

For an arbitrary r ;_ satisfying r1 < r ; < r2 , let

Then by ( 10.4), we have (1 0.6)

for all sequences approaching a boundary poi nt of D lying in the annulus r ;_ ::; r S r 2 • We wi sh to conclude th at

( 1 0.7)

in the domain D ' == D

n Ir

;_ < r < r2 1.

The result then follows by

letting r ;_ tend to r 1 • To p rove ( 10. 7), it suffices by Lemma 10.1 to show that (10 .6) holds at every boundary point of D '. Si nee it already holds at

boundary points of D, it suffices to sh.ow th at ( 10. 7) holds at in· terior points of D which lie on the circle r = r ; . Suppose not.

94

A SURVEY OF MINIMAL SURFACES

Then the function F(x 1 , x 2) - G( r ; ri) would have a maximum

M1

>

M

at sorne point (a 1 , a 2) on this ci rcle, i nterior to Do

+ f

By Lemma 1 0. 1 , F( x 1 , x 2 ) - G( r ; r;) :S M 1 throughout D '. On the

other hand , F has a fini te gradient at the point ( a 1 , a 2 ), whe reas by ( 1 0 . 2),

Since F - G

aa�: r�- \ =

, r= r 1

=

-

oo •

M 1 at ( a 1 , a 2) , it follows that F - G

>

M 1 for ali

points in D ' sufficiently near ( a ' a 2) and the origin. Thus the

l

s upposition that {10.7) did not hold leads to a contradiction, and th e lemma is proved.

•

We shall give several applications of this lemma. The first is a generalized maximum princ iple for solutions of ( 10. 1) THEOREM 1 0 . 1 . Let F(x 1 , x 2 ) be a solution of the minimal

s urface e quation ( 1 0 . 1 ) in a bounded domain D. Suppose that for e very boundary point ( a 1 , a 2) of D, wi th the possible exception of a finite number of points, the relation

( 10.8)

are known to hold. Then ( 10.9)

throughout D. Proof: It is sufficient to prove the right·hand inequality in ( 1 0.9), since the other then follows by considering the function -F.

NON·PARAME TRIC MINIMAL SURFA CES IN E

We proceed by contradiction. Suppose that sup F

=

3

95

M1

>

M. Then

there exists a sequence of points in D along which F tends to M 1, and a subsequence of these points must converge to one of the e xceptional boundary points. (If F assumes its maximum at an interior point of D, it must be constant, since x3 is a harmonie function in local isothermal paramete rs.) We may suppose by a t ranslation that this point is at the origin, and we may choose r2

>

0 so that none of the other exceptional boundary points lies

in the disk r :;; r2 • Choose r 1 arbitrarily so th at 0 let D ' be the intersection of D wi th Ir 1 =

sup F for points of D on the circle r

M2

<

<

=

<

r1

<

r2 and

r < r2 1. Finally let M 2

r2 • We assert that

M 1 . Namely, given a sequence of points on the circle r

=

r2

along which F tends to M 2, they have a point of accumulation which either lies on the boundary of D, in whi ch case M 2 S M, or e ls e interior to D, in which case F w is e F

=

M1

>

<

M 1 at th at point, si nee other­

M contradicting (10.8). Combining ali these facts,

we conclude that

a t a il boundary points of D ' for which r

>

r 1 , where

By Lemma 10.2, it follows that

throughout D '. Now for each fixed r, it follows from (10.2) that G(r ; r 1) for 0

<

->

r

0 as r 1

<

->

0, and we de duce that F(x 1, x2) S 1'43

r2 • But this is in contradiction with

F

->

<

M1

M 1 along a

s equence of points tending to the origin, and the result fo llows. t

96

A SURVEY OF MINIMAL SURFACES

Let us note that in the statement of Theorem 10. 1 , each of the e xceptional points may either lie on a boundary con tinuum, or else be an i solated boundary point. ln the latter case the above proof s im plifies considerably, and a mu ch stronger result is in (act true. THEOREM 10.2.* A

solution of the minimal surface equation cannot have an isolated singularity. Proof: Let F(x 1 , x2 ) be an arbitrary solution of equation (10. 1 ) in a punctured disk : 0 < r < e. We wish t o show that F( x l x 2) ex­ '

tends continuously to the origin , and that the resulting surface

x 3 F(x l ' x 2) t

< E.

is a regular mi ni mal surface over the whole disk

r2 < e, and let M = max ! F(x 1 , x2 )1 for xi + � r� . B y Theo rem 10. 1, 1 F(x 1 , x 2 )1 :S M for 0 <_ xi + x� < r� . Ac­ cording to Theorem 7 . 2 the re exists a function F ( x 1 , x2) defined and continuous in the disk r ;; r 2 , satisfying (10. 1) in r < r2 , and takin g on the same v alues as F for r = r2 • Then I F 1 ;; M also for r ;; 2 • We shall show that in fa ct F F for 0 < r < r2, and : },ence F is the desi red extension of the solution F . To this end C hose

r2 ,

0

<

=

=

we recall t hat in the notation p

=

aF aF ax l ' q dx 2 ' =

the equation (10.1) can be written as 0

( 1 0. 1 0)

where ( 1 0. 1 1) •see Appendix 3, Section 5.

ifJ = 9. w

NON-PARAME TRIC MINIMAL SURFACES IN E

(This is merely the special case

n

=

97

3

3 of equation (3. 14), but it

c an e asily be verified directly.) We now introduce the corres ponding quantities for the function

F,

and consider the difference of these two functions in the annu­

D À : À :; r ,:; r 2 • If ;:e let CÀ be the circle r À, then by v ir­ tue of the fact that F - F 0 on the ci rcle r r2 , Green 's theo· rem applied to the domain DÀ yie lds lus

=

=

�CA(F - F ) [(cp - � )dx 2 - (tfr-�)dx 1] JfvÀ[(p- p)(cp - �) + (q- q)(tfr - �)] dx 1 dx2 ,

( 1 0. 1 2)

=

where we have applied equation (10.10) to both functions

F. Now we note from (10. 11) and the definition of W th at cp 2 + tfr 2 < 1 and simil arly � 2 + � 2 < 1 . Since \ F - F \ :; follows that the left-hand side of ( 10.12) tends to zero as

F

and

'2M, it

À O. ....

But the inte grand on the right is non-negative, as one sees easi ly by applying Lemma 5 . 1 to the function E ( p, We consider

p

q) .j 1 + p 2 + q2• =

and q as independent variables , and we find aE oq

_

'"5- -

{J

:1. W

tfr .

The inte grand on the right of (10.12) then reduces to the expres­ sion on the l eft of (5.2). Thus if

(p - p) 2 + ( q - q) 2 � 0

at sorne

point, it follows that the integrand on the right of (10. 1 2) i s posi­ tive at sorne point, hence in a neighborhood of that point, hence

À tends to zero. Thus we 0 < r < r 1 , and hence also

the inte gra l could not tend to zero as c onclu de

F F

=

p- p

0,

q- q

=

0 for

0, which proves the theorem . , t

98

A SURVEY OF MINIMAL SURFACES

The proof jus t given , together with the proofs of Theorem and Lemma

10.2

cial case of Finn 's lemma of Theorem

10.1

[ 5] .

are due to Finn

([5] ,

p.

Lemma

139) .

10.2

Theorem

10.1

i s i n fact a s pe­

10.2

and the case

in which the exceptional boundary points are iso­

l ated are ori gi n ally due to Bers

[1].

The form of Theorem

s tated above i s due to Ni tsche and Nitsche

[ l] .

10. 1

I t was later gener­

a lized to allow the exceptional set to be an arbitrary set of linear H ausdorff m easure zero (Nitsche

[ 5]).

Si milarly, Bers ' theorem o n re­

m ovable sin gularities has the followi ng extension: * li D is a bound­ ed domain, and E a compact subset of D wh ose linear Hausdorff m easure is zero, then every solution of the minimal surface e qua­ tion in D - E extends to a solution in ali of D. This result was obtained ind ependently by Nitsche chia

[l].

[ 5]

and De Giorgi and Stam pac­

The s ituation is quite different for the case n > 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\ <

tT/2,

and tends

at every boundary point except for the vertices .

O bviously if D i s any domain interior to this s quare whose bound­

ary i s tangent to one of the s ides, then F is a solution i n D which

tends to infi nity at the point of tangency. We next prove a result complementary to Theorem 10.4, which shows that infinite boundary values cannat occur along a whole convex a rc .

A SURVEY OF MINIMAL SURFA CES

1 02

T H E O REM 1 0 . 5 . Let D be a plane domain, y an arc of the boundary of D, L the line segmen t joining the endpoin ts of y. 1 f y and L form the boundary of a subdomain D ' of D, th en no

solution of the minimal surface equation in D can tend to infin i ty a t each poin t of y. P roof: Let P be a point of D ' and let C be the cücle through P a nd the end points of L. Then there exists a subdomain D " of D ' which lies outside C and is bounded by parts of C and y. If

F were a solution of the minimal surface equation in D which ap­ pro ached infinity at each point of y , then Lemma 10.2 applied to

- F would i m ply F

oo

in D -� which is imposs ible.

t

C ombining Theorems 10.4 and 1 0. 5 , one sees the fol l owing: if a solution of the minimal surface equa tion tends to infini ty along a whole arc of th e boundary, then that arc is a s traigh t· line s egment. T he question then arises whether one can solve the Dirichlet prob­ lem w ith i nfinite boundary values along a given line segment on the b oundary . In the paper of F i nn

[ 4],

wh ich also con tains Theo­

rem 1 0.5, the fol l owing resu lt is proved: 1 et D be a con v ex domain whose boundary contains a s traight-line s egment L. Let cont inuous boundary values be prescribed arbi trarily on the complementary part of th e boun dary.

Then there exists a solution of the minimal sur·

fa ce equation in D wh i ch takes on these boundary values, and which tends to i n finity at each inner poin t of L. A deta iled study of the question of the Dirichlet problem witb

i n fi n ite boundary values bas been made recently by H.

J.

Serrin

[ 1, 2].

J enkins

and

Necess ary and sufficient conditions are given for

th e exis te nce and uniqueness of soluti ons taking on prescribed

1 03

NON-PARAME TRI C SURFA CES IN E 3

boundary values which are plus infinity, minus infinity, and con­ tinuous , respectively, on three given sets of boundary arcs. Let u s note one example of their results: Le t D be a convex quadri­

la teral. Let one pair of opposite sides have total length L, and th e o th er pair have total length

M. Necessary and sufficient that

there e xis t a solution o f the minima l surface equation in D takin on th e b oundary values o th er pair i s that L

=

+ oo

on one pair of sides and

- oo

on the

M. When the solution exists, it is unique u

to an additive constant.

APPLICATIONS OF 1\RAMETRIC M ETHODS

105

A PPLICA TIONS OF P ARAME TRIC METHODS

§ 1 1 . Application of parametric methods to non-parametric problems�i W e conclude our discussion of minimal surfaces in

E

3

with

s evera l examples of how results on solutions of the minimal sur­ face equation can be obtained by expressing the corresponding sur faces in parametric form.

x3 F(x 1, x 2) define a minimal surface x i + x� < R2• Let P be the point on the surface

THEOREM 1 1 . 1 . Let in th e disk D :

=

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 boundary. Then the ine quality

I KI

( 1 1 .1 )

<

c

- d2W20

holds, where c is an absolute constant, and the origin o f w

2

=

aF )2 '\Jx 1

1 f

.;

W�

is the value at

( dx; aF ) 2

R EM ARK. lnequality ( 1 1 . 1) may also be expressed in the form o f a bound at the origin for the second derivatives of an arbitrary s olution o f ( 10.1) in the disk D. P roof: We may represent the surface parametrically by

1 (1

<

l , the orientation being chosen so that the unit normal is

q

U sing the representation (8.2), we have by (8.8) that ( 1 1.2)

x((),

1 06 If C is

A SURVE Y O F MINIMAL SURFA CES

an

arbitra ry curve in 1 �� < 1, then the length of its image

on th e s urface is given, according to (8. 7), by

In particular, if we consider aU di vergent paths C in 1 ( 1 < 1 whi ch s tart at the origin, then we have

d = inf

( 1 1 .3)

c

where C

1

f

c

i\

1 d(l :;;

f

1 f( () 1 1 d( l ,

cl

is any particul ar such path. We choose the path C

1

as

follows. Set w = G( () = J f ( () d(, where G(O) = O. Then G '(()

f ( () :ft 0 in 1 (1 < l, by Lemma 8 .2, and following the reasoning of Lemma 8 .5, the inverse function ( = H( w) defined in a neighbor­ hood of the origin will be defined in a largest ci rcle 1 wl <-p on whos e boundary there wil l be a point w 0 over which the inverse c annot be e xtended . The radius L : w( t ) = t w0, 0 ::; t < l, wil l map onto a d ivergent pa th C i n 1 (1 < 1 which starts at the origin. 1 B ut

( 1 1 .4)

f

l f( () J ! d( l =

Ct

f l dwl L

=

p

•

By S chwarz ' lemm a, applied to the map H(w) of l wl < p into l t;] < 1, we h ave J H '(O) j ::; 1 /p. Thus l f(O)I = I G '(O) J = 1/I H '(O) J 2: p . Comparing with ( 1 1 . 3) and ( 1 1 . 4) yields ( 1 1 .5 )

d � J t( O) I

•

N ext we apply Schwarz' l emma to the map g ( () of 1 ( 1

! g(()J < 1, and we find

<

1 into

APPLICA TIONS OF PARAME TRIC METHODS

j g '( O) j � 1 - j g ( O)j 2 .

( 1 1 .6)

F inally, combining ( 1 1 . 2) , ( 1 1 . 5) and ( 1 1 . 6) with the expression (9 .4) for the Gauss curvature, we fi nd

vi K J d < -

(

4 1 g '(O!l__._ < 4{ 1 - J g (Qltl__ j g ( O) j 2) 2 - 0+ j g ( O)j 2 ) 2

w hich proves the theorem . C O ROLLARY

.•

+

Un der the same hypotheses, th ere exists an ab­

solu te constant c 0 such that

( 1 1 .7) Proof: Obviously d ,2: R, and W ,2: l. 0

+

Let us note that the inequalities (1 1 . 1) and ( 1 1 .7) may be re­ garded as quantitative forms of Bernstein' s theorem. Namely, i f

F(x1, x ) is defined over the whol e ( x1, x ) plane, then we may 2 2

c hoose a disk of arbitrarily l arge radius R about any point, and it follows from (1 1 . 7) that the Gauss curvature of su ch a surface i s identic ally zero, which for a minimal surface implies that it i s a p lane. The first results of thi s type were given by Heinz [ 1], where one finds inequality ( 1 1 . 7) . Thi s was later sharpened by E. Hopf

[ 1], who gave

à

differen t proof and found that one could insert the

factor w2 in the denominator. The inequality ( 1 1 . 1) is in Os serman [ 2]. It has the advantage th at it remains true if the disk D is replaced by an arbitrary do­ m ain in the (x11 x ) -plane . Concerning the constant c, one can

2

show that if the s urface has a horizontal t angent plane at the then W0

=

1 , and

•see Append1X 3 , Sections

2. 6IB.

and 6IIA.

1 08

A SURVEY O F MINIMAL SURFACES

F urthermore , this inequal ity is sharp , e quality being atta i ned i n the c ase o f Enneper's surface (the do ma in D no longer be ing a disk i n th is case). By a refinement of the argu me nt g i ve n above o ne can show that the constant c i n (1 1 . 1 ) satisf ies

64 < c < 8 . -9 T hese results are co nta ined in the paper Osserman

[2] .

A co mpletely different proof of inequality ( 1 1 .7), us i ng purely para metr ic methods is g i ve n in Finn and Osserman

[ 1] .

It is shown

there that if the surface has a hor izo ntal tange nt plane at the or i· g in, then o ne has

! KI a nd that th is inequality is best possible , with Sherk 's surface play­ i ng the role of an e xtrema ! . It is also shown that the constant c 0

in ( 1 1 .7) satisfies

2 2 -

11 < --

c

o

< 6

•

T hese considerations po i nt obviously to the followi ng quest ion ,

PROB L EM . What are the precise values of the consta nts a nd c

0

c

in ( 1 1 . 1 ) and ( 1 1 . 7)? For a g i ve n value of W ca n one de­

sc ribe the e xtremal surface for these inequal ities? We tu m next to solutions of the minimal surface equation de ­ f ined in the

ex terior

due to Bers

[ l].

of a disk. We note first the followi ng result,

1 09

APPLICATIONS OF P ARAMETRIC METHODS

x 3 R 2•

T H EO R EM 1 1 . 2. Let

in th e r egion

x� + x;

>

=

F( x 1 , x 2)

define a minimal surface

Th en th e g radien t of

F

ten ds to a

limit at in finity. Proof: The normal s to such a surface are contained i n a herni· s phere , whereas the length of ali curves on the s urface which d i ­ v erge to infinity is clearly infinite. It fol lows b y Theorern 9 . 5 that the n ormals tend to a lirnit at infinity , which is the desired result. t Actu ally , Bers obtained rnuch more precise res ults on the be­ h avior of the solution

F( x 1 , x 2 )

in the nei ghborhood of infinity .

F or this purpose h e us ed a pararnetric representation analogous to

(8 . 2 ), together with the followi ng lernma. L E MM A 1 1 . 1 . L e t

in th e region

x i + x�

th e punctured disk: 0

F( x , x 2 ) 1 R 2 • Th en S

x3 > <

==

1 Çl

<

define a minimal surface S is con formally equivalen t to

l, wh ere

Ç

t ends to zero as

xi + x �

tends to infinity. Proof: This result follows by the s ame reasoning that was used

t

in the proof of Theorern 9 . 5 .

The precise cond itions in pararnetric forrn for a surface to be of th e kin d we wi sh to consi der are the fol l owing. L EMM A 1 1 . 2 . Let a mi nimal surface S be repres ented by the

equations

( 1 1.8)

in the punctured disk li : 0

<

1 Ç!

<

l, wh ere

1 10

A

SURVEY OF MINIMAL SURFA CES

( 1 1 .9)

Then the following statements are equivalent: 1 . There exists p

to 0

<

1 (1

<

>

0 such that the part of S corresponding

p can be solved in the form x3 = F( x 1, x ) , where 2

F(x 1 , x 2) is a bounded solution of the minimal surface equation

in the exterior of some ]ordan curve ï.

' II . There exists p > 0 such that in the punctured disk 11 ' : ' 0 < 1 (! < p , the functions f(() , g ( () are analyti c and have the

fol/owing properties:

a) g ( () extends to an analytic function in

fying \ g ( () l

<

1, g (O) = 0, g '( O) = O.

b) f ( () 1: 0 in 11

'

,

1 (!

<

' p , satis­

and f ( () has a pole of arder two at the

origin with residue zero. Proof: 1

=;>

' Il. Let p = p. A surface in non-parame tric form

i s necessarily regu lar, and satisfies 1 g 1

<

1, bence f 1: 0, by

L em m a 8.2. Since g is a bounded analytic function in a punctured disk, it bas a removable singularity at the ori gin. The value of

g ( O) determines the limiting position of the gradient at infinity; if g ( 0) 1: 0, then by ( 1 1 . 2) the gradient would tend to a limit dif­ ferent from zero and F(x 1, x 2) would not be bounded. Thus g (O) "" O. Furthermore, x3 = F( x l ' x ) bounded at infinity means that

2 x 3 is a bounded harmonie function of ( in a nei ghborhood of the

o rigin, and hence also bas a removable singularity. Thus

is analytic i n the full disk, and the same is true of the function

111

A PPLICA TIONS OF PARAMETRIC METHODS

f( (> g ( Q 2 •

From this it follows first of all that 1 cannot have an

essential singularity at the ori gin. Further, from (11.8) and (11 .9), we h ave

(11.10) from which we deduce that 1 must have residue zero at the origin in order for x 1 and x2 to be single-valued functions of ' On the other band, 1 cannot be regu lar at the origin, since otherwise x 1 a n d x2 would b e bounded. Thus 1 has a po l e o f order at least two, and fg analytic at the origin implies g bas a double zero; i . e. , g '( 0) "' O. F inally, we must show that

/((;)

cannot have a pole of

order greater than two. But note that by (11. 10) we may write

(1 1.11) where l'

H( (;)

(11 .12)

"" j f( (;) d(; ,

are s in gle-valued analytic functions in A '. I f

f((;)

bad a pole of

order greater than two, then H((;) would have a pole of order greater th an one, and the image of 1 (;1

=

e:

for all small

a round the o rigi n more than once. But IH((;)I , and th us the image of 1 <;[

>

e:

would wind

1 G(Ç)[

on 1 <:1

==

e:

in the x l ' x2-plane would also wind around the origin more than once. But this contradicts the

for small

e:

fact that for 0 < result i s proved.

1<:1

=

e:

< p, the map <; -+ ( x l ' x 2) i a one-to-one, and the

II =9 I . Conditions a) and b) guarantee that the functions G ( (; ),

H((;) defined by (11 .12) are single-vàlued in A •, and that G((;) i s analytic a t the origin, whi le H((;) bas a simple pole. Thus, by (11.11) i

A SURVEY OF MINIMAL SURFACES

1 xi

+

.S

tends to i nfi nity as ( tends to zero. For the Jacobian of

1 the map ( 1 1 .1 1) we have the e xpression

1

! (1 1.13)

J

1

/ Since ! 8 /

•1 1

<

=

él ( x l' x 2) él( (p If2)

1, we have ]

<

0 everywhere, and the map (11. 11) is

a local d iffeomorphism. We wish to show that it is a global diffeo­

. m orphism in 0 < / (! < p, for suitable choice of p. To this end, we 1 firs t choose r > 0 so that H( () is univalent in 0 < j ( l .:S r, which 1

!

is possible since H(() has a simple pole. Let M

1

==

max / G(()/ for

/ ( .:S r, and choose r1 .:S r so that / H( ()/ > 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

< <

1 (!

<

+

x�

=

Mi

is taken on precisely

r 1 • Namely, given any such point (a1, a2),

r1 such that j H( () / > 2j a1

+

ia2!

+

M

M2 for

1� '1 r2 • / x1 ix2/ j a1 ia2/ / (! r2 , 1 ( 11 . 1 1) r2 / (/ r1, l (a l' a 2) . (a1, a2) � 1 '1 r2 • 1 1 1 .3. Let F( x l ' x2) satisfy the minimal surface equa· 2 x� R • lf F( x1, x2) is tion in the e xterior of some circle xi 1 rounde" th en i t tends to a limit at infini ty, and .:S

Then

>

+

on

+

=

=

is a local diffeomorphism of the annulus

and the map

<

<

s uch that the image of the bound ary bas winding number equal to

one with respect to the point The point has , ·t herefore exactly one pre-image in this annulus and none for :$

!

This completes the proof of the !emma .

+

L EMM A

+

( 1 1.14)

1 .,

(

1

ÈF a xl

+

i q_f_ ax 2

1

:$

xi

=

M

+

x;

n a nei,ghborhood of infinity, for a sui table constant M.

113

A PPLICA 'lYONS OF PARAME TRIC METHODS

Proof: By Lemmas 1 1 . 1 and 1 1 . 2 w e have the representation ( 1 1 .8), ( 1 1 .9), where f and g satisfy conditions II a) and b) of Lem ma 1 1.2 . As we pointed out in the proof of Lem ma 1 1 . 2, x 3 tends to a limi t as � tends to zero, bence as x �

+

x� tends to

infinity. F rom the representation (1 1 . 1 1) and the properties of G and H, it follows immediately that ! x 1

+

ix2 1 1 � 1 tends to a finite

l imit as � tends to zero. Finally, from (8.8) one may compute 2 Relgl

2 I mlg l I- l al 2

I- l a l 2

and s in ce g ( �) has a double zero at the origin,

in a neighborhood of infinity.

+

THEOREM 1 1 .3. If the exterior Dirichlet problem has a bound·

F ed solution, then it is unique. Specifically, if F(x 1 , x 2), (x 1 • x 2) are bounded solutions of the minimal surface equation in the do­ 2 2 m ain � + x � > R , if they are continuous for x� + x � � R and 2 2 agree for XI2 + x 2 .. R , then F( x 1 , x2 ) = F(x 1 , x 2). Proof: Let us use the same notation as in the proof of Theorem 1 0.2, but applied to the annulus D : r 1 :5 r :S r2 , where r 1

>

R. We

obtain formula (10. 12), but with two boundary integrais on the left. The one around the outside boundary tends to zero as r2 tends to

infinity by virtue of ( 1 1 . 4) and the fact th at both F and F are boun� ed

The one on the inside ci rcle tends to zero as r 1

-+

R because F - F

:

j ·

tends to zero, whereas the quantities cp, cp, 1/J, 1/J are ali l ess then 1

i n absolute value. Thus the reasoning of Theorem 10.2 shows that F and F have the same gradient, bence differ by at most a con stant , an and since they coïncide on a ci rcl e they are identi cal .

+

1 :

1 14

A SURVEY OF MINIMAL SURF A C E S

R E M ARK. The above proof was suggested by R. Finn. Obvi­ ously the boundary need not be a circle, since the reasoning is quite general . The fact that this theorem was true was pointed out to the author by D. Gilbarg, who also raised the question of whether there a lways exists a bounded solution of the exterior Dirichlet problem. We show that there does not in the following theorem.

There exist continuous functions on the cir­ cle x� + x� = l which may be chosen to be arbi trarily smooth ( e.g., real analytic functions of arc length) such that no bounded solution of the minimal surface equation in the exterior of this cir­ c le takes on these values on the boundary. Proof: We first construct a particular surface S which will T H EOREM 1 1 .4.

p lay a role s imilar to that of the catenoid in the arguments of the previous section. Thi s surface is defined by setting i( (;)

=

2 '

i n ( 1 1 .9), for 0 < 1 (;1 :5 l. The transformation (11. 11) takes the form ( 1 1 . 1 5) O ne can easily veri fy that the image of 1 (;1 a

•

l under ( 1 1 . 15) is

Jordan curve r. and it follows by the same reasoni ng as in the

proof of Lem ma 1 1. 2 that ( 1 1 . 15) is a diffeomorphism of 0 < 1 (;1

onto the e xterior of r. Thus the surface non-parametric form the equation

x3

x 3 = 2Ç1 ,

=

S

<

can be expressed in

F (x l, x 2) in the exterior of r. Using

one may easily verify the following facts:

i) th e -m ap ( 1 1 . 15) takes the axes into the axes and symmetric p oints into symmetric points;

l

115

APPLICATIONS OF PARAMETRCC METHODS

ii)

F( x 1, x2)

is symmetric with respect to the

x 1 -axis,

and anti­

x 1 < 0,

symmetric with respect to the x2 -axis ; i t is positive for n egative for x 1 > 0, and tends to zero at infinity; iii) r m ay be expressed in polar form:

r =

h (()),

where

r

de­

c reases monotonically from 4/3 to 2/3 as () increases from 0 to TT/2 (the rest of r being obtained by symmetry); v i) the gradient of

F

is infinite along r

xi + x�

We now let C be the unit circle boundary values a)

( x 1 , x2)

•

= l, and we assign

on C s atisfying



is symmetric wi th the respect to the x c axis, antisymmet­ ric w ith respect to the x2 - axi s .

b) for points on c outside r in the left half-plane

( x 1 , x2) < F( x l ' x2) . Suppose now that F(x 1, x2) were a bounded h ave 0

<

xl

<

0 , we

solution of the

m inimal surface equation in the exterior of C, taking on the bound­

(x 1, x 2). Th en by property a) of , the function F ( x l ' x2) - F(- x l ' x 2) would be a bounded solution taking on the s ame boundary values, and by Theorem 1 1.3, F(x 1 , x2) F (x l' x 2). ln particular F(O, x2) F (O, x2) - F(O, x2), so that F(O, x2) = O. Hence the limit of F at infinity mus t be zero. Thus, for any E > 0, we can choose R sufficiently l arge so that I FI < E , 2 I F 1 < E on x� + x� R • We now let D be the domain in the 2 left half-plane, lying inside xi x� R , and outside the curves c and r. On the vertical axis we have F F = o. On xi + x� 2 R , we h ave F < F 2E , and for points on c outside r, F = < F. Thus the equality F < F + 2E holds at all boundary points of D ary v alues

=

=

=

+

=

+

except possibly for those on r. However, along r the grad ient of -

F

'

is infinite, and by the same re asoning as was used in

116

A SURVEY OF MTNIMAL SURFACES

Lemma 10.2, the inequality F < F

+

2 €. mus t also hold for bound­

ary points o n r. Finally, if we let D' be the cres cent-shaped do­ m ain in the left half-plane lying inside r and outside the circle C, then the inequality F <

F + 2 E on r i mplies, again by Lemma

1 0.2, a bound of the form F :5 M on the part of the boundary of D

•

on the circle C. This means that if the bound ary values cp ( x 1 , x ), 2 which have not yet been prescribed o n the part of c interior t o r,

are chosen so that cp > M at sorne point, then the exterior boundary problem cannot be solved. This proves the theorem.

+

W e conclude this discussion with the following remarks. F irst of all , one could remove the requi rement of boundednes s and ask if there exists

any

solution of the exterior Di richlet prob­

lem. However, one woul d then have to sacrifice uniqueness, since e ven for constant boundary values on the circle one bas distinct s olutions in the catenoid and a horizontal plane. N ext we note tha t the choice of a circ le as boundary curve is of no specia l i mportance . Obviously the a rgu ment works in great generality . It would be interesting to give an argument for an arbi­ trary exterior domain. F inally, we note that according to an observation of D. Gi lbarg, one c an prove by standard methods in the theory of uniformly ellip­ tic e quations that a solution of the exterior Di richlet problem will a lways e xist if the boundary data are sufficiently small. We thus have, by virtue of the non-existence for certain sufficiently large b oundary v alues, one more striking example of how the non-linearity of the m inimal surface equation affe cts the behavior of its solutions.

PARAMETRIC SURFACES IN E n ; GENERAL IZED GAUSS MAP

117

§ 12. Parametric surfaces in En; generalized Gauss map. In this section we shall show how the results of sections 8 and i

9 may be e xtended to minimal surfaces in En. For this purpose we must define the generalization to n dimensions of the classical Gauss map, a nd study its properties in the case of minimal sur­ iaces. This was first done for n

=

4 by Pinl [ 1]. We may also note

here that Pinl bas obtained a number of results on parametric min­ i m al surfaces in En which we are not able to discuss here. (See P inl [ 2, 3] and further references given there. Note also the papers i of Beckenbach [ 1] and Jonker [ 1].) For arbitrary n, the Gauss map of minimal surfaces was first studi ed

by Chem [ 1]. The present

section is d evoted to the results in that paper, together with those given in Osserman [5] and in Chem and Osserman [ 1]. This is a difW e begin by discussing the Grassmannian G 2 , n ferentiable manifol d whose points are i n one-to-one correspondence •

with the set of all oriented two-dimensional linear subspaces of En, i . e., all oriented planes through the origin. Let ll be any such p lane , and let

v, w

be a pai r of orthogonal vectors of equal length

which span n, and such that the ordering orientation of ll. We then set zk thus obtain a map (12.1) Suppose next that

v, w

' ' v , w

...

=

k

v

+

v,

w

agrees wi th the

i wk , k

=

1, ... , n, and we

z f. en .

are another such pair of vectors span­

n ing ll, and z ' is their i mage under the map (12. 1). Then one veri· fies easily thaf there exists a complex constant c su ch that z�

c z, for all k. Thus the map (12. 1) induces a map (12.2)

=

A SURVEY OF MINIMAL SURFACES

hich assigns to each plane n a point in the (n - 1) -dimension al

t:complex projective space.

Furthermore, we have

n

zf = I vf - I wf - 2i i vk wk = O. kI =l Thus, if we introduce the complex hyperquadric

Qn _ 2 = , z f pn -l(C) : Î k= 1

( 12.3)

zf = 1

the m ap ( 1 2.2) actually takes the form <12.4) c onversely , for any point then the vectors the point

z.

v, w

z

<

Qn _ 2 , if we write zk = vk

iwk ,

wi ll span a plane n which correspond s to

We thus see that the m�;�p (12.4) defines a one-to-one

correspondence between ali planes ll and ali points of thQs identi fy

+

Qn- 2

with the Grassmannian

G2,n

•

Qn _ 2 • We

Suppose now that we have a regular cr-surface S defined by a map 1

( 12.5) where M is an oriented 2-manifold. Then at each point we have a

1 tangent plane ll ( p). eters

u l, u2

This may be defined in terms of local param­

ax;au 1, ax;au2 . We may then assign

as the plane spanned by the vectors

,and îs independent of choîce of parameters.

the poin t z( p) obtained by the map ( 12.4) , and the resulti ng map ( 12.6) where

g : S ->

Qn - 2

g : x ( p) f-+ ll ( p)

...-.

z ( p)

PARAM E T RIC SURF ACES IN

p;ll ;

GEN E:R ALIZED G AUSS M AP

119

i s called the generalized Gauss map of the surface S. In the case n

=

3, we have a one-to-one correspondence between oriented planes

ll and their unit norma ls N, and one-to-one correspondence between the points of Q 1 and those of the unit sphere obtained by setting

z /(z 1 - iz ) and fol lowing by stereographie projection. The 2 m ap (12.7) then reduces to the classical Gauss map: x (p) _,. N( p). w

We will simply refer to the map ( 12.7)

as

the "Gauss map" for arbi­

trary n. We recall next that the immersion (12.5) defines a natural con­ form ai structure on M, where local parameters on M are those which yield i sothermal parameters on S. L EMMA 12. 1. A surface S defined by ( 12.5) is a minimal sur­ face if and only if the Gaus s map (12.7) is anti-analytic. Proof:* In a neigh borhood of any point the surface S may be represented by x ( ') i n terms of isothermal parameters '

Ç 1 + iÇ • 2 are orthogonal, equal in length

Then the vectors à x/àÇ 1 , à x/à Ç 2 and span the tangent plane. Thus we may define the Gauss map by s etting

( 1 2.8)

By Lemma 4.2 we have: S a minimal surface <

xk harmonie <

>

z k analytic i n '

·

t

L EMMA 12.2. A surface S in E" lies in a plane if and only if its image under the Gauss map reduces to a point. Proof: The image under the Gauss map is a point if and only if the tangent vectors à x/àu l ' àx/ou

2

may be expressed as linear

•This proot is not complete as 1t stands. For a discussion and complete proof, see pp.

7-10 of Hoffman and Osserman ( 1 ].

1 20

A SURVEY OF MINIMAL SURFACES

combinations of two fi xed vectors v,

w,

which is equivalent to

x(p) - x(p0) e xpressible as a linear combination of v and a li points p and a fixed point

Po .

t

w

for

TH EOR EM 12. 1 .* Let S be a complete regular minimal surface In En . Then either S is a plane, or else the image o f S under the Gauss map approaches arbitrarily c losely every hyperplane in pn - l ( C) .

P roo f: We may first of ali pas s to the universal covering sur­

fac e S of S, whi ch affects neither the hypothesis nor the conclu­ s ion. Suppose that the image under the Gauss map does not come a rb it ra ri ly close to a certain hyperplane !. ak zk th at > E

(12.9)

>

=

O. This means

0

everywhere on the image. Let S be defined by x( () : D .... En , where D may be either the unit disk or the whole (-plane. Then in the notation { 12.8), the in­ equality {12.9) takes the form n

{ 1 2 . 1 0)

where !/!( ()

!.

ak cpk(() is analytic and different from zero in D.

B ut the metric on S is given by gl. J.

= A2ô . . , where A2 = lh !. l cf;k( ( ) 1 2, l}

and by ( 12 . 1 0) log À bas a harmonie ma jorant in D . Since S is

complete , it follows from Lemma 9 . 2 that D cannot be the unit cir­ c le, but m ust be the entire plane. But in that c ase, it follows from •see Appendix 3, Section 4 .

121

P ARAMETRJC SURFACES IN E"; GENERAL!ZED GAUSS MAP

( 1 2.10) that each of the functions

if>k(IJ/1/J (�)

function, bence constant. Thus the image of

S

reduces to a point, and by Lemma 1 2 .2 c omplete,

S

S

is a bounded entire under the Gauss map

lies in a p lane, and being

t

coïncides with this plane.

The normals to a complete reguler minimal sur­ are everywhere dense unless S is a plane.

C O RO L LARY .

face S in P roof: least

a

En

Suppose that the normals to S all make an angle of at

with a certain unit vector (Al'

• . •

, \).

One can show that

this is equivalent to the i nequality

\ I Àk zk\ 2 I \ zkl 2

( 1 2 . 1 1)

which means that �he image of S under the Gauss map is bounded away from the hyperplane

I \ zk

=

O.

The result therefore follows

from Theorem 12. 1 . To obt ain (12 . 1 1) one need only choose a pair

v, w

o f orthogonal unit vectors point, and decompose n onnal at the point. If

À·N

=

INI 2,

À w

which spÇln the tangent plane at a

i n the form

À

av

+

bw

is the angle between

+

N

N, and

where

À,

N

is a

then since

we have

( À · v) 2 + (A. w) 2

=

a2

+

b2

=

l- I NI 2

which is precise! y ( 1 2 . 1 1), where

zk

=

l c os -

vk + i wk .

2w � l

-

c os

2a

t

The above theorem and its corollary may be regarded as the first steps in the direction of settling the following basic problems:

Describe the behavior of th e Gauss map for an arbitrary com­ plete minimal surface in E". 1.

2.

i tself.

Relate this behavior to geomet�ic properties of the surface

122

A SURVEY OF MINIMAL SURFACES

In sections 8 and 9 we have given a number of results of this type for the case n = 3. We now review briefly the situation for a rbitrary n. First let us make the following definition. D EFIN ITION . The Gauss map is degenerate if the image lies in a hyperplane. L EMMA 12.3. Let S be a minimal surface in En whose image

under the Gauss map lies in a hyperplane (12. 12) If

H is a "real " hyperplane; i .e. , the ak are real, then S it­

s elf li es in a hyperplane of En . If H is a tangent hyperplane to Q _ 2 , i. e. , n ( 12. 13) then there exists a change of coordinates in En, say xi = I bj kxk , where B = ( bj k) is an orthogonal matrix, such that x 1 + ix 2 is an analytic function on S. Proof: In the case where the ak are real, equation (12. 12) sim­ ply says that the tangent vectors v and w are both orthogonal to the v ector a = (a l ' . . . , a ) ' where zk = vk + iwk. By integration , n the surface lies in the hyperplane orthogonal to the vector a . In the s econd case, if we set ak = ak + i{3k , the vectors a and are real orthogonal vectors of equal length, by (12. 13), and we may assume them to be unit vectors . We may then complete them o an orthonormal set of vectors which we use to form the columns f the matrix B. Then introducing the functions rpk in terms of ocal isothermal parameters , equation (12. 12) takes the form

PARAMETRIC SURFACES IN En; GENERALIZED GAUSS MAP

123

But this is just the Cauchy-Riemann equations for the function x 1

ix , which proves the lemma. + 2 Let us n ote the following concerning the two cases which arise

+

in the lemma. ln the first case we can make

orthogonal change

an

of coordinates so that S lies in a hyperpl ane x 1 �onsta�t, or _ equivalently cp 1 = O. In the second case we have cp 1 = icp , or 2 cp- 21 + cp- 2 = O. In general , we are led to make the following defini2 tion. =

D E FIN ITION .

A minimal surface in

En

is decomposable if,

with respect to suitable coordinates , we have (12. 14)

m

� cp ( (,) 2 k 1 k=

=

rn <

0, for sorne

n .

N ote th at equation (12. 14) implies th at also n

k

�

=

m

This means that if we write

+

1

En

cp ( (,) 2 k =

=

0.

Em $ En - m ,

then the projection

of the surface into each of these subspaces is again a minimal sur­ face . although one of them may degenerate into a constant mapping. In particular, one can always manufacture minimal surfaces in h igher dimensions out of pairs of minimal surfaces in lower dimen­ s ions by this procedure. As a special case, if

are analytic functions of a variable (, where ,

n

is even, then the

1 24

A SURVEY OF MINIMAL SURFA CES

s urfa ce

is a (decomposable) minimal surface in En . The ex­

x ( ()

a m ple given at the end of section 2 is an illustration of thi s . L EM M A 1 2 . 4 .

For a minimal surface S in E 3, th e following are

equivalen t: a ) S is decomposable; b) the Gauss map of S is degenerate; c) S lies on a plane.

Proof: C learly a) is equi valent to c) , because if S is decompos­ able , either m

=

l or

m

=

2, in which cas e either

cp 1

=

0 or cp 3

= O.

If the Gauss ma p is degenerate, then the i m age of S lies in a hyper­

2

plane H of P ( C) and in the quadric fy th at H n

Q

1

Q • But one may easily veri­ 1

consists of either one or two points, and sin ce the

i mage is conn ected , it must reduce to a point, and S lies in a plane by Lemma 1 2 . 2 .

+

If one now passes to h i gher dimension s , then no two of these c onditions are equivalent, even if one replaces condition c) by the n atural generali z ation, "S lies in a hyperplane. " We have already s een th at the latter condition corresponds to a special type of de­ generacy or decomposabi lity . As for the first two conditions , the e xample given at the end of section

5

is a surface for which cp

2

=

2icp , so tha t the Gauss map is degenerate, but one can easily 1

verify th at it is not decomposable.

In the above discussion , as well as in condi ti on (12.9) in Theo­ rem 1 2 . 1 , we have described the Gauss map in terms of the image lying in certain s ubsets of

pn- 1(C) . For many purposes it is more

n atural to characterize the set of hyperplanes in

pn - l( C) which in­

tersect the image. One can easily verify , for example, th at equation 12.9) is e quivalent to the statement: " the image under the Gauss

PARAMETRIC SURFA CES IN E";

125

GENERALIZED GAUSS MAP

m ap does not intersect any hyperpl ane sufficiently near the hyper­ plane �

ak z k

=

"If S

O. " Thus Theorem 1 2 . 1 can be reworded :

is a complete minimal surface, not a plane, then its image under the Gauss map intersects a dense set of hyperplanes. "

•

For a c loser study of the Gaus s map from this point of view we note the followi ng. L E M M A 12 .5.

Given a minimal surface S in

map g(S), and a hyperplane in

H,

or else

H

E" ,

the Gauss

in pn- l( C) , then ei ther g (S) lies

H

intersects g (S) at isolated points with fixed mul­

tiplicities. Proof:

If H is given by �

ak z k

choose local isothermal parameters Then either

t/J ( ()

=

0, then near any point of S

=

(,

and set

t/1 (()

0, i n whi ch case g (S) lies in

by a nalyticity, globally, or else

g ( ()

=

� ak cpk (() .

H loca lly, and

bas isolated zeros whose

m ultiplicities are independent of the choice of parameter

.

(

We are th us led to a study of the "meromorphic curve "

+

g (S),

which is precisely the type of object treated in detail in the book of Weyl

[ 1].

lt is advantageous for the general theory, and particu­

larly for the applications we wish to make, to introduce a Rieman ­ nian metric on P" - 1 (C) . We may do this by defi ning the element of arclength do of a di fferenti able curve

(12 . 1 5)

( ç!_C!_ )2 dt

=

2 2 i z( t) /\ z '( t) l 4 l z ( t) i

z ( t) ,

by the expression

=

U p to a multipli cative factor, thi s is a standard me tric for complex projective s pace. Suppose now that we have a mi ni mal surface of local is othermal parameters •see Appendlx 3, Section 4.

(, zk

=

cpk( ().

S,

and in terms

Then (12. 15) takes

126

A SURVEY OF MINIMAL SURFACES

the form

(12. 16) where

À

(12. 17)

I n p articular, the image of a domain � in the (-plane has area

( 1 2. 18)

A

2 rJ

�

2

.;_;__--'----'-:� -

aç1 aç2

w ith respect to this metric. We next compute the Gauss curvature. The formula (9.5): K - � log ,\f,\2 , applied to the metric on S :

=

(12.19) yields the expression

(12.20)

K

In particular, for the total curvature, we obtain

In other words : the total curvature of any part of the surface is the

negative of the area of the image with respect to the metric (12. 15). We thus have a precise generalization of the situation in three d imensions , represented by (9. 1 1), and we may exam i ne the form in which various theorems generalize. The methods of inte gral

PARAMETRIC SURFACES IN En; G E!VERAUZED GAUSS MAP

127

geometry allow one to relate the number of times the image under th e G auss map intersects various hyperplanes with the area of the image, bence with the total curvature of the surface. We do not go into the details here, but refer the interested reader to the discus­ s ion in the p aper of Che rn and Osserman [ 1].

APPENDIX 1 LIST OF THEOREMS

T H EO RE M 5 . 1 . Let f ( x 1 , x ) 2

==

( f ( x ' x ) , . . . , f ( x , x ) ) be a 3 l 2 2 n 1

solu tion of th e minimal su rface equation (2. 8) in th e whole x ' x l 2 p lane. Then th ere exists a non singular linear tran sforma tion x u , 1

x

2

==

au

1

+

bu , 2

b

>

=

==

0, such that ( u , u ) are (global) i so­ 1 2

th ermal parameters fo r the surface S delin ed by x k

k

1

=

f ( x ' x ), k l 2

3, . . . , n . C O R O L L A R Y 1 . ln th e case n

=

3, the only solution of the

minimal surface e quation in the whole x ' x -plan e is the trivial l 2 s olu tion, f a linear fun ction of x • x • 1 2

C o R O L LARY 2 . A bounded solution of e quation (2. 8) in the whole plane must be cons tan t (for arbi trary n).

C O RO L L A RY 3. L e t f( x , x ) be a solution of (2.8) in the 1

2

whole x , x -plane, and let S be th e surface defin ed by x 1 k 2

Ïk( u 1 ,

u ), k 2

=

==

3, . . , n, o btained by referring the s urface S to the .

i sothermal parameters given in Th eorem 5. 1 . Th en the lunctions

k

are analy tic lun ction s of u

1

+

iu

2

sati s fy the e qua tion

1 29

3, . . . , n ,

in the whole u , u -plane and 1 2

APPEND IX 1 LIST OF THEOREMS

T H EO REM 5 . 1. Let ! ( x l ' x ) 2

=

(f (x , x ), 3 1 2

• . •

, f ( x , x ) ) be a 2 n 1

solu tion o f th e minimal surface equation (2. 8) in the whole x ' x l 2 p lane. Then there exists a non singular linear transformation x u , x 1 2

=

au

1

+

bu , 2

b

>

=

3,

• . .

==

0, such that ( u , u ) are (global) iso­ 1 2

th erm al parameters for the s urface S defin ed by x k

k

1

=

f (x , x ), k 1 2

, n.

C O R O L L A R Y 1 . l n th e case n

=

3 , the only solution of the

minimal surface e quation in the whole x , x -plan e is th e trivi al 1 2 solution, f a linear functi on of x . x • 1 2

C O R O L L ARY 2 . A bounded solution of e quation (2. 8) in the whole plane must be cons tant (for arbi trary n) .

C o RO L L A RY 3 . L e t f( x , x ) be a solution of (2.8) in the 1

2

whole x , x -plane, and let S be th e surface defin ed by x 1 k 2

Ëk( u l ' u 2 ),

k

=

=

3, . . , n, o btain ed by referring the s urface S to th e .

i sothermal parameters given in Th eorem 5. 1 . Th en the fun ctions

k

are analy tic functions of u

1

+

iu

2

sati s fy the e qua tion

129

=

3, . . . , n ,

in the wh ole u , u -plane and 1 2

130

A SURVEY OF MINIMAL SURF ACES

c = a - ib . Converse/y, given any complex constant c -

=

a - i b wi th b

-

>

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

=

<

'h , there exists a simply-connected

c, bounded by r.

Complete surfaces of constant m e an curvature are studied i n K lotz a n d Osserman

[ l]

using methods s i mi lar t o those of section

8 . The follo wing result is proved: Let S be a complete surface o f constant m ean curvature.

If K > 0 on S, then S is a plane, 135

136

A SURVEY OF MINIMAL SURFACES

a sph ere, or a right circular cylinder. If K :5 0, then S is a cylin­ der or a min imal surface. If one makes no restriction on the sign

of the Gauss cu rvature K , then there exist complete su rfaces of constant m ean cu rvature which are none of the types listed . C oncernin g non-parametric surfaces, we have fi rst of ali the followin g result, which may be considered the natural generaliza­ tion of Bemstein' s theorem. Let l of constan t mean curvature H

=

c

x 3 F(x 1, x 2) =

>

0 in a disk

define a surface

xi x� +

<

R 2•

Then R :5 1/c, wi th equality holding only if the surface is a hemi ­

l sphere. The first statement in the conclusion is due to Bernstein hl and the second to Finn [ 5] . The theorem of Bers on removable singularities (Theorem 10 .2) goes over without change (Finn [ 1] ),

and indeed so does the fact that every set of vanishing linear Haus­ d orff m easure is removable (Serrin [2]). Finally, there is an exis­ tence theore m corresponding to Theorem 10.3. (See li B , below. ) B . Quasiminimal surfaces. C onsid e ring minimal surfaces in non-parametric form from the point o f view of partial differentiai equations, one is led to ask for m ore gene m l equations whose solutions behave in a similar manner. Various classes of equations were studied by Bers [ 3] and Finn [ 2] with a view toward generalizi n g results such as Bemstein's

1th eorem and Theorems 10.2 and 1 1 .2 of Bers. The underlying idea

•

i s to replace conformai maps by quasiconformal maps at sorne point ·

n the theory . ln particular, Finn introduces a class of equations

of " m inimal surface type" whose solutions have the property that their Gaus s map is quasiconformal. With varying additional restric­ ions, these equations have been studied in Finn [ 2, 3, 4] , J enkins 1, 2], and J enkins and Serrin [ 1].* It is shown in the se papers th at Appendix 3. Section 6IB.

1 37

GENERALIZATIONS

a l a rge part of the theory of minimal surfaces which we have pre­ s ented h ere can be carried over, and in fact sorne resu l ts were firs t proved in this wider context before they were known for the special case of minimal surfaces. Amon g the most interesting, let us note the Harnack inequality and the convergence theorems of J enkins and Serrin

[ 1].

If one considers parame tric surfaces with the single require­ m en t that the Gauss map be quasiconformal , one is led to the c lass of surfaces introduced in Osserman

[4]

as "quasiminimal

surfa ces." Certain results , for example Theo rem 9.2, can be shown to continue to hold in this genera lity . It would be interest· in g to know whether Bemstein 's theorem is true for qu asiminimal surfaces, or whether sorne of the additional restrictions imposed by F inn a re really necessary. • II. Hypers urfaces in

En .

A . Minimal hypersurfaces. F rom m any points of view the most natural generalization ;:,f surfaces i n

E3

is to hypersurfaces in

En .

If one asks for the

condition th at a hypersurface minimize volume with res pect to all hypersurfaces having the same boundary, then one is led to th e geometrie c ondition that the mean c u rvature be zero , and the ana· lytic condition that the surface in non-parametric form

F(x 1, . . . , x _ 1 ) n

x n

s atisfy the equation

(1)

w

It h as lon g been an open problem whether Be mstein' s theorem generalizes to this case, i . e . , whether every solution of (1) de•see Appendix 3, Section 6IB.

138

A. SURVEY OF MINIMAL SURFA CES

fined for all x 1 , . . . , x _ 1 is linear. This was proved i n the case n n 4 by de Giorgi [ 1] , for n 5 by Almgren [ 2] , and for n 6, 7, 8 =

by Simons [ 1]. Weaker forms of Bernstein's theorem have been proved for arbitrary n, where the conclusion that F is linear is o btained by imposing additional restri ctions, such as

W

bounded

(Mos er [ 1]) or F positive (Bombi eri , de Giorgi , and Miranda [ 1]). T he latter paper also contains references t o a number o f other re· "' cent contributions to the theory of minimal hypersurfaces. A n analogue of Theorem 7 . 2 is given in a recent paper of M iranda [ 1], while de Giorgi and Stampacchia [ 1] gave the follow­ ing theorem on removable singulari ties: Every solution of equation (1) in a domain D from whi ch a compact subset E of (n - 2) ·dimen·

sional H ausdorll measure zero has been removed ex tends to a so· lution in ali of D. + U p till now it has not been possible to extend the results of s ections 8 and 9 on parametric surfaces to the case of hypersur· faces. F in ally , J enkins and Serrin [ 3] have obtained the following re· markable generalization of Theorem 10.3: Let D be a bounded

domain in En - l wi th C 2 boundary, and let H be the mean curva· ture of the boundary with respect to the inner normal. Th en neces· sary and suflicient that th ere exist a solution of equation (1) in D taking on arbitrary assigned C 2 boundary values is that H 2 0 e verywhere on the boundary.

*

It appears th a t th e problem of Bemstein's theorem h as just been settled

in th e m os t unexpected fashion. Bombieri

[ 1]

h as announced th e construc·

tion of a counterexam p le showing that although Bemstein's theorem is tru e for

n

Bomb

< 8, i t fails to hold for

�ri,

n>

de Giorgi, and G iusti

8. The details will appear in a paper of

[ 1].

tSee Appendtx 3. Sections 61IA and 6IIB.

139

G E N ERALIZATIONS

B. O th er hypersurfaces. The result of J enkins and Serrin just referred to has been fur­ ther extended by Serrin [3] to a large class of equations which in· e lude the equation for surfaces of constant mean curvature. His result is the following: Under the same hypotheses as before,

n ecessary and sufficient that there exist a hypersurface of con­ stant mean curvature c corresponding to arbi trarily prescribed C 2 boundary values i s that H .2: c(n - 1)/(n - 2) at each point of th e bormdary. We note also the fol lowing result, contained in a recent paper of C hem [ 2]. If x

n

•

F(x 1,

constant mean curvature, H

• • •

=

,x

n-1

) defines a hypersurface of

c, over the whole space of xl'

•

.,,

xn - l ' th en in !act H = O. III. Minimal varieties in

En .

We come now to the general case of k-dimensional minimal vari· eties in

En, where 2 :5 k :5 n - l. Very little progress had been

m ade for the intermediate values of k, 2 < k < n - l until recently, when there appeared the papers of Reifenberg [1] and Federer and F lemin g [ l]. ln both cases one gets away from parametric methods (includin g the "non-parametric methods" discussed earlier) and in troduces s ui table classes of objects which are handled chie fly by measure-theoretic methods. In particular, Reifenberg [ l, 2, 3] s howed that there always exi sts a compact set X having a given boundary A in a certain precise sense, such th at the set X- A, outside of a possible subset of (k - 1) - dimensional H ausdorff mea­ sure zero, is a k-di mensional mini mal variety. These methods have been carried fjlr ther by Almgren [ l] and by

140

A SUR VEY OF MINIMAL SURF ACES

M orrey [ 3] . The book of Morrey is an excellent general reference to this theory, viewed in terms of differentiai equations and the c alculus o f variations. A m ajor ques tion conceming these methods is whether the pos ­ s ible singular sets actually oc cur . Simons [ 1] showed that in the c ase o f hypersurfaces in dimension s n .$ 7, the set X - A is a real a nalytic m inimal variety without singularities . He also gave an ex­ ample for n

=

8 of a cone having a singularity at the origin which

provides a relative minimum in Pl ateau's problem. Now Bombieri [ :l] has announced a series of resul ts which prove that Simons ' cone represents an absolute minimum , and hence a counterexample to the regularity of the solution .

Details are gi ven in Bombieri,

de Giorgi and Giusti [ 1]. IV. Minimal subvarieties of a Riemannian manifold.

The th eory of two dimensional minimal surfaces in a Rieman­ n ian m anifold has been studied by various authors (for example, Lumiste [ l] , Pinl [2]). Morrey [ 1] showed that the problem of Pla­ teau could be sol ved also fo r the case of a Riemannian manifold. M ore recently Morrey [ 2, 3] has extended the work o f Reifenberg from the euclide an to the Riemannien case. Almgren [ 1, 2] and Frankel [ l] have studied topological properties of minimal sub­ varieties. We have noted earlier that a complex-analytic curve in en , considered as a real two-di mensional surface in E2n , is al ways a minimal surface. This generalizes to the statement that a Kahler s ubmanifold of a Klfuler mani fold is a minimal variety. For this and related statements we refer to the recent paper of Gray [ 1] . A m ajor contribution t o the theory o f mini mal varieties has been m ade by Sim ons [ 1]. He deri ves an elliptic second order

141

GENERALI ZATIONS

e quation w hich i s satisfied by t h e s econd fundamental form o f a m inimal variety in an arbi trary Riemanni en m anifol d. There are a n umber of appl ication s , including the extension of Bemstei n ' s theo rem mentioned above (in liA), and a l arge pa rt o f the earlier theory is uni fied and generalized.

Minimal s ubmanifolds of spheres have been the object of par­ ticularly i n te ns i ve research in the past two or three years. In ad­ d ition to s evera! of the above·menti oned papers we note the work of C alabi

[ 2]

which applies complex-variable methods to 2-dimen­

s ional minimal surfaces in an n-sphere , and the papers of Chern , do C armo, and Kobayashi Ot suki

[ 1],

Rei lly

[ 1] ,

[ l] ,

Hsi ang

Tak ahashi

[ 1] ,

[ 1, 2] ,

Lawson

[ 1, 2, 3] ,

and Takeuchi and Kobayash·

[ 1] . T o all those whos e work h a s not been mentioned, o r barely s kimmed over in this brief survey , we apologize. To the novice in th is fie ld we hope at least to h ave given s orn e idea of the range a nd the depth of cu rrent activity in the field of minimal surface s .

APPENDIX 3 DEVELOPMENTS IN MINIMAL S URFACES, 1970-1985 1.

Plateau 's Problem

Many basic questions concemlng Plateau's problem have been set­ tled in the past decade. The fundamental existence theorem of Douglas, Theorem

7. 1,

may now be sharpened to the form: let r be an arbitrary

Jordan curve in E3• Then there extsts a regula:r simply-connected minimal surface bounded by r. (See Osserman

Gulliver, Osserman, and Royden l i ], and Ait

1 1 1 1.

Gulltver I l l.

[ 1 ]. ) Among the further

questions that arise are whether or not the surface obtal ned is actually embedded ( i .e. , without self-Intersections), and when the solution is unique. If the curve r lies on the boundary of a convex body, then

Meeks and Yau I l l proved that the surface is indeed embedded, whUe, independently. Almgren and Simon

[ 1 ] and Tomi

and Tromba I l l proved

related embedding results. Earlier. Gulliver and Spruck I l l proved the result under the additlonal assumptlon that the total curvature of r was

at most 41T.

In ali the above results we cannot properly speak of the surface bounded by r (or the Douglas solution) slnce one does not ln general have unlqueness. An interesUng example ls provided by Enneper's

surface. In the standard representation of Enneper's surface ( see p. 65 , above), If one denotes by r, the image of the circle

[10]

has shown that for

1 <

r

< 1

1'1

=

r, then Nltsche

+ e there exist at !east three distinct

solutions to Plateau's problem wlth boundary f,. In the other direction, Ruchert I l l proved that for 0

<

r ,.:;

1,

the solution ls unique. Ruchert's

proof makes use of a sharpened form of a unlqueness theorem due to

143

APPENDIX 3 DEVELOPMENTS IN MINIMAL SURFACES, 19 70-1985 1. Plateau 's Problem Many basle questions concernlng Plateau's problem have been set­ tled ln the past decade. The fundamental existence theorem of Douglas, Theorem 7. 1, may now be sharpened to the form: Let r be an arbitrary

Jordan curoe in E3• Then there exists a regular simply-con nected minimal surface bounded by r. (See Osserman 1 1 1 1 . Gulliver

I l l.

Gulllver, Osserman, and Royden [ 1 ] , and Alt [ 1 ]. ) Among the further questions that arise are whether or not the surface obtalned is actually embedded ( i . e . . wlthout self-Intersections ), and when the solution is unique. 1f the curve r lies on the boundary of a convex body. then

Meeks and Yau [ 1 1 proved that the surface is lndeed embedded. while. independently. Almgren and Simon [ 1 ] and Tom! and Tromba [ 1 ] proved related embedding results. Earlier. Gulllver and Spruck ! 1 1 proved the result under the additional assumption that the total curvature of r was

at most 4?T.

In ali the above results we cannot properly speak of the surface bounded by r (or the Douglas solution) since one does not ln general have uniqueness. An interesting example is provlded by Enneper's

surface. In the standard representation of Enneper's surface (see p. 65 . above), If one denotes by r, the image of the clrcle

[ lOI has shown that for 1

<

r

<

1

+

1'1

=

r, then Nitsche

e there exlst at least three distinct

solutions to Plateau's problem with boundary r,. In the other direction, Ruchert [ 1 ] proved that for 0

<

r .;; 1, the solution is unique. Ruchert's

proof makes use of a sharpened form of a uniqueness theorem due to

143

144

A SURVEY OF MINIMAL SURFACES

Nltsche [91: a real analyttc Jordan curve in E3 whose total curvature ts at most 4 1T bounds a untque simply-con nected minimal surface. An lnterestlng counterpart ls a construction of Almgren and Thurston

I ll.

who show that gtuen any E and any positive integer g. there extsts a Jordan curve r ln E3 wlth the property that the total curvature of r ts at most 41T + E. and any embedded minimal surface bounded by r must haue genus at least g. If one drops the requlrement of simple connectivlty, then one has extensive results due to Gulllver

[2] on

the

regularlty of Douglas' solutions of hlgher topologlcal type (where the topologlcal type ls speclfled ln advance), and the basle theorem of Hardt and Simon

I l l provlng

the existence of a regular embedded minimal

surface spannlng an arbltrary glven set of Jordan curves ln E3 • When we go to En, n ;;. 4, although Douglas' basle existence the­ orem 7. 1 stlll holds , both regularlty and unlqueness tend to break down. An Important example ls the surface S ln E4 deflned non­ parametrtcally by

lx1

+ tx2l

E;

3

x

=

Re{R(x1 + lx2l4},

x4

=

Im{R(x1 + tx2J4}.

1, R a positive constant. As was noted at the end of §2, S ls

a minimal surface. Its boundary r ls a Jordan curve. Federer

I l l showed

that S ls ln fact the unique orlented surface of !east area bounded by r. Thus ln this case, Douglas' solution is unique. On the other hand, Morgan

I l l has shown

that there exlsts a non-orlented minimal surface

s• bounded by r. and that for sufflclently large R there exists a one­ parameter family of isometrles of E4 mapplng r onto ltself and taklng s•

jonto a one-parameter family of mutually distinct m inimal surfaces of !east area ali bounded by r. Federer's Theorem also gtves examples where Douglas' solution ls not regular. but has branch points. Even for non-parametrlc solutions, uniqueness may fail. Thus, ln

ftheorem 7.2. the solution is unique when n

�

hereas for n

;;.

3 by Lemma 10. 1 ,

4 there exlst arbitrarily smooth boundary values o n the

DEVELOPMENTS 1970-1985

145

unit circle for which the solution is not unique ( Lawson and Osserman [ l ]l. A paper of White 1 2 1 gtves a generlc regularity result : for almost

every smooth closed curve in En, the unorlented area-minimizing sur­ faces that lt bounds have no singularltles. Returning to E3, Morgan [31 gave another example of a continuum of distinct minimal surfaces havlng the same boundary. In his example, the boundary ls the union of four disjoint circles. On the other band, in the paper referred to above. Hardt and Simon i l l show that if r is an arbitrary collection of sufflclently smooth Jordan curves ln E3, then r can bound only a fini te number of oriented area-mtntmtzing surfaces. Thelr result uses basic earlier work of Bôhme and Tom! [ 1 1. as weil as Tomi [ 11. For further work on f1niteness. we refer to the papers of Tromba [ l J. Morgan ( 2 , 5 ] , Nttsche [III]. Beeson 121. Kolso I l l . and Bôhme and Tromba [ 1 ). Another area of major progress is that of regularity at the boundary. This was settled by a number of authors followlng the Initial ground­ breaking work of Hildebrandt [3]. For details we refer to the book of Nltsche

lill (Chapter V, §2. 1 ) and to the paper of Hardt and Simon [ 1 ].

Finally we note that a totally different approach to regularlty ls gtven by Beeson I l l. who applies varlational arguments in a neigh­ borhood of a branch point. For background on the subject of branch points on minimal surfaces, see Osserman I I I and Nitsche [II], V. 2.2. For more on Plateau's problem, tncludtng a long list of open problems, see the lecture notes of Meeks [ I ]. 2. Stabiltty There ts a class of minimal surfaces that falls between that of the area-minimizing surfaces and that of the totaltty of ali minimal sur-

46

A SURVEY OF MINIMAL SURFACES

faces: stable minimal surfaces. There are various r.ouous of stability, but basically a stable surface is area-minimiztng relative to nearby surfaces with the same boundary. Thus, stable minimal surfaces are precisely the ones that one would expect to obtain in physical experi­ ments. In terms of our discussion ln §3, mlnlmality of a surface ls equivalent to the vanishing of the first variation of area A'(OJ under ali deformations of the surface leavlng the boundary flxed. Thus, mlni­ 1

mality ls a necessary condition for a surface to be area-mlnlmizing. However, If for sorne deformation the second variation is negative.

A"(OJ < O. then there are nearby surfaces of smaller area. and the surface ls called unstable. The property of stability has assumed increasing Importance in recent years. Among the results obtalned are the following: Barbosa and do Carmo [ 1[ showed that for a surface S in E3, if the image of S under the Gauss map has area less than 21T, then S is stable: the second variation of area is positive for ali deformations fixing the boundary. This result has found important applications in Nttsche's theorems [9,IIII on uniqueness and fintteness. It was later generallzed to minimal surfaces ln En by Spruck [ 1 [ , replacing the Gauss map by the eneralized Gauss map ( 12. 7). Spruck's result was ln turn lmproved by arbosa and do Carmo 121. who showed that the assumpt!on ffiKidA < 411'/3 implied stability for simply-connected minimal surfaces in En. The above results ali involve sufficient conditions for stabiUty. In the other direction. the assumptlon of stability often has important consequences. One example ls a recent result of Schoen [ 1 ], who obtains an analog of inequality ( 1 1 . 1 ) for stable surfaces. Specifically, Schoen shows that there ls a universal constant c such that if S is a stable

!minimal suiface ln E3, thenjor any point p ln S the Gauss curvature K

1Q{ S at p satisfles IK!pll � c/d2, where d

ls the distancefrom p to the

147

DEVEWPMENTS 1970-1985

boundary of S. Since a non-parametric minimal surface can easily be shown to be area-mintmizing with respect to its boundary and hence stable, Schoen's result is an extension of Heinz inequality ( 1 1 . 7 ) , which in turn tmpltes Bernstein's Theorem. It also tmplies the earlier result of Fischer-Colbrie and Schoen

I l l and do Carmo and Peng [ li that, if S

is a

complete globally stable minimal surface tn E3, then S ts a plane. The assumptton "S globally stable" means that every relatlvely compact domain D on S ts stable. Another class of surfaces that are area-mtnimiztng, and therefore globally stable, is complex holomorphie curves in (:n, considered as real surfaces in E2n (Federer [ 1 ] ) . A paper of Micallef [ l ] makes a number of important contributions to the study of stability, including the proof that a complete global!y stable minimal surface in E4 must be a holo­ morphie complex curve with respect to sorne orthogonal complex struc­ ture on E4 (which is thereby identlfied with C2), provlded that sorne further condition is satlsfied, such as a restriction on the Gauss map. finite total curvature, or that the surface is non-parametric. This last condition was obtained independently by Kawa! [ 1] . It shows in par­ tlcular that the surfaces, such as ( 5 . 19), obtained as global solutions of the minimal surface equation must be unstable, even though they are non-parametric. The paper of Schoen [ 1 1 applies to minimal surfaces not only in E3 but also in a three-dlmenslonal Rtemannian manifold. The study of stable minimal surfaces ln Rlemanntan manifolds has been a very fruitful one in recent years and has led to sorne of the most Important advances. These will be discussed below in Sections 6.IV.A and 6.V. 3 . Isoperimetric tnequaltttes There are two roles that lsoperimetrtc inequalities have played in recent work on minimal surfaces. First, isoperimetric inequallties for

148

A SURVEY OF MINIMAL SURFACES

domains on minimal surfaces have been applied ill a .nümber of wa�. (For a discussion of such applications, see §4 of Osserman [III]. ) Second, isoperimetric tnequalities on arbitrary surfaces have been applied to the theory of minimal surfaces. For example, the stability results of Barbosa and do Carmo f l . 2, 31 and Spruck

Ill depend on the use of iso­

perimetric inequalities on the image of a minimal surface u nder the (generalized) G auss map. It bas been a long-standing conjecture that the classlcal lso­ perlmetric inequality L2 ;ae 41TA should hold for an arbttrary domain on a minimal surface in E11, where A is the area of the domain, and L the length of the boundary. Until recently that was only known for an oriented domain bounded by a single Jordan curve. (New and simpler proofs of that have been given by Chakerian I l l and Chavel [ 1 ]. ) Then 1t was proved for doubly-connected domains, ftrst in E3 (Osserman and Schiffer [ 1 )) and then in general ( Feinberg [ 1 ]). That i n turn implied the result for minimal Môbius strips ( Osserman ( 13 ]). Except under further hypotheses, the sharp isoperimetric inequality for domains of arbitrary topological type on minimal surfaces is still not known. (See LI, Schoen. and Yau

I l l for certain hypotheses that suffice. ) For more details, see

Nltsche !III. §323 , and Osserman [III ] , §4.

4. Complete minimal surfaces One of the most strlking results of recent years is a theorem of Xavier [ 1 ] that goes a long way toward answering the questions on p. 74, above, concerning the gap between Theorems 8. 2 and 8.3. What Xavier shows is that the normals to a complete minimal surface in E3, not a

plane, can omit at most six directions. The method of proof is qui te different from that used in the proof of Theorem 8.2, and it does not seem amenable to reducing the number of omitted values to four, which

DEVELOPMENTS 1970-1985

14

by Theorem 8.3 would be best possible value. Thus, it sttll remains an '1

open question to determine the precise number of values that the normals to a complete non-planar minimal surface ln EJ can omit. Xavier's Theorem represents a very strong general!zation of Bern­ stein's Theorem (Corollary 1 of Theo rem 5 . 1 ). Two other general!zatlons of Bernstein's Theorem ln different directions have been proved re­ cently. One !s the theorem on complete globally stable surfaces dis­ cussed above. The other is a theorem of Schoen and Simon [2). Instead of a condition on the Gauss map or on stability, an assumption is made on area growth. Speclfically, they prove the followtng: let S be a stmply­

connected complete minimal surface embedded in E3• For someftxed point p in S, let Sp(r) denote the component containing p of the intersection of S with a bal! of radius r centered at p. Let A(r) be the area qf SP( r). If A(r) < Mr2 for someftxed M. then S must be a plane. Using the area-m!nim!zlng property of non-parametric minimal sur­ faces. and comparing Sp(r) with the area of a domain on the sphere of radius r having the same boundary. one sees easily that the classical Bernstein Theorem is a consequence of the Schoen-Simon Theorem. An excit!ng recent development has been the discovery of the first new complete embedded minimal surface of finite genus in over two hundred years . The previously known examples were the plane, the catenoid, and the hel!coid. A paper of Jorge and Meeks [ 1 1 studied necessary conditions on the Weierstrass representation for the resulttng surface to be a complete embedded minimal surface. Using those condi­ tions, Costa I l l showed that there exists a constant c such that the choice off= p g = c/p' in the Weierstrass representation (8.2), (8.6), ,

where p is the elliptic p-function of Weierstrass based on the unit square, gtves a complete minimal surface of genus one ln E3, wlth three embedded ends. Hoffman used computer graphies to obta!n excellent

150

A SURVEY OF MINIMAL SURFACES

plctures of Costa's surface, showing clearly that it was embedded. (See Peterson [1]. ) Hoffman and Meeks

I l l gave an analytic proof of the

embeddedness, and subsequently [2] found complete embedded mi nimal surfaces of every genus. Returning to the distribution of normals, we note that Gackstatter

I l l showed that the normals

E3,

to

a complete abelian minimal suiface in

not a plane, can omit at mostjour directions. Complete abelian

minimal surfaces are a generalization of complete surfaces of fin !te total curvature for which the functions 'Pk of (4.6) extend to be meromorphic on a compact Riemann surface, but the

X�c

=

ReJ'P1c need not be single­

valued. Since the surfaces of Theorem 8.3 are abelian, the nurnber "4" of Gackstatter's theorem is sharp. Finally, we refer to the papers of Meeks 13], Jorge and Meeks Gulliver 13], and Gulliver and Lawson

I l l.

I l l for further results on complete

minimal surfaces of finite total curvature in Es.

Complete minimal surfaces ln En have been studied by various authors. Among recent results, we note the following. Gackstatter ( 2 ] studied relationships between vartous quantities associated with a complete minimal surface En. Let

S of finite total curvature ln

m be the dimension of the smallest affine subspace of En

containing

S. Let k be the number of boundary points of S. and g its

genus. Then the total curvature of S satisfies f f KdA � ( 3 s

- m

-

k

4ghr.

This result complements inequality (9.22) in Theorem 9.3, which holds in En as weil as Es (Chern and Osserman

[ 1 )). Combining the two, one

can prove the following result. complementing Theorem 9.4: if f f KdA = -

4Tr,

s

then either S ts stmply-connected and m � 6, or else S ts

doubly-connected fg

=

0,

k

2)

and m � 5. This theorem was first

151

DEVELOPMENTS 1970-1985

proved by C. C . Chen [31 using other arguments. One can in fact give a complete characterizatlon of the surfaces with total curvature

- 411'

( Hoffman and Osserman [ 1 ]). The doubly-connected ones are particularly interesting. They are a kind of generalized catenoid, in the sense that they are ail generated by a one-parameter family of ellipses (or circles) that are obtained by in tersecting the surface with the set of ali hyper­ planes parallel to a given fixed one. In that connection we note another result of C. C. Chen [ 2 ] :

a minimal suiface S tn En that ts tsometric to a

catenoid is tnjact ttseif a catenoid lytng in a three-dimensional affine subspace. Finally we note that the complete minimal surfaces of total curva­ ture

211' have a very simple characterizatlon:

let (z.w) be coordinates

in IC2 For each real c > 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 <11.jor sorne 11.>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
also obtains analogous results in En, generalizing those of Chern and Osserman ( 1 ).

A paper of Osserman [ 14] shows that a surface in En gtven by a polynomial map must have finite total curvature; if tt is in addition a regular complete surface, then it must be conformally the plane. IL Hypersurfaces in En A Minimal hypersurfaces

One of the major advances of the past decade was a paper of Schoen, Simon, and Yau i l l in which they obtain pointwise curvature estimates generalizing Heinz' inequallty ( H . 7) to non-parametric mini­ mal hypersurfaces in En for n < 6. An immediate consequence was a

156

A SURVEY OF MINIMAL SURFACES

new proof of Bernstein's Theorem in those dimensions. But more important were the methods used, which have led to many further results (see IV.A, below). A number of improvements on the original paper were given by Simon 1 1 .3 1. A higher-dimensional version of the parametric Bernstein Theorem, Theorem 8. 1 , has been given by Solomon

I l l. ln fact, he gives a finite

version, analogous to the parametric version of Theo rem ( 1 1 . 1 ) (Osser­ man 121) in the following form: let M be a smooth area-minimizing

hyperswjace of E n. S uppose that thefirst cohomology class of M is zero, and that the Gauss map of M omtts a neighborhood U of some great (n - 3)-dimensional sphere on the (n - 1)-sphere. Then there exists a constant c depending on n and U such thatfor any point p of M. if d is the distancefrom p the boundary of M and if K1,

•

•

•

,

Kn - t are the principal curvatures of M at p. then 2 2 K + ••• +K � c/d2 . l n- I

A paper of Morgan 1 5 1 gives a number of finiteness theorems for the set of solutions to Plateau's problem for various classes of minimal hypersurfaces in En , for n

�

7.

ln another direction, the paper of Simon

121 sharpening the remov­

able singularity theorem of de Giorgi and Stampacchia holds in ali dimensions. A question that has been studied by a number of au thors is the following: given a Riemannian metric, when can it be realized on a minimal hypersurface in En? The case n Curbastro Ziller

=

3 was treated by Ricci­

I l J. while higher dimensions were considered by Pin! and

I l l. and Barbosa and do Carmo 1 4 1. Further conditions,

as

weil as

a review of the whole subject, are contained in Chern and Osserman

121.

DEVEWPMENTS 1970-1985

157

B. Other hyperswjaces ln the papers of Simon [ 2 , 3 ) referred to above, he shows that the generalized Heinz inequality, Bemstein's Theorem , and the de Giorgi-Stampacchia Theorem ali hold for broader classes of hypersur­ faces. ln particular, he extends the results of Jenkins [ 1 ) for parametric elliptic functionals in two dimensions to three dimensions, and he shows that a Heinz inequality and the Bernstein Theorem hold through dimension 7 for the non-parametric Euler-Lagrange equation of any integrand whose associated parametric functional is close enough to the area integrand. A higher-dimensional version of Hopfs question referred to earlier (6. l.A) was answered by W. -Y. Hsiang [3], who showed that for ali n ;;. 3 there exist non-standard immersions of the n-sphere in En+ I with constant mean curvature. (Hopf had shown that to be impossible when n = 2. ) Ill. Mi n i m al vartettes tn En A major breakthrough was the paper of White [ 1 ], solving the classical Plateau problem in higher dimensions. Specifically, any smooth map of the ( n - 2 )-sphere into En, for 4 � n � 7, extends to a Lipschitz map of the ( n - 1 )-ball that minimizes ( n - 1)-dimensional area among ali such maps. In fact, in ali dimensions n ;;. 4, the infimum of areas obtained from Lipschitz maps (the parametric problem) is the same as one gets using integral currents or singular chains. This result con­ trasts strongly with the situation for E3 , where, for example, a higher­ genus surface spanning a given Jordan curve may have much less area than the parametric ( Douglas) solution, obtained by mapping a disk. Sorne of the most interesting other results have concerned minimal graphs of arbitrary dimension and codimension. An m-dimensional

158

A SURVEY OF MINIMAL SURFACES

graph in En ts gtven by a set of functions xk =fk (x1 ,

•

•

•

, Xml• k = m + 1 ,

. . . , n, o r by a vector functtonf(x) , where x = (xl, . . . , Xm l . f= ifm + I • . . . . fnl· The corresponding submantfold is minimal in En if and only if f satisfies the minimal surface equation

(*)

where (g!l) = (gu l -

1

iJf

iJf

·a a . The equation (2.8) is the special and gu = 8 u + X 1 x1

case m = 2 of this equation. (See Osserman [ 9 [ , p. 1099, and [V]. ) One of the surprising results for minimal variettes of higher co­ dimension concerns the Dirichlet problem. Lawson and Osserman [ 1] showed that one can prescribe a set of three quadratic polynomtals on the boundary of the unit hall in E4 which are not the boundary values of any solution to the minimal surface system over the intertor of the hall. Thus, Theorem 7. 2 , which holds for dimension 2 and arbitrary co­ dimension, as well as for arbitrary dimension and codimension 1 . fatls for dimension 4 and codimension 3. Another result in the same paper is that there is a ( specifically given) minimal cone of dimension 4 and codimension 3 , whtch represents a solution of the minimal-surface sys tem everywhere except at the ortgtn, where tt is continuous but has a non-removable stngularity. A Bernstein-type theorem in arbttrary d imension and codimension was proved by Hildebrandt, Jost. and Widman [ 1 ] : Letf(x) be a solution

Q/ the minimal surface equation ( * ) over all Q/ Em . Suppose there exists a number j30

< 1 /cos P( 7r/2

n = m + 1 , K = 2 if n

>

�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


;;;;

O. But that means precisely that the mapj is conformai. Since

the image of a conformai harmonie map is a minimal surface, tt follows that

N

the !mage of any harmonte mapj: 52 --�> N is a mtntmal swjaee in

(Chern and Goldberg [ I l, Prop. 5 . 1 ) . Another link between harmonie maps and minimal surfaces i s the

fact that

a map j: M --�> 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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A SURVEY OF MINIMAL SURFACES

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A SURVEY OF MINIMAL SURFACES

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AUTHOR INDEX Ahlfors, 73

Feinberg, 148

Almgren, 59, 138, 140, 143, 144,

Finn, 86. 89, 98, 100, 102, 114, 136, 155

159, 160. 163 Alt. 143

Fischer-Co1brte, 147, 161

Aminov, 163

Fleming, 58, 59, 139

Anderson, 159

Franke1, 140, 163

Barbosa, 146, 148, 154, 156, 161

Fujimoto, 152

Beckenbach , 1 1 7

Gackstatter, 150-152

Beeson, 145

Gage, 164

Bernstein. 1. 100, 136

Galloway, 163 Garabedian, 55

Bers, 48, 98, 108, 109, 136, 152,

Gehring, 164

153, 155 Bôhme, 145

Gllbarg, 114

Bombieri, 138, 140, 163

Giusti, 138, 140

Calabi, 48

Glaser, 166

Chakerian , 148

Go1dberg, 164, 165

Chavel, 148

Gray. 140

Chen, C. C., 151, 152

Gromov, 162, 164

Chen, Y. W., 48

Gulllver, 143, 144, 150

Chern, 1 17, 127, 139, 140, 150,

Haeflinger, 162 Hardt, 144, 145

155, 156, 159, 164-166 Costa, vUi, 149 , 150

Harvey, 153, 159, 162

Courant, 55, 58

Hass, 163

da Sllveira, 154

Heinz. 40, 107, 135, 147, 153,

de Georgi, 98, 138. 140, 153, 156,

155. 157

157

Hildebrandt, 145, 158, 165

Din, 166

Hoffman, vlil, 149-151, 153,

do Carmo, 141, 146-148. 154, 156,

161, 164 Hopf, E. , 107

161 Douglas, 55, 58

Hopf. H .. 50, 154, 157

Eells, 154, 165, 166

Hsian_g . 141. 157, 161

Enneper, 65

Huber, 81, 89

Federer, 59, 139, 144, 147

Jenkins, 102, 136-138, 155 . 157

201

A SURVEY OF MINIMAL SURFACES

Jonker, 1 1 7

Nirenberg. 72

Jorge. 149, 150

Nltsche, J. C . C., 33, 40, 98, lOO,

Jôrgens, 40

143-146, 148. 153

Jost, 158, 165

Nltsche, Joachim, 98

Kamber, 165

Oliveira, 152

Kawa!, 147

Osserman, 37. 72-74, 89, 98,

Kenmotsu, 154

107. 108, 1 1 7. 127. 135, 137.

Klotz, 89, 135

143, 145, 148, 150-156, 158,

Kobayashl, 141

159. 1 6 1 , 163, 164

Kolso, 145

Otsukl, 1 4 1

Kommerell, 1 9

Palmer, 154

Kunert, 1 5 1

Peng, 147

Lawson, 1 4 1 , 145, 150, 152, 153,

Peterson, 150

158-163

Picard, 152

Lemaire, 154, 165

Pin!, 1 17, 140, 156

Lévy, 58

Pltts, 160

Lewy, 35. 55

Rado, 33, 55, 59, 63

Li, 148

Reifenberg, 58, 59, 139

Llouvllle. 165

Reilly, 141

Lumlste, 1 40

Ricci-Curbastro, 156

MacLane, 89

Rinow, 50

Meeks, viii, 143, 145, 149, 150,

Royden, 143

161, 162

Ruchert, 143

Mlcallef, 147, 161

Ruh, 154, 165

Mllnor, 164

Rummler, 162

Miranda, 138

�acks. 160. 164, 165

Misner. 166

Sa..tpson. 165

Moore, 163

Sario, 89

Morgan, 144, 145, 156

Schiffer, 148

Mort, 1 6 1

Schoen, 146-149, 153, 155, 160,

Morrey, 140

162, 163

Moser, 138, 159

Serrin, 102, 136, 137-139, 155

Müntz, 33

Simon, 143-145, 149, 153-157,

Nagano, 161

160, 162, 163

Nakauchi, 163

Stmons, 138-140, 161, 163

Nevanlinna, 152

Smyth, 1 6 1

AUTHOR INDEX

Solomon, 156

Vllms, 165

Spruck. 143, 146, 148

Voss, 73, 89

Stampacchla. 98, 138, 153 , 156.

Weierstrass, 65

157

Wente, 153, 154

Steffen, 153

Werner, 135

Stora. 166

Weyl. 125

Sullivan, 162

White, 145, 155, 157. 163

Takahashl, 1 4 1

Wldman, 158, 165

Takeuchl. 1 4 1

Wolfson, 166

Thurston. 144

Wood, 166

Tom!, 143, 145

Xavier, 148

Tondeur, 165

Yau, 143, 148, 155. 160. 162, 163

Tromba, 143, 145

Zakrzewskl, 166

Uhlenbeck, 160, 164, 165

Zlller, 156

SUB]EC T INDEX

Dirichlet problem for minimal

area, 6

surface equation

first variation, 22

exterior, 1 13-1 16

second variation, 146

higher dimensions, 130, 158

atlas, 43

intertor. 6 1 . 100, 102

Bemstein's Theorem, 34, 38, 69, 92, 107, 138, 147, 149,

divergent path. 50

155-158. 165

divergent sequence, 53 embedded surface, vtll , 143, 144,

black hole, 163

149

Blanc-Fiala Theorem, 89

Enneper's surface, 65, 87, 108,

boundary values, 53

143

branch points, 47, 48 capaclty, 70. 7 1 . 74

Euler characteristic, 85, 86

catenoid, 18, 87, 92, 93, 149, 1 5 1

exterior product, 3

Cohn-Vossen Theorem, 86

flnlte total curvature. 147

complete Rlemannlan manifold,

foliations. 1 6 1 , 165 fundamental form

50, 5 1 complex curve, 1 9 , 147

flrst. 1 0

conformai structure, 43, 45

second. 12, 7 5 , 141 Gauss curvature, 75

convex hull, 53, 54

Cr

lnequalltles for, 105-108 map, 43

ln isothermal parameters, 7

structure, 43

in terms of Gauss map. 77,

surface ln En, 4, 44

126 Gauss map, 66

curvature vector, 1 1

degenerate, 122

curve complex analytic, 19

generallzed, 1 19

meromorphic, 127

of a minimal surface, 66-74, 77, 87-89, 1 19-127

regular, 8

generalized minimal surface, 47

decomposable minimal surface,

Gras::�mannlan, 1 17- 1 18

123

gyrotd, fronttsplece

degenerate Gauss map, 122

205

SUB]ECT INDEX

Dirichlet problem for minimal

area, 6

surface equation

first variation, 22

exterior, 1 13- 1 1 6

second variation, 146

hlgher dimensions, 130, 158

atlas, 43

lnterior, 6 1 , 100, 102

Bemsteln's Theorem, 34, 38, 69, 92, 107, 138, 147, 149,

divergent path, 50

155-158, 165

divergent sequence, 53 embedded surface, viii , 143, 144,

black hole, 163

149

Blanc-Fiala Theorem, 89

Enneper's surface, 65, 87, 108,

boundary values, 53

143

branch points. 47, 48 capacity, 70, 71. 74

Euler characteristlc, 85, 86

catenoid. 18, 87, 92, 93, 149, 1 5 1

exterlor product, 3

Cohn-Vossen Theorem, 86

flnrte total curvature, 147

complete Rlemannlan manifold,

foliations, 161, 165 fundamental form

50, 5 1 complex curve. 1 9 , 147

flrst, 10

conformai structure, 43 , 45

second, 1 2 , 75, 141 Gauss curvature, 75

convex hull, 53, 54

C'

lnequallties for, 105-108 map. 43

ln lsothermal parameters,

structure, 43

in terms of Gauss map, 77,

surface ln En, 4, 44

126

curvature vector. l l

Gauss map, 66 degenerate, 122

curve complex analytlc, 19

generallzed, 1 19

meromorphlc. 127

of a minimal surface, 66-74, 77, 87-89, 1 19-127

regular. 8

generalized minimal surface, 47

decomposable minimal surface,

GraSl>mannlan, 1 17- 1 18

123

gyrold, frontlsplece

degenerate Gauss map, 122

205

206

A SURVEY OF MINIMAL SURFACES

harmonie maps, 154, 164-166 Harnack tnequality, 137

non-parametrtc, 9 1 - 1 1 6 , 144, 147, 152

hellcold, 17. 149

regular, 45, 48, 60

Henneberg's surface. 45

Scherk's, 18, 74, 89, 101, 108

Hesstan matrix. 34 hyperboUc (Riemann surface), 52 infinite boundary values , 101- 103 isoperimetric inequalities, 147, 159

minimal surface equation. 17, 24, 25, 158 in E3, 9 1 minimal vartetles, 140- 141 in spheres. 141

isothermal parameters, 27, 37

monotone map, 56

Kahler manifold, 140

non-orientable surface, 45

logarlthmic capacity, 70, 71. 74

non-parametric form, 7, 15, 23

manifold, 43

normal curvature. 12

Kahler. 140

for a minimal surface. 75

orientable, 43

normal space, 1 1

Riemanntan, 49

normal vector. 1 1

maximum principle. 94 mean curvature, 13. 44

to a minimal surface i n E3, 66, 68

constant, 135, 139, 153

orientation, 43, 44

geometrie interpretation, 22

parabolic (Riemann surface), 5 2

ln isothermal parameters, 27

parameters, independence of, 5

in terms of Laplacian, 28

Plateau problem. 53, 58, 59, 6 1 ,

meromorphic curve, 125 meromorphic functlon, 46 minimal graphs, 152 minimal hypersurfaces, 137-138, 155 minimal surface, 14, 23, 45

135, 139-140, 143 , 145. 153, 157. 163 principal curvatures. 13 for a minimal surface. 75 quasiminimal surfaces. 136-137, 154

catenotd, 18, 87, 92

reflectlon principle, 54

decomposable, 123

regular curve. 8

Enneper's, 65, 87, 108

regular surface, 4, 44

generalized, 47

relativity, 163

gyrotd, frontlsptece

representation theorem

helicoid, 17 Henneberg's , 45 non-orientable. 44, 45, 144, 152

for constant mean curvature, 154 for doubly-connected sur­ faces, 109- 1 10

207

SUB]ECT INDEX

Enneper-Welerstrass, 63-65

stereographie projection. 46

for entire solutions of the

structure

minimal surface equation,

conformai, 43

38, 40, 42

cr, 43

Riemannian metrtc. 49 Riemann sphere, 46 Scherk's surface. 18. 74, 89, 1 0 1 , 108 second fundamental form, 12 for a minimal surface, 75, 141 s lngularlties, 140 removable, 96, 98, 138, 152, 156 stabtlity, 145, 148, 161, 162

Rtemanntan, 49 surface in E• global, 44 local, 3 tangent plane, 9, 44 total curvature, 77, 85, 87, 89, 1 26, 150, 1 5 1 , 155, 159 uniqueness, 143, 144, 152 unlversal coverlng surface, 5 1 variational formula, 2 2