Complete and compact minimal surfaces 0792303997, 3819895183, 1401441491, 9780792303992

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Mathematics and Its Applications

Managing Editor:

M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdtun, The Netherlands

Editorial Board:

F. CALOGERO, Universita degli Studi di Roma, Italy Yu. I. MANIN, Steklov Institute ofMathemlJtics, Moscow, U.S.s.R.. A. H. G. RINNOOY KAN, ErasmllS University, Rotterdtun, The Nether1ands G.-C. ROTA, Ml.T., Cambridge, Mass., U.sA.

Volume 54

Complete and Compact Minimal Surfaces by Kichoon Yang Department of Mathematics, Arkansas State University, U.s.A.

KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON

Library of Congress Cataloging in Publication Data

Yang. Klchoon. Complete and compact mlnlmal surfaces I by Klchoon Yang. p. cm. -- (Mathematlcs and lts appllcatlons) Includes blbllographlcal references. ISBN 0-7923-0399-7 1. Surfaces. Mlnlmal. I. Tltle. II. Serles: Mathematlcs and lts appllcatlons (Kluwer Academlc Publlshers. CA644.Y38 1989 518.3'62--dc20 89-15578

ISBN 0-7923-0399-7

Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands, Kluwer Academic Publishers incorporates the publishing programmes of D, Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, WI Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands.

printed on acidfree paper

All Rights Reserved © 1989 by Kluwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Printed in The Netherlands

To My Parents

SERIES EDITOR'S PREFACE

One service mathematics bas rendered the

'Et mai•...• si j'avait su comment en revenir, jc n'y serais point aIlC.'

human race. It has put common sense back

Jules Vcmc

where it belongs. on the topmost shelf next to the dusty canister labelled 'discarded non· sense'. Eric T. Bell

The series is divergent: therefore we may be able to do something with it. O. Heaviside

Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and nonlinearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered computer science ...'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'ctre of this series. This series, Mathematics and Its Applications, started in 1977. Now that over one hundred volumes have appeared it seems opportune to reexamine its scope. At the time I wrote "Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as 'experimental mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large-scale order', which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics." By and large, all this still applies today. It is still true that at first sight mathematics seems rather fragmented and that to find, see, and exploit the deeper underlying interrelations more effort is needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will continue to try to make such books available. . If anything, the description I gave in 1977 is now an understatement. To the examples of Interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modular functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more) aU come together. And to the examples of things which can be usefully applied let me add the topic 'finite geometry'; a combination of words which sounds like it might not even exist, let alone be applicable. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And, accordingly, the applied mathematician needs to be aware of much more. Besides analysis and numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability, and so on. In addition, the applied scientist needs to cope increasingly with the nonlinear world and the vii

SERIES EDITOR'S PREFACE

viii

extra mathematical sophistication that this requires. For that is where the rewards are. Linear models are honest and a bit sad and depressing: proponional efforts and results. It is in the nonlinear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreciate what I am hinting at: if electronics were linear we would have no fun with transistors and computers; we would have no W; in fact you would not be reading these lines. There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace and anticommuting integration, p-adic and ultrametric space. All three have applications in both electrical engineering and physics. Once, complex numbers were equally outlandish, but they frequently proved the shortest path between 'real' results. Similarly, the first two topics named have already provided a number of 'wormhole' paths. There is no telling where all this is leading fortunately. Thus the original scope of the series, which for various (sound) reasons now comprises five subseries: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything else), still applies. It has been enlarged a bit to include books treating of the tools from one subdiscipline which are used in others. Thus the series still aims at books dealing with: - a central concept which plays an important role in several different mathematical and/ or scientific specialization areas; - new applications of the results and ideas from one area of scientific endeavour into another; - influences which the results, problems and concepts of one field of enquiry have, and have had, on the development of another. Minimal surfaces, both with a given boundary and without boundary, are a panicularly esthetically pleasing subject of mathematics. Partly this comes about because of the manifold interrelations of the subject with various parts of mathematics such as local and global differential geometry, the calculus of variations, the theory of functions, the theory of partial differential equations, topology, measure theory and algebraic geometry. On the other hand, something (natural) gets minimized and that almost immediately and inevitably means that there are interesting applications in the (physical) sciences. Lastly, the resulting geometrical shapes simply do tend to be beautiful. It is also true that a great deal has happened in the theory of minimal surfaces in the last decennia. The present volume gives an account of the exciting developments in recent 'years for the case of minimal surfaces without boundary, together with a brief look at the applications of these results to that powerful and fascinating program that goes under the name of twistor theory. The shortest path between two truths in the real domain passes through the complex

domain. J. Hadamard

La physique De naus donne pas seuIement ('occasion de resoudre des problemes ... eIle nous fait pressentir Ia solution. H. Poincare

Bussum, July 1989

/

Never lend books, for no one ever returns them: the only books I have in my library are books that other folk have lent me. Anatole France The function of an expert is not to be more right than other people. but to be wrong for more sophisticated reasons. David Buder

Michiel Hazewinkel

Table of Contents Series Editor's. Preface

vii

Preface

xi

Chapter I. Complete Minimal Surfaces in Rn 1. Intrinsic Surface Theory

2. 3. 4. 5. 6. 7.

Immersed Surfaces in Euclidean Space Minimal Surfaces and the Gauss Map Algebraic Gauss Maps Examples Minimal Immersions of Punctured Compact Riemann Surfaces The Bernstein-Osserman Theorem

Chapter II. Compact Minimal Surfaces in Sn 1. Moving Frames

1 2. 3. 4. 5.

Minimal Two-Spheres in So The Twistor Fibration Minimal Surfaces in Hp! Examples

Chapter III. Holomorphic Curves and Minimal Surfaces in Cpn 1. Hermitian Geometry and Singular Metrics on a Riemann Surface 2. Holomorphic Curves in (po 3. Minimal Surfaces in a Kahler Manifold 4. Minimal Surfaces Associated to a Holomorphic Curve

1

Chapter N. Holomorphic Curves and Minimal Surfaces in the Quadric

3 8 13 20 31 35

41 46 47

52 64 71

76 80

82 86 95 103 110

1. Immersed Holomorphic Curves in the Two-Quadric

110

2. 3. 4. 5.

119 122 131 132

1

Holomorphic Curves in Q? Horizontal Holomorphic (furves in SO(m)-Flag Manifolds Associated Minimal Surfaces Minimal Surfaces in the Quaternionic Projective Space

Chapter V. The Twistor Method 1. The Hermitian Symmetric Space SO(2n)/U(m) 2. The Orthogonal Twistor Bundle 3. Applications: Isotropic Surfaces and Minimal Surfaces 4. Self-Duality in Riemannian Four-Manifolds

1

137 137

140 144 149

Bibliography

153

Index

169

PREFACE These notes contain an exposition of the theory of minimal surfaces without boundary.

There have been many exciting recent developments in the study of

minimal surfaces in various Riemannian manifolds and we have tried to present some of these materials from a consistent perspective. Main tools we have used are the method of moving frames and the theory of compact Riemann surfaces. Chapter I gives a, more or less, standard treatment of minimal surfaces in

IRn with a distinct emphasis on complete minimal surfaces with finite total curvature.

In §6 we prove an immersion theorem for punctured compact

Riemann surfaces resolving one of the questions raised by Osserman in his book

[06]. In Chapters II, III, and IV we concentrate on the case where the target manifold is a compact Riemannian homogeneous space (Sn, (pn, Qn , and other related spaces), and the technique of complex differential geometry becomes particularly effective.

The works of Calabi, Bryant, Eells-Wood,

and

Chern-Wolfson are presented in Chapters II and III from a uniform point of view.

Chapter IV represents a generalization of these works to include minimal

surfaces in complex quadrics and flag manifolds. A cursory look at the twistor method and its applications to minimal surface theory is given in the final chapter. Of interest is our construction of 21, 'Y

= n(n-l)/2,

many almost complex structures (these include the usual almost

complex structures J+ and JJ on the orthogonal twistor bundle over a 2n-dimensional Riemannian manifold.

These structures may prove to be useful

in studying minimal surfaces. xi

PREFACE

xii

Many people contributed to the completion of this writing project by sending me their preprints/reprints and by providing me with helpful comments and stimulating conversations.

In particular my thanks go to N. Ejiri, H.

Fujimoto, M. Guest, D. Hoffman, G. Jensen, R. Osserman, M. Rigoli, and S. Salamon. I also want to thank R. Liao who has read the entire manuscript and caught several mysterious statements. Finally it is my pleasure to thank Dr. M. Hazewinkel for writing Series Editor's Preface and for giving me several helpful suggestions; Dr. D.J. Larner and his staff for their expert handling of the manuscript.

Kichoon Yang Arkansas State University 1989

Chapter I. Complete Minimal Surfaces in RD Let M be a compact oriented smooth manifold with boundary aM (possibly

aM

= 0).

Also let f: M

(N, ds 2 ) be an immersion into a Riemannian manifold

-+

N. By a smooth variation of fwe mean a smooth map

F: I i) Ft = F(t, .): M

ii) Fo

-+

x

M

-+

N, I

= (-1,

1) such that

N is an immersionj

= fj

iii) Ft laM

= fl aM for

every tel.

Fix a variation F of an immersion f and let d volt denote the volume element of M with the metric induced by F t" Define the map AF: I AF(t) Definition.

f: M

-+

= 1M

-+

IR by

d volt"

(N, ds2) is called a minimal immersion (or minimal

submanifold) if with resepect to an arbitrary variation F of f, AF(O)

= o.

A minimal immersion f is said to be stable if in addition to AF(O) have AF'(O) > 0 for every variation F of f.

=0

we

A stable minimal submanifold has

the least volume amongst nearby submanifolds with the same boundary. (Historically speaking, the theory of minimal submanifolds arose in an attempt to find the surface in 1R3 of least area among those bounded by a fixed curve.

This

problem, called the problem of Plateau, was given a solution by Douglas and Rado in the thirties.

The Plateau problem for higher dimensional submanifolds

of IRn was given a satisfactory treatment only quite recently by Federer, Fleming, Almgren, De Giorgi, and Reifenberg.

See [F] for a detailed account. Also see

[G].) Given an immersion f: M

-+

(N, ds 2 ) from any smooth manifold M we let

CHAPTER I

2

flu'

F denote a smooth variation of neghiborhood in M.

where U is a relatively compact oriented

Call f minimal if relative to an arbitrary F of fl

u we have

= O.

A;(O)

So called the first variational formula states that

A;(O)

=c

J

d volo'

where H denotes the mean curvature vector of f and c is a dimensional constant. It is an immediate consequence of this formula that an immersion into a

Riemannian manifold is minimal if and only if the mean curvature vector vanishes identically. Hereafter we deal exclusively with the case where M is a smooth oriented two-manifold without boundary and N

= IRD

with the standard metric dS~. Take

an immersion f: M .... IRD and let A denote the Laplace-Beltrami operator of (M, f *ds E2 ). Notation. ds 2

= t ds E2.

Local coordinates (x, y) on (M, ds 2) are called isothermal if ds 2 = h (dx2

+

dy2) for some local function h >

o.

Make M into a Riemann surface by decreeing that the I-form dx type (1,0), where (x, y) are any isothermal coordinates. holomorphic coordinate z

=x +

+

idy is of

In terms of the

iy we can write

A

= =tPf/lnaz).

The Gauss map of f is defined to be

~: M ....

(pD-l,

where [(.)) denotes the complex line in

~(z) (D

= [(:)), through the origin and (.).

A

straightforward computation shows that f is minimal if and only if Af = 0, i.e.,

COMPLETE MINIMAL SURFACES IN R"

3

an immersion into Rn is minimal if and only if it is harmonic relative to the This result coupled with the above formula for the

induced metric.

Laplace-Beltrami operator gives Theorem. f is minimal if and only if its Gauss map is holomorphic. The maximum principle for harmonic functions implies that there does not exist a compact minimal surface in IRn and we are lead to study conformal minimal immersions from punctured compact Riemann surfaces.

(Note that an

immersion from an oriented two-manifold into IRn is, by design, conformal relative to the induced complex structure.)

A conformal minimal immersion f: M

->

IRn

from a Riemann surface M is said to be algebraic (to be more precise, its Gauss map is algebraic) if i) M is conformal to a compact Riemann surface M' punctured at a finite set Ej

ii) the Gauss map

~

extends holomorphically to all of M'.

A fundamental theorem of Chern-Qsserman states that

conformal minimal immersion J M curvature is finite.

->

a complete

IRn is algebraic if and only if the total

By virtue of this theorem the theory of compact Riemann

surfaces and algebraic curves is brought to bear on the study of complete minimal surfaces in IRn of finite total curvature.

§1. Intrinsic Surface Theory Let M be a connected oriented smooth two-manifold. Riemannian metric ds 2 .

Equip M with a

By a well-known theorem of Korn-Lichtenstein (see

[CherI] for a proof) we have, in a neighborhood of any point, coordinates x, y

CHAPI'ERI

4

such that

(1) (x, y) are called isothermal coordinates. Notation. z = x

+ iy.

(1) can be rewritten as ds 2

(2)

= h(z)

dz·dz (symmetric product).

Suppose we have another pair of isothermal coordinates orientation class of (x, y). Writing z

= x+

iy we have ds 2

= ii dz· di

ds 2 is globally defined we must have h dz· dz

= ii

(x, y) in the dz.d~. Since

From this we compute

that (Ex.)

(3)

dz

= Adz,

where A is a ( *-valued smooth local function on M. So z and z are conformally related.

Therefore on M there is a naturally defined complex structure, namely

the complex structure whose local holomorphic coordinates are given by x with (x, y) isothermal and positively oriented.

+

iy

Hereafter given a Riemannian

two-manifold (M,' ds2) we will think of it also as a Riemann surface using the aforementioned complex structure.

The associated hermitian metric on M is

simply h dz

®

dz.

This time we let M be an (abstract) Riemann surface. Pick a Riemannian metric ds 2 on M and locally orthogonalize it.

(4)

ds 2

= (rpl)2 +

(rp2)2.

(rpl, rp2) = (rpi) form a local orthonormal coframe on M.

the complex structure of M if rpl

ds 2 is compatible with

+ irp2 is a type (1,0) form on M.

It is

COMPLETE MINIMAL SURFACES IN RD

5

straightforward to see that any metric conformally equivalent to ds2 (Le., the two metrics are smooth multiples of each other) is again compatible with the complex structure and conversely any metric compatible with the complex structure of M is in fact conformally equivalent to ds 2.

Thus on a lliemann surface M there

exists a naturally defined conformal class of lliemannian metrics.

Note that

given a Riemannian two-manifold its metric is in the conformal class defined by the canonical complex structure. Let (M, ds 2) be a lliemannian two-manifold. frame e

= (ei ) = (e l , e2)

Given a local orthonormal

in M define I-forms (w~) by

(5)

Ve.

1

= e.J ® J,1

where V denotes the Levi-Civita connection. Convention.

Hereafter by a frame (coframe) we mean an orthonormal frame

(coframe) unless it is clear otherwise. Digression (Fundamental Theorem of lliemannian Geometry). Let N be a smooth n-manifold. A connection on N is a IR-linear operator V: r(TN)

->

r(TN

®

T *N)

satisfying the Leibnizian rule. That is to say,

e2) = Vel + Ve2, V(fe) = df· e+ f ve, f E COO(N). V(e l

+

Suppose N is a lliemannian manifold with the metric ds;.

Then a Levi-Civita

connection on (N, ds;) is a connection which is symmetric and metric compatible, Le.,

e

= Ve2(e l ) - Ve l (e2), e2> =

a

and 0 = denotes the Euclidean inner product.

a

af Q

a

af P

af Q 2

Now

af Q 2

0, ,\ as in (6). So ii

~ h. Then ii dz·dz also defines a

complete metric in D since the length of a curve in (D, of the same curve in (D, h dz·dz). Let lift of ,\ to

ii

dz· dz) is at least that

0 denote the universal cover of D. The

0 is the real part of a holomorphic function A. Consider w(z)

a function on

D.

I:~ I

=

IeA I

= JZ eA(e)de, Zo

= e'\ = ii.

Thus the map w:

0 ..... ( has an

inverse defined in a maximum disc a(R) = {w: Iwl < R} c (. The complet~ ness of

ii

dz·dz implies that R

holomorphic covering map

= ( or it

7r.

= (I),

i.e., w is bijective and

0 = (. Consider the

0 ..... D c (. Picard's theorem says that either 1r(0)

misses a single point p E (. D

Lemma 2. Let D be an annular region given by

D and also let ds 2

(7)

= {z E (: 0 < r < Iz I
0 and m, an integer. Now ds2 and this means a path approaching p.

H

J

we must have m

~

m

=

complete in

0 has an infinite length.

~j\{O}

It follows that

1. In fact we must have

(17) If m

order terms,

~

2.

1 then for suitable constants 0 "I (c a) E

fa

(n

.,pa = ~ -

¥a

would be

holomorphic at zero. Thus Re

c~og z = Ref(:a - .,p~

would be a well-defined harmonic function at But E (c a)2 Digression.

= 0 and

this would say ca

= ra z = O.

dz

= 0 for

every

Re f .,padz So each c a must be real.

It.

A canonical divisor on a compact Riemann surface Mg (g

genus) is , by definition, the divisor of a meromorphic I-form on M. g

=

the

Applying

so called the Riemann-Hurwitz relation one can show that the degree of any canonical divisor is equal to 2g - 2.

(See [Y5] Chapter IV §4 for a proof.)

In

other words given a meromorphic I-form on Mg the total number of zeros counted with multiplicity minus the total mumber of poles counted with multiplicity equals 2g - 2. Proposition.

Let f: M ..... IRn be a complete minimal surface with finite total

curvature. Also let d denote the number of ends and g, the genus. Then

(18)

Tf

5 471"{I - g - d).

Proof. Identify M with Mg\{pl' "', Pd} and note that each (a = :adz gives a meromorphic I-form on M. g

Put m. = the maximum order of the poles of ((a) J

31

COMPLETE MINIMAL SURFACES IN R"

at Pj' 1 ~ j ~ d. Picking suitable constants (c~ E 1/

(n

the meromorphic I-form

= E ca,a

has a pole of order exactly m. at each p.. J

J

Have 2g - 2

= (the

number of zeros

of 1/) - (the number of poles of ,,). So the number of zeros of 1/

= E mj + 2g -

2 ~ 2g - 2

+ 2d

by (17). But -Tf

= 271"' (the number of zeros of 1/)

since the number of zeros of where Hc ~

(pn- 2

is the hyperplane given by the equation E caw a

proof of the Chern-Osserman theorem.

Remark.

,,= I()(Mg) n Hc I,

Suppose f: M

-+

= 0 as

in the

D

IRn is a complete minimal surface that is embedded.

Then we must have equality in (18), i.e.,

(19)

Tf

= 471"' (1

- g - d).

For a proof see (J-M].

§5. Example. Example 1. The plane c 1R3 is a complete minimal embedded surface with total Gaussian curvature equal to O. By (19) it has one end. Indeed 1R2 ~ ( is biholomorphic to

(pI

= ( U {oo}

with one puncture.

Example 2. The Catenoid in 1R3 is given by the Weierstrass representative

{J'

= 12 z

on M

= (\{O},

dz, IP(z)

= z}

where z is the Euclidean coordinate. An explicit parametrization

of the Catenoid is

CHAPTER I

32

(u, V)

1-1

(sinh-Iu, (1

+ u2)1/2sinv, (1 + i)I/2cos v).

It is a surface of revolution obtained by revolving the Catenary x3 = coshxl

about the xl_axis.

From the above parametrization one computes the total

curvature which is -47r.

The surface is easily seen to be embedded with two

ends. In fact Shoen [S] has shown that a complete minimal surface in 1R3 of finite total curvature of any genus with two ends must be the Catenoid. Example 3. The Helicoid is given parametrically by 1R2

--+

1R3; (u, v)

1-1

(ucosv, usinv, v).

It is the conjugate surface to the Catenoid, hence locally isometric to the

Catenoid. The Helicoid is a complete embedded ruled minimal surface of infinite total curvature. Example 4. Take M

= (,

=

J.L

1R3 is called Enneper's surface.

dz, rp(z)

=

z.

The resulting minimal surface in

For z E ( = M, (x Fuv + (1 + 1Ful2)Fvv

= o.

The classical Bernstein theorem states: if F: 1R2 ..... IR is a function whose graph is a minimal surface then F is linear, that is, S is planar.

Remark. The above theorem fails in the higher codimension case in that S does not have to be planar. For example, given any entire function w S = ({z, w(z): z E

q ( (2 =

= w(z):

( ..... (,

1R4 is a minimal surface.

We now give a proof of the Bernstein theorem following Chern [Cher3l. Proof of the Bernstein theorem. Suppose S is a minimal surface.

= ({u,

v, F(u,v)} E 1R3: (u, v) E 1R2}

Given local isothermal coordinates (x, y) on S we have

from §1 Ll where ds 2

=

a2 (P = -1 ~-2 + -2)' ax ay

K

= 21 Lllog

h,

h(dx2 + dy2) is the induced metric on S ( 1R3, Ll is the

Laplace-Beltrami operator of (S, ds 2), and K is the Gaussian curvature of (S, ds 2). Put J = (1 +

F~ + F;)1/2. Then a standard calculation gives

42

CHAPTER I

= ~ log(J!I).

K On S introduce a new metric ds 2

= e;-I)2

ds 2. ds 2 is conformally equivalent to

ds 2 and its Gaussian curvature is identically zero. Since 1 5 e;-l) 5 2 and ds 2 is complete we see that ds 2 is also complete. It follows that (S, ds 2) is isometric to the uv-plane with the flat metric du2 -(...£2

au

+ "'£2)

+

dv2. Since K 5 0 we obtain

log(J!I) 5

lJv

o.

Note that the Laplace-Beltrami operator of (S, ds 2) is a multiple of (...£2

au

+

"'£2) lJv

since ds 2 and ds 2 are multiples of each other. Now the above inequality says the function log(J!I) is a subharmonic negative function on the uv-plane.

The

parabolicity of the uv-plane then implies that log(J!I) must be a constant. Hence K :: 0 and S is planar.

Theorem (Osserman [05]).

0

Let f: M - 1R3 be a complete conformal minimal

immersion of finite total curvature.

Then if the Gauss map omits more than 3

points of (pI, f(M) is a plane. Proof. Identify M with Mg\{pl' ... , ~d}' where Mg a compact Riemann surface of genus g.

We have {p

= dF,

cp}, the Weierstrass representation pair of f.

extends to Mg giving a holomorphic map .

I

.

cp: Mg - (P, CPIM

,/

= cpo

Applying a rotation to f(M) if necessary we may (and do) assume: i) supp(~)

lR(n+1)2 be the matrix coordinates.

X-1dX with the usual tangent space

The exterior differentiation of both sides of the equation dX =

XO leads to dO

=

-0 A 0 which, written out in components, gives the

Maurer-Cartan structure equations of GL(n+ljlR). GL(n+ljlR), say G

=

simply by restrictions.

For a closed subgroup G
1 we

obtain a nonlinear horizontal holomorphic embedding (>(f,g): (pI

-+

{p3,

78

CHAPI'ERn

~(f,g)

(4) If n

= 2 then

= t[l,

az, (1 - n)bzD+I /2, b(l

+ n)zD /2a].

~(f,g)((pl) c (p3 is a rational normal

cUnJe.

By a theorem from

§4 the maps ~(f,g) in turn give confomal minimal immersions into S4 upon projection (p3 __ "pi = S4. Let S3

=

i}.

= {(z,

w) E (2: Izl2

+

Iwl2

=

I} and T

T C S3 is called the Clifford torus.

= {(z,

w): Izl2

=

Iwl2

It is the unique non-equatorial

minimal surface of constant curvature (= 0) in S3.

Lawson generalized this

example and constructed a compact embedded minimal surface of an arbitrary genus in S3.

In the following we will give a sketch of this construction.

details we refer the reader to [L1].

Let st: S3 __ 1R3

U

For

{oo} denote the

stereographic projection coordinatizing S3 away from the north pole.

Under st

the south pole corresponds to the origin E 1R3 and the equatorial hypersphere of S3 corresponds to the unit two-sphere in 1R3.

We also note that the standard

metric in S3 becomes ds 2

= 4(dx2 + dy2 + dz2)/(1 + x2 + i + z2)2,

where (x, y, z) E 1R3. From this we observe that straight lines through the origin and great circles of S2 are geodesics.

Fix g E 7l+.

We construct a broken

geodesic, denoted by 1g , in 1R3 = S3\{oo} as follows: Start with two radii of S2 C 1R3 lying in the xy-plane and close the curve by taking the shortest great circular arcs on S2 from the end points of the radii to

(0,0,1T=

S2 n {the z-axis}. From

[11] we have Fact. There exists a conformal embedding (continuous at the boundary) f 1: ~

= the

unit disc __ S3

= 1R3

U {oo}

which represents a solution to Plateau's problem for the closed curve 19· Moreover, this surface f

1

(~)

can be smoothly continued as a minimal surface by

79

COMPACf MINIMAL SURFACES IN Sa

geodesic reflection across each of its boundary arcs.

After a finite number of

geodesic reflections f'Y(t~.) then yields a closed embedded surface of genus g in S3. Lawson also showed that a compact embedded minimal surface in S3 divide S3 . into two diffeomorphic components, and conjectured that components are of equal volume.

[K-P-S).

these two

This conjecture was recently disproved by

Chapter III. Holomorphic Curves and Minimal Surfares in (po Much attention has been paid to the investigation of minimal surfaces in (pn (and in more general spaces) in recent years.

Inspired by the work of

physicists Din and Zakrzewski [D-Z] Eells and Wood [E-W] first gave a rigorous mathematical treatment of the classification of minimal two--spheres in (pn.

[R]

later gave a classification of minimal two--spheres in the complex Grassmannian (G 4 ,2'

[U, C-W2] gave a description of minimal two--spheres in (Gn,k in

general. [Gl, Y4] constructed a class of minimal surfaces in "pn. Two features stand out in the works mentioned above (and in many other works not mentioned).

One is the invocation of the Riemann-Roch theorem to

say that certain symmetric holomorphic differentials vanish on 82; the other is to classify

minimal

two--spheres

holomorphic objects.

in

terms of holomorphic curves,

or other

Compact homomorphic curves in (pn are well-understood

objects in Algebraic Geometry.

Indeed the space of all nondegenerate

holomorphic two--spheres in (pn of degree d is parametrized by (G d+1,n+1 and combining this with the classification theorem of [E-W] we see that the space of all nondegenerate minimal two--spheres in (pn is parametrized by

d~n ((Gd+l,n+l

x

Consider a conformal minimal immersion

7l~+I)'

i; 52 = (pI ..... 82m.

Have

7r: 82m ..... IRP2m, i: IRP2m .... (p2m, where i is induced by the inclusion IR c (.

As the map i is totally geodesic the

map i07rof: 82 ..... (p2m gives a totally real conformal minimal immersion. Therefore, as we will show in §4 of this chapter, to a minimal two--sphere in 82m one can associate a holomorphic curve in (p2m, which is nothing but its directrix 80

HOLOMORPmc CURVES AND MINIMAL SURFACES IN CP"

81

curve mentioned in §2 of Chapter II. Moreover, the minimal two-sphere can be reconstructed from its directix curve. In §§1-2 we give a rather complete description of holomorphic curves in (pn.

It is important to understand the holomorphic case as the bulk of known

minimal surfaces in (pn "comes" from holomorphic curves. Given a holomorphic curve h: M

-+

(pn there arises a U(l)n+l-principal bundle, .!Y -+ M, and sections

of this bundle, in turn, give rise to (generalized) minimal surfaces in (pn. This process is reversible, i.e., a minimal surface in (pn is attached to some holomorphic curve via the above process only when M ~ S2 or certain holomorphic symmetric differentials on M vanish.

Often times such a minimal

surface is called a supermini mal surface or a pseudo-holomorphic curve. The above mentioned process of producing minimal surfaces from a holomorphic curve can be generalized to manufacture a minimal surface from another minimal surface.

Repeating the process one obtains an infinite sequence

of (generalized) minimal surfaces in (pn.

However, in general none of these

surfaces thus obtained is (::I:)holomorphic. No doubt an interesting remaining problem in the theory of minimal surfaces in complex projective space is to give a classification of higher genus minimal surfaces

CHAPrERm

82

§1. Hermitian Geometry and Singular Metrics on a Riemann Surface

Let N denote an n-dimensional complex manifold equipped with a hermitian metric ds2N

(1) where each

= E a Q® SQ'

1< a -< n , -

a is a local type (1,0) form on N. (a Q

Q

),

collectively, are called a

unitary coframe. The type (1,0) metric connection V on the holomorphic tangent bundle, T(l,O)N ..... N, is given by the skew-hermitian connection matrix

da = Q

--e;

+

II afJ

(a;) satisfying

TQ,

where (T~ are type (2,0) forms, called the torsion forms. Remark.

The above V is called a type (1,0) connection because its connection

matrix written relative to a holomorphic coframe consists of type (1,0) forms. Of course, a unitary coframe is not holomorphic. The Kahler form of N is defined to be -~ times the imaginary part of ds~,

~ E a QII SQ. (N, ds~) is called a Kahler manifold if its Kahler form is closed. Exercise. Show that N is Kahler if and only if the torsion forms vanish. Hereafter, we assume that N is Kahler. We thus have

da =

(2)

Q

--e;

II afJ.

Let (e) denote the unitary frame dual to (SQ). Then (2) becomes Ve

(3)

Q

= efJ ® afJ. Q

The curvature forms, (x~, are defined by

x; = da p + a~ II ap.

(4a)

(N, ds~) is said to be of constant holomorphic sectional curvature c if (4b)

Xp

=

i (a

Q

II efJ

+ t5 p E a1 II e 1 ).

83

HOLOMORPIDC CURVES AND MINIMAL SURFACES IN CP"

Example. Take N

= (pn.

U(n+1) acts transitively on (pn by

A·[v] = [Av], A E U(n+1), v E (n+\{O}.

= t[1,0, .. ·,0]

The isotropy subgroup at 0 include Go

c+

U(n+1) by (a,A)

(pn is Go

E

= U(1)xU(n),

where we

(~:X). Let m denote the Ad(Go)-invariant

H

complementary subspace to go (= the Lie algebra of Go) given by

m=

{[~:_t~]:

_tX]

0 We identify m with (n via [X: 0

H

X E

(n}.

X. n = (n AB ), 0

~ A,B ~ n, denotes the

u(n+1)-valued Maurer-Cartan form of U(n+1). The m--component of n is given by n~

where

f

ot

=

t(O " ... 0, 1"0 ... , 0)

1 ~ a ~ n,

® fOt'

E

(n

=

The forms (nOt) 0 pull back to give

m•

type (1,0) forms on (pn The Fubini-8tudy metric on (pn normalized so that the holomorphic sectional curvature equals 4 is given by

(6)

ds;

where w~

= s*n~ and s is

= E w~ ® w~,

a local section of U(n+1)

Index Convention. 0 ~ A,B,C ~ nj 1 ~

(eOt

= w~)

1 ~ a ~ n,

a,/3, 1

-+

(pn.

~ n.

form a local unitary coframe on (pn. Using the Maurer-Cartan

structure equations dn de

ot

=-

nil n we compute that

ot * ot cOlO = dwoot = s*dn o = s (-np + 0p no)

II

"'0'8

It follows that (7)

We also compute that de;

= -s*(n~ II n~)

+ w~

II

wg + 0; E wJ II wJ.

84

CHAPTERID

Therefore, (8)

-

WoQ A

wf!.0 + If..{J ~

-1 Wo1 A w o' and (pN, ds~) is of constant holomorphic sectional curvature 4. Q

X{J -

Let M be a lliemann surface now.

A hermitian metric on M can be

written as ds 2

(9)

= h(z)

dz

®

dz,

where z is a local holomorphic coordinate. So h(z)

= z > o.