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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 of MathemlJtics, 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 ofM athematics,
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
'Et mai• ...• si j'avait su comment en revenir, One service mathematics bas rendered the
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·
The series is divergent: therefore we may be sense'.
able to do something with it. Eric T. Bell
O. Heaviside
Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non
linearities 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 com
puter 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, modu
lar 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
viii SERIES EDITOR'S PREFACE
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 non
linear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreci
ate what I am hinting at: if electronics were linear we would have no fun with transistors and com
puters; 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 fre
quently 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 sub
series: 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 subdis
cipline 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 cal
culus 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 (physi
cal) 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 decen
nia.
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 Never lend books, for no one ever returns
real domain passes through the complex / them: the only books I have in my library
domain. are books that other folk have lent me.
J. Hadamard Anatole France
La physique De naus donne pas seuIement The function of an expert is not to be more
('occasion de resoudre des problemes ... eIle right than other people. but to be wrong for
nous fait pressentir Ia solution. more sophisticated reasons.
H. Poincare David Buder
Bussum, July 1989 Michiel Hazewinkel
Table of Contents
Series Editor's. Preface vii
Preface xi
Chapter I. Complete Minimal Surfaces in Rn
1. Intrinsic Surface Theory 3
2. Immersed Surfaces in Euclidean Space 8
3. Minimal Surfaces and the Gauss Map 13
4. Algebraic Gauss Maps 20
5. Examples 31
6. Minimal Immersions of Punctured Compact Riemann Surfaces 35
7. The Bernstein-Osserman Theorem 41
Chapter II. Compact Minimal Surfaces in Sn 46
1. Moving Frames 47
12. Minimal Two-Spheres in So 52
3. The Twistor Fibration 64
4. Minimal Surfaces in Hp! 71
5. Examples 76
Chapter III. Holomorphic Curves and Minimal Surfaces in Cpn 80
11. Hermitian Geometry and Singular Metrics on a Riemann Surface 82
2. Holomorphic Curves in (po 86
3. Minimal Surfaces in a Kahler Manifold 95
4. Minimal Surfaces Associated to a Holomorphic Curve 103
Chapter N. Holomorphic Curves and Minimal Surfaces in the Quadric 110
1. Immersed Holomorphic Curves in the Two-Quadric 110
12. Holomorphic Curves in Q? 119
3. Horizontal Holomorphic (furves in SO(m)-Flag Manifolds 122
4. Associated Minimal Surfaces 131
5. Minimal Surfaces in the Quaternionic Projective Space 132
Chapter V. The Twistor Method 137
11. The Hermitian Symmetric Space SO(2n)/U(m) 137
2. The Orthogonal Twistor Bundle 140
3. Applications: Isotropic Surfaces and Minimal Surfaces 144
4. Self-Duality in Riemannian Four-Manifolds 149
Bibliography 153
Index 169
