Minimal Submanifolds in Pseudo-riemannian Geometry

184 Pages·2010·1.369 MB·English

by  Henri Anciaux

Minimal Submanifolds
in Pseudo-Riemannian
Geometry
HENRI ANCIAUX
World Scientific
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Minimal Submanifolds
Pseudo-Riemannian
in
Geometry
7542 tp.indd 1 10/1/10 8:59 AM
Minimal Submanifolds
Pseudo-Riemannian
in
Geometry
Henri Anciaux
University of São Paulo, Brazil
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
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Published by
World Scientific Publishing Co. Pte. Ltd.
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USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
MINIMAL SUBMINIFOLDS IN PSEUDO-RIEMANNIAN GEOMETRY
Copyright © 2011 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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ISBN-13 978-981-4291-24-8
ISBN-10 981-4291-24-2
Printed in Singapore.
LaiFun - Minimal Submanifolds.pmd 1 9/21/2010, 4:15 PM
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September21,2010 16:18 WorldScientificBook-9inx6in MinimalSubmanifolds
Foreword
I met Henri Anciaux in 2000, at a time when I was reading an interesting
paper where he solved partially a conjecture of Yong-Geun Oh. Since that
timeIhavefollowedhisscientificdevelopmentandIhaveobservedhisskills
to make his results more understandable by illustrating them with many
examples.
In my opinion this book is a consequence of his particular vision of
geometry. It also fills a long-standing gap, because many texts of pseudo-
Riemannian geometry make use of physics to approach their topics, which
sometimes implies that to learn basic concepts is a hard job for graduate
students.
The introduction to pseudo-Riemannian and pseudo-K¨ahlergeometries
is enjoyable and easy to follow. The treatment made by the author about
minimal,complexandLagrangiansubmanifoldsisclearanditwillbeuseful
for young researchers interested in these topics. The great quantity of
examples not only helps to make clear the theory, but also allows aneasier
comprehension and a pleasant approachto the topic.
F. Urbano
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Preface
About 1755,the Turinese mathematician Lagrangederived the differential
equationthatafunctionsatisfieswhenitsgraphminimizes theareaamong
all surfaces with the same boundary. This achievement may be considered
asthebirthofthetheoryofminimalsubmanifolds,althoughEulerhaddis-
coveredafew yearsbefore the firstnon-planarexampleofminimalsurface,
the catenoid. This is a surface of revolution (actually the only non-planar,
minimalone)whichowesitsnametothefactthatitsgeneratingcurveisthe
catenary,thecurveobtainedbyhangingfreelyachainwithuniformweight.
ThenextstepwastakenbyMeusnierwhogaveageometriccharacterization
of the minimal surface equation: the sum of the two principal curvatures
of the surface vanishes at any point. He also re-discovered the catenoid1
and did discover the helicoid, the surface made up by the trajectory of a
straight line subject to a helical motion.
Sincethenthissubjecthasbeenenjoyinganenduring—althoughnotat
aconstantrate—developmentuntiltoday. Ithasbecomeanimportantone,
withconnectionsnotonlywiththeanalysisofpartialdifferentialequations,
but more surprisingly with complex analysis and even algebraic geometry,
and has received contributions of major mathematicians, such as Poisson,
Riemann, Weierstrass,Calabi... to name a few. While evensome problems
regarding the original setting, i.e. two-dimensional surfaces in Euclidean
three-dimensional space, have proved hard to handle (to give an example,
it was proved only in 2005 that the only non-planar minimal surface of
Euclidean three-dimensional space which is embedded, complete and sim-
ply connected is the helicoid, see [Meeks, Rosenberg (2005)]), the theory
has been generalized in several directions: instead of surfaces of Euclidean
three-dimensionalspace,onecanconsiderhigherdimensionalsubmanifolds,
1
Inthosetimesfreeof”publishorperish”ideology,successivediscoverieswerefrequent.
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