Differential geometry, Valencia 2001 : proceedings of the international conference held to honour the 60th birthday of A.M. Naveira, Valencia, July 8-14, 2001

Differential GeometTy, Valencia 2001

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Differential Geometry, Valencia 2001 Proceedings of the International Conference held to honour the 60th Birthday of A M Naveira Valencia

July 8-14, 2001

Editors

Olga Gil-Medrano Vicente Miquel

5jj World Scientific lb

New Jersey London • Singapore • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

DIFFERENTIAL GEOMETRY, VALENCIA 2001 Proceedings of the International Conference Held to Honour the 60th Birthday of A M Naveira Copyright © 2002 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, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-02-4906-3

Printed in Singapore by Mainland Press

Preface The Congress Differential Geometry Valencia 2001 took place in Valencia from 8 to 14 July 2001. It was organized on occasion of the 60 t/l birthday of Antonio Martnez Naveira, as an homage of his friends and disciples. Some of them constituted the Organizing Committee: Manuel Barros (Univ. de Granada), Francisco Carreras (Univ. de Valencia), Ximo Gual(Univ. Jaume I), Maria Luisa Fernandez (Univ. Pais Vasco), Angel Ferrandez (Univ. de Murcia), Olga Gil-Medrano (Univ. de Valencia), Luis Hervella (Univ. de Santiago de Compostela), Vicente Miquel (Univ. Valencia) and Salvador Segura (Univ. de Alicante). There were one hundred and nineteen participants at the conference. The program featured 15 invited lectures (six of fifty minutes and nine of forty minutes), 27 twenty minutes talks and 17 posters. The present volume contains a part of the lectures, talks and posters presented at the Conference and some other contributions of friends of A. M Naveira that, for different reasons, could not attend the conference. The contributions collected in this volume are mainly on the areas of Riemannian (and pseudo-Riemannian) geometry (curvature properties, variational problems and curvature deformation,..), submanifolds of Riemannian and Lorentzian ambient spaces (with a very special attention to minimal and constant mean curvature submanifolds), structures on manifolds (twistor spaces, G-structures, natural bundles,...). The first paper, by A. Ferrandez and L. Hervella, which serves as introduction to the book, remarks the influence of A. M. Naveira in the Differential Geometry in Spain, together with some aspect of his scientific and human personality. It is a pleasure for us to thank the other members of the Organizing Committee, and all the participants who contributed to the success of the Congress. We are specially indebted to the geometers who accepted to give an invited lecture: L. Alias, T. Aubin, J. P. Bourguignon, M. P. do Carmo, K. Grove, D. Blair, A. Borisenko, F. Brito, P. B. Gilkey, D. L. Johnson, and L. Vanhecke. We extend our warm thanks to S. Montiel, A. Ros and S. Salamon for contributing to these Proceedings. This Congress was organized under the auspicious of the Department of Geometry and Topology of the University of Valencia, and we give special thanks to the people there who helped in different ways for the preparation and performance of the Congress: F. Mascaro, R. Sivera, P. M. Chacon, A. Hurtado, A. Villanueva and S. Jorda. The Conference would not have been possible without the support of the

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institutions listed below; to all of them we must express our acknowledgement. Finally, we are specially indebted to the referees for their careful work and to World Scientific for giving us the opportunity to publish these Proceedings.

March, 2002 The Editors

List of Sponsors Ministerio de Ciencia y Tecnologia Generalitat Valenciana Universitat de Valencia Facultat de Matematiques R.S.M.E. C.A.M.

List of Participants

Aledo, Juan Angel Alias, Luis Jose Aubin, Thierry Badura, Marek Barros, Manuel Bartoll-Arnau, Salud Beltran, Jose Vicente Bergqvist, Goran Binh, I r a n Quoc Bivia, Carles Blachowska, Dorota Blair, David Blazic, Novica Boeckx, Eric Bonome, Agustin Borisenko, Alexander Bourguignon, Jean Pierre Brasil Junior, Aldir Brito, Fabiano Cabrerizo, Jose Luis Calvaruso, Giovanni Carreras, Francisco J. Carriazo, Alfonso Chacon, Pablo M. Chaves, Rosa Maria Czarnecki, Maciej de Andres, Luis Carlos do Carmo, Manfredo Domingo-Juan, M. Carmen Druetta, Maria Josefina Etayo, Javier Fernandez-Andres, Manuel

U. Castilla-La Mancha U. Murcia U. Paris VI U. Lodz U. Granada U. P. Valencia U. Valencia Malardalen U. U. Debrecen U. Valencia U. Lodz Michigan S. U. U. Belgrade K . U . Leuven U. Santiago de Compostela U. Kharkov I.H.E.S. U. F. Ceara U. Sao Paulo U. Sevilla U. Lecce U. Valencia U. Sevilla U. Valencia U. Sao Paulo U. Lodzki U. Pais Vasco I.M.P.A. U. Valencia U. N. Cordoba R.A.C.E.F.N. U. Sevilla

VII

Spain Spain France Poland Spain Spain Spain Sweden Hungary Spain Poland U.S.A. Yugoslavy Belgium Spain Ukraine France Brazil Brazil Spain Italy Spain Spain Spain Brazil Poland Spain Brazil Spain Argentina Spain Spain

VIII

Fernandez-Lopez, Manuel Fernandez, Marisa Ferrandez, Angel Ferrer, Leonor Fornari, Susana Galvez, Jose Antonio Garcia-Rio, Eduardo Garcia, Alicia Nelida Gil-Medrano, Olga Gilkey, Peter B Gimenez, Fernando Gimenez-Pastor, Angel Girbau, Joan Gonzalez-Davila, Carmelo Gonzalez, Maria del Mar Grove, Karsten Gual, Ximo Herbert, Jorge Hernandez, Luis Hervella,Luis Hullet, Eduardo G. Hurtado, Ana Javaloyes, Miguel A. Jelonek, Wlodzimierz Johnson, David L. Kamissoko, Dantouma Koh, Sung-Eun Koiso, Miyuki Koufogiorgos, Themis Lluch, Ana Lozano, Maria Teresa Lucas, Pascual Lusala, Tsasa Marchiafava, Stefano Marinosci, Rosa Anna Martin del Rey, Angel Martinez Antonio Mascaro, Francisca Mencia, Jose. J. Mercuri, Francesco

U. Santiago de Compostela U. Pais Vasco U, Murcia U. Granada U. Minas Gerais U. Granada U. Santiago de Compostela U. N. Cordoba U. Valencia U. Oregon U. P. Valencia U. Murcia U. A. Barcelona U. La Laguna Priceton U. U. Maryland U. Jaume I U. F. Ceara CIMAT U. Santiago de Compostela U. N. Cordoba U. Valencia U. Murcia Cracow U. T. Lehigh U. U. Bretagne 0 . U. Konkuk Kyoto U. E. U.Ioannina U. Jaume I U. Zaragoza U. Murcia T. U. Berlin U. Roma U. Lecce U. Salamanca U. Granada U. Valencia U. Pais Vasco U. Campinas

Spain Spain Spain Spain Brazil Spain Spain Argentina Spain USA Spain Spain Spain Spain USA USA Spain Brazil Mexico Spain Argentina Spain Spain Poland USA France Korea Japan Greece Spain Spain Spain Germany Italy Italy Spain Spain Spain Spain Brazil

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Mikes, Josef Min-Oo, Maung Miquel, Vicente Mira, Pablo Montaldo, Stefano Monterde, J u a n Montesinos Amilibia, Angel Moraes, Simone Morales, Santiago Naveira, Antonio, M. Nikitin, Igor Nuho-Ballesteros, Juanjo Oh, Seungtaik Orengo, Javier Oubifia, Jose Antonio Palmer, Vicente Pereira, A n a Piu, Paola Pokorna, Olga Rakic, Zoran Reventos, Agusti Romero-Fuster, M. Carmen Rybicki, Tomasz Rylov, Alexander A. Salvador, Beatriz Salvai, Marcos Sambusetti, Andrea Sanabria, Esther Sanchez, Miguel Santisteban, Jose Antonio Savo, Alessandro Scherfner, Mike Segura, Salvador Simon, Miles Simon, Udo Sivera, Rafael Suh, Dong Youp Swift, T i m Tarri'o, A n a Tomas, Jiri

Palacky U. McMaster U. U. Valencia U. P. Cartagena U. Cagliari U. Valencia U. Valencia U. F . Vigosa U. G r a n a d a U. Valencia G.N.R.C. for I.T. U. Valencia K.A.I.S.T. U. Castilla-LaMancha U. Santiago de Compostela U. J a u m e I U. do Minho U. Cagliari Czech U. Agriculture U. Belgrade U. A. Barcelona U. Valencia AGH Finance Academy U. C. Madrid U. N. Cordoba U. R o m a U. P. Valencia U. G r a n a d a U. Pais Vasco U. R o m a T. U. Berlin U. Alicante U. Freiburg T. U. Berlin U. Valencia K.A.I.S.T. U. West of England U. A Coruiia T. U. Brno

Czech Republic Canada Spain Spain Italy Spain Spain Brazil Spain Spain German Spain South Korea Spain Spain Spain Portugal Italy Cezch Republic Yugoslavy Spain Spain Poland Russia Spain Argentina Italy Spain Spain Spain Italy Germany Spain Germany Germany Spain South Korea England Spain Czech Republic

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Valerio, Barbara Vanhecke, Lieven Walczak, Pawel Wegner, Bernd Wiehe, Martin Woo, Jeongsoo Yim, Jin-whan

U. Sao Paulo K.U. Leuven U. Lodzki Zentralblatt MATH T. U. Berlin K.A.I.S.T. K.A.I.S.T.

Brazil Belgium

Poland Germany Germany South Korea South Korea

Contents Preface

v

List of Participants

vii

A Tour on the Life and Work of A. M. Naveira Luis M. Hervella and Angel Fernandez

1

Some Rigidity Results for Compact Spacelike Surfaces in the 3-dimensional de Sitter Space Juan A. Aledo and Jose A. Gdlvez

19

Hypersurfaces with Constant Higher Order Mean Curvature in Euclidean Space Luis J. Alias and J. Miguel Malacarne

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Uniqueness of Spacelike Hypersurfaces with Constant Mean Curvature in Generalized Robertson-Walker Spacetimes Luis J. Alias and Sebastian Montiel

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Best Inequalities of Sobolev Type on Complete Riemannian Manifolds . . . 70 Thierry Aubin Critical Points of Willmore-Chen Tension Functionals Manuel Barros

72

Some Generalizations of Twistor Spaces David E. Blair

84

Biharmonic Immersions into Spheres R. Caddeo, S. Montaldo and C. Oniciuc

97

The Gauss Map Spacelike Rotational Surfaces with Constant Mean Curvature in the Lorentz-Minkowski Space Rosa M.B. Chaves and Claudia Cueva Cdndido

106

The Intrinsic Torsion of SU(3) and G2 Structures Simon Chiossi and Simon Salamon

115

iJ-Hypersurfaces with Finite Total Curvature Manfredo P. do Carmo

134

Null Helices and Degenerate Curves in Lorentzian Spaces Angel Ferrdndez, Angel Gimenez and Pascual Lucas

143

Minimal Discs Bounded by Straight Lines Leonor Ferrer and Francisco Martin

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XII

Volume and Energy of Vector Fields on Spheres. A Survey Olga Gil-Medrano

167

Spacelike Jordan Osserman Algebraic Curvature Tensors in the Higher Signature Setting Peter B. Gilkey and Raina Ivanova

179

Stability of Surfaces with Constant Mean Curvature in Three-Dimensional Space Forms Miyuki Koiso

187

Pseudo-parallel Surfaces in Space Forms Guillermo Antonnio Lobos

197

Rotational Tchebychev Surfaces of 5 3 (1) Tsasa Lusala

205

On Holomophically Projective Mappings onto Riemannian Almost-product Spaces Josef Mikes and Olga Pokornd

211

A Characteristic Property of the Catenoid Pablo Mira

217

Convexity and Semiumbilicity for Surfaces in M5 Simone M. Moraes and Maria del Carmen Romero-Fuster

222

The Gauss Map of Minimal Surfaces Antonio Ros

235

The Fermi-Walker Connection on a Riemannian Conformal Manifold . . . 253 Beatrix Salvador Allue On the Volume and Energy of Sections of a Circle Bundle over a Compact Lie Group Marcos Salvai On Minimal Growth in Group Theory and Riemannian Geometry Andrea Sambusetti

262 268

Deformation of Lipschitz Riemannian Metrics in the Direction of Their Ricci Curvature Miles Simon

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Some Classification Problems on Natural Bundles Related to Weil Bundles Jin Tomds

297

A T O U R O N T H E LIFE A N D W O R K OF A. M. N A V E I R A

Departamento

LUIS M. HERVELLA AND ANGEL FERRANDEZ de Xeometria y Topoloxia, Universidade de Santiago de Compostela 15782 Santiago de Compostela, Spain Departamento de Matemdticas, Universidad de Murcia SO 100 Espinardo, Murcia, Spain E-mails: [email protected], [email protected] Dedicated t o A. M. Naveira

In this paper we will summarize a lecture we gave with pleasure on the life of our advisor, professor Antonio Martinez Naveira, at the "Differential Geometry, Valencia 2001, an International Meeting on the occasion of the 60th birthday of A. M, Naveira" 1

Introduction

It should be quite easy for us to write this paper, because Professor Naveira is our teacher and friend. However, we are not sure to be able to communicate you, t o get into your souls, a series of facts absolutely necessary t o understand why t h e Naveira phenomenon and why many of us love him. We are sure t h a t anyone of the Organizing Committee was able to prepare a very nice article on this subject. However, we have been chosen according t o t h e following three reasons: (i) we are who less time have spent t o organize this event; (ii) we are probably the closest t o Antonio and his family; and (iii) we are Naveira's oldest, but not the best, disciples. Be sure t h a t we will do our best to reach our goal. In this article, a t r i b u t e t o our friend, we intend t o illustrate his virtues as a person and as a mathematician, trying t o give a t r u e image of who Antonio is. We will t r y t o keep it as close as possible to the presentation we gave, in the inaugural lecture of t h e International Conference, t h a t was organized by all of his disciples in his honour. 2

S o m e b i o g r a p h i c a l d a t a of A . M . N a v e i r a

Antonio Martinez Naveira was born in La Coruna, but he spent his childhood in t h e village of Churio, in t h e corunian county of Aranga where he spent his childhood. His parents, farmers, had scarce economic resources. For t h a t reason, Antonio begins his studies in the Unitarian school of his town, where there

1

is only one teacher for all children regardless of their age, Dona Raquel Rey de Castro, who would have a decisive influence in his life. During the first year of high school (there were seven grades in all) he studied in the town of Aranga, having to take exams at the high school of La Corufia. At that time his teacher was Professor Mosquera. This professor's son still remembers perfectly that studious boy whom they tried to help so that he could continue his studies in some school of La Corufia as an intern student. The brother of Dona Raquel, Don Gumersindo hflfcHSb?" \4fil , 4 Rey de Castro was one of the owners and the principal of the Academia Galicia school which had the prestige of being one of the best ttV^M »***' schools of La Corufia, but it had a very high tuition, especially for the means of people dedicated to agriculture. Antonio is received in the school as a direct relative of the principal with the right to free education. A great friendship blos••V soms between them, to such a point 1 that, when talking about Antonio, Don Gumersindo's mother refers to him as her grandson. The confiAntonio in Santiago (1960) dence bestowed on him by the Rey de Castro brothers is highly rewarded. Thus, in the year 1959, he graduates with honours as a Bachiller. While he was in elementary school, Antonio studied to be a school teacher (in those days you could do it after completing four years of high school education) and when he passed the admission exam to go to university, he also obtained a teacher's degree. We remember that Antonio always says that the subject that most trouble gave him was calligraphy, which had to be done with quill and ink. Anybody that has seen his writing realizes how hard it must have been to him to pass.

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A sample of Antonio's writing

In June of 1960 he passes the exam which will allow him to begin his studies in mathematics, and, in October of that same year, the first course, common for all science students, begins. Antonio spends this course in a typical boarding house of Santiago, which later would become the San Clemente's college, a men only facility at the time, it becomes his residence until completing his studies. There are numerous anecdotes of his life at San Clemente's college that are recalled by his mates. Thus, very famous, were Antonio's broadcasts of imaginary soccer games, which were told so passionately that one could even see the ball run. He also stood out by being the most methodical student of the entire College which helped him enormously in his studies. Since all the rooms of the College were double, even if the room was full of people, he would take a nap or go to bed early, not mind- i* j> ! ing the noise all around him. W Concerning t h e mathe- f m a t i c a l s t u d i e s of t h e t i m e , *

' •'"*,•• I /* ,.....-• > ,,.. _, • " " *•""••-"••" >"— i v--™ •• • •-•••••••^.. «, *

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Antonio with t h e Director of his College

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it is worth mentioning that A

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and the Rector of his University (1960)

J v Antonio s class was the third ' since the creation of the mathematics section of the School of Science. The number of students in his class was nine, something unthinkable these days; but the number of professors was also very low, standing out amongst them, Prof. Garcia Rodeja (in charge of algebra), Prof. Aguilo (analysis), Prof. Vidal Abascal (geometry) and Professor Aller (astronomy). It is with Professor Enrique Vidal Abascal that Antonio clicks with and he will be Antonio's advisor for his doctoral dissertation and one of the most influential people of his mathematical development. Antonio graduates in June of 1965. Around that time D. Enrique was working in foliations theory and suggested Antonio

4

to work on a related problem in this area. As a result, his doctoral thesis, defended in January 1969, titled "Foliated manifolds with almost-fibred metric". The board of examinators was composed of Dr. Vidal Abascal, Dr. Etayo, Dr. Vaquer, Dr. Viviente and Dr. Garcia Rodeja who granted it with the highest grade of outstanding cum laude. It was the second thesis defended in Santiago, the first one being that of Professor Echarte, currently a Full Professor at Sevilla. While he prepared his doctoral thesis, Naveira had a position as a hire professor in the mathematical section from October 1965 until the year 1973, when he is appointed Associated Professor in a famous scholarly ceremony where all the candidates for Associated Professor had to go to Madrid to swear loyalty to the National Movement in the Royal Theater. Afterwards he prepares exams for a position as an Full Professor and in the year 1975 he obtains the position of Full Professor in differential geometry at the University of Granada. ^ **j«. ^% *uw w*v^^^viM«v^*rtui^^v^vwvv«'V^^v^ W* }*v»fl{viwv)(vwwv«»i'v A n i

Antonio's doctoral dissertation. Prom right to left: Profs. Viviente, Vaquer, Vidal-Abascal, Etayo and Rodeja.

In 1976 he obtains via a merit contest the position of Full Professor at the University of Valencia, appointment that he currently holds. In the year 1996 he is appointed President of the Royal Mathematical Society of Spain. Thanks to his effort we can now say that the RSME (its Spanish acronym) is alive once again and tries to occupy its corresponding place. Conclude with that office in the year 2000.

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Antonio with the Prince of Spain and other members of the RSME (2000)

3

N a v e i r a a t Santiago, by L. Hervella

My relationship with Antonio Martinez Naveira begins when I was a student on my second year in mathematics, in the academic course of 1966-67. He was our Analysis teacher and one of my classmates was Isabel Vazquez Paredes, today Antonio's wife, who at the time was his girlfriend, and with whom I was bound in great friendship. This friendship quickly led to one with Antonio and at the end of the course, together with another friend, we were the three of us bringing serenade to Isabel. I must say that Antonio is a good mathematician but as a singer he still needs more practice. Once Antonio and Isabel were married, when she finished her third school year, their house was my favorite restaurant. They invited me uncountable times so that I could forget how bad I ate in my Residence. Once I graduated, I started working in the Department of Geometry and Topology which, at the time, was run by Professor Vidal Abascal, be-

6

ing Naveira his closest collaborator. I remember that D. Enrique handed me Haefliger's thesis on foliations theory, so that I would read it and start getting involved with the subject. I told Antonio that it wasn't what I liked and he began studying the work of Alfred Gray on Almost Hermitian Geometry to advise me on this subject. That is how I completed my doctoral dissertation, which I defended in July 1974 and which was officially under the supervision of D. Enrique Vidal because in those days only full Professors could appear advisors, but in the thesis I acknowledged Dr. Naveira for his unvaluable help. This is why Antonio always says that I was his first student. I am thankful for and share this thought. Prom 1970, the year I graduated, my relationship with Antonio grows stronger. I must point out, among other things, the following: i) His ability to work. I have never met anyone capable of surpassing him. When I was working on my thesis, we would stop discussing at 11 p.m. and already he would ask me to arrive at nine in the morning the next day with all the computations finished. I should point out that he would always bring them solved to compare them with mine. ii) His hopes to solve any problem he would find himself working on. I will always say that the Royal Mathematical Society of Spain is alive again because of the luck they had to be able to appoint Antonio as their President. Besides, he transmits this hope to everyone that surrounds him. During the five years we worked together, he left in 1975 to Granada, I could tell many stories of things we've gone through together. For instance, those occurred on the travels to the congress we attended. Since in those days life at the University was not as good, we had to travel in our own cars and, for that, Antonio's Seat 600 helped us a great deal. But then again, we had to leave at three in the morning to go to the Piedrafita pass before all the trucks did. Also, during spring of 1973 we went to Paris to go to the presentation of his third cycle thesis at the University of Paris VI, supervised by Prof. Rene Deheuvels, and which received the highest grade. Besides the scientific work done in that journey, I remember that Antonio would subject his wife and me to long walks. He had a theory: if you want to get to know a city you have to walk it. Today I am thankful of it, but in those days I hoped he would sprain an ankle so that we could sit down and keep exploring Paris via subway. To finish my journey through Antonio's life, I would like to relate the Durham congress of 1974 to which we both went to. We were going, basically,

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with the sole purpose of meeting Alfred Gray, since he had been the referee for our work on Schur's lemma on Nearly Kahler manifolds, published in the Proceedings of the A.M.S. He had asked us to go. I believe this trip was fundamental for differential geometry in Spain, since besides meeting Alfred and beginning a strong relationship with him, we also met Lieven Vanhecke who has done so much in helping geometers from Santiago, Granada, La Laguna... . Also in this conference, Professor Gray showed us the work of one of his collaborators, which Naveira and I refuted since the new structure under consideration were in reality nothing more than Kahler ones

-

.

; , Antonio and Luis in Paris (1973)

We attended this congress invited by Professor Willmore who was a great friend of the University of Santiago, who in numerous occasions had visited us to participate in congresses organized by Professor Vidal. One year after these events, Prof. Gray visited Granada and from 1976 we started stays at Maryland extending for periods of six months or more, most often by Antonio's students. Antonio himself also visited Maryland several times, and his classification of Almost Product structures was produced while in that american university. 4

N a v e i r a a t G r a n a d a a n d Valencia, by A. F e r r a n d e z

I decided to look for amenity, keeping of course seriousness, to explain the excellent work of Prof. Naveira, which is nothing but the answer of someone devoted himself to studying; someone always having blind faith of his doing. And everything done with a bit of geniality that makes him unique, singular and unreproducible. Antonio, as Spain, is different". I will point out a series of key facts all through his life which characterize Prof. Naveira and become him a great guy. I always like to name him El Maestro (The Teacher), and this fact is so important for me that I will forever proclaim it provided that I keep a little °a touristic slogan of the 70's in Spain

8

bit of breath. On ocassion of his 60 birthday, I, we, dedicated him our best paper 6 . The organizers of this conference believe that the scientific work of Prof. Naveira is so important that I am sure that it is our duty to spend some of our time talking on it. Furthermore, I think you will agree with me, the main project carried out by Antonio was the creation of a Differential Geometry School. This is, probably, the very reason of this meeting. The fact is that this tribute to Prof. Naveira is a pretext to talk about some recent advances in Differential Geometry. However, if this were the only aim of this meeting, I confess that probably I would not have attended it. I strongly bet for this celebration, its inspiration and philosophy, because two, among the principles governing my life, are, on one hand, to be happy with the success of my friends; and, on the other hand, to apply myself that saying of Sancho to Don Quijote: "It is only well born who is able to give thanks" (free translation). So then, besides to speak highly of Naveira's work in the past ten years, I will point out some details, moments, sentences and anecdotes that we will bring ourselves near to the human profile of Prof. Naveira, so that you can find him more understandable and you can then understand why certain unconditional adhesions. The core of my speech is going to anybody that want to know the very personality of Antonio. I met Naveira in September 1975 in the University of Granada. A few young men of the Department of Geometry and Topology had decided to study Geometry in order to make a thesis. It seemed, at that time, an impossible task because we had no direction, no open problems, no objectives, no ..., no ... However, we found a never open pair of books: K-N I & II C . The first time we talked about working with Prof. Naveira we spoke different languages. Otherwise, we thought he were coming from another world. He arrived, to the department, the first, at 8 a.m.; he leaved it the latter; he had a cup of coffee in 3 minutes; he took lunch in 45 minutes; and so on. As a matter of fact, we cannot arrive until 11 a.m., because we had to close all pubs in the Campo del Principe d. Antonio was seeking desesperately for any young mathematicien to start a thesis. He came to Granada with a lot of open problems, and almost solved. But, please, believe me, we were afraid of the working way of Antonio and such an amount of strange things such as

b

M. Barros, A. Ferrandez and P. Lucas; Conformal tension in string theories and M-theory, Nuclear Physics B 584 (2000), 719-748. C S. Kobayashi and K. Nomizu, Differentail Geometry, vol. I and II, Interscienece 1963, 1969. d a well known square of Granada

9

Kahlerian, nearly, almost, quasi, Gi and G2, don Enrique, Gray, Vanhecke, Santiago, Paris, Lovaina, Maryland,... It seemed we were in Hollywood. As a matter of fact, we yearned for working in our thesis. However, I wish to point out a key fact: we had an unbreakable agreement, accordingly Manolo Barros should be the first one to begin a thesis with Antonio. It was a little difficult to explain it to Antonio, since he had not yet met Barros. In January 76, Manolo came back from military duties and, hardly a year, those open problems, that Antonio brought himself, were nicely well solved. This was the first thesis of Naveira out of Galicia, which was finished in the University of Valencia, althought its reading took place in Granada about the middle of December 76. It deserves a special mention, since this is the root, under the direction of Prof. Barros, of the recent and brilliant history of the Granada Differential Geometry team. Following Barros' tracks, A. Ramirez, F. J. Carreras and myself went to Valencia looking for a promising future. The time has been said to us that we were right. I do not pretend to talk about Granada University, but Naveira is directly concerned with the success of the current Differential Geometry group. I like to speak of "Granada before Naveira" and "Granada after Naveira". But how did Naveira land in Granada? In a Boeing 747? In an Airbus? Look at the right Antonio changed radically our way of life. However, he also learnt from us. I attended his classes, took personal notes, put in a clean and easy to read form and he asked me for them. It was for me a great pleasure. He never used to have personal notes to teach, up to those taken from Prof. Deheuvels classes in Paris. They have a sad history: we need them for t j***^' my thesis and a bad day, cleanfiling people throw them out the ^i.t .-. rubbish. *" We followed a postgraduate course where Prof. Naveira taught us the first classification of almost-hermitian manifolds. There he felt completely at home, and very soon we were able to get up and down with the almost-

10

complex structure, i.e., with the familiar J. In such a way that in our argot we used to say "to dance the J", like a quite typical Spanish dancing. It is well known that the best choreography of the J-dancing has been, without doubt, carried out by the couple Gray-Herveila. We found Antonio's genialities anywhere and anytime. For instance, also, in that glorious time (the 70's), Antonio had a favourite sentence: "I introduced the curvature in Spain". Indeed, it is easy to imagine the following situation

Sk /

Sf C L 4 of a 2-dimensional connected manifold M is said to be a spacelike surface if the induced metric via ip is a Riemannian metric on M, which, as usual, is also denoted by (,}. The time-orientation of Sf allows us to choose a timelike unit normal field N globally defined on M, tangent to Sf, and hence we may assume that M is oriented by N. We will denote by H = -trace(vl)/2 the mean curvature of M, where A stands for the shape operator of M in Sf associated to N. The choice of the sign — in our definition of H is motivated by the fact that, in that case, the mean curvature vector is given by H = HN. Therefore, H(p) > 0 at a point p G M if and only if H(p) is in the time-orientation determined by N(p). On the other hand, the Gaussian curvature of M is given by K = 1 - det(^l).

21

We will say that a spacelike surface tp : M2 —> Sf C L 4 is a linear Weingarten surface if there exist constants a,/3,/i£ R such that

aH + pK = n being a and /? not both zero. Finally, recall that every compact spacelike surface in Sf is diffeomorphic to a 2-sphere (see, for instance, [4]).

3

Surfaces with a constant principal curvature

It was proved by Shiohama and Takagi [9] that the only compact surfaces of genus zero with a constant principal curvature in the 3-dimensional Euclidean space are the totally umbilical round spheres. Following their ideas, we have the following result, where the hypothesis of genus zero can be removed: Theorem 1 The only compact spacelike surfaces in Sf with a constant principal curvature are the totally umbilical round spheres. Proof: Let ip : M2 —> Sf C L 4 be a compact spacelike surface in Sf with a constant principal curvature Ai = R > 0 (up to a change of orientation). If there exists a non umbilical point p £ M, then we can consider local parameters (u, v) in a neighborhood U of p without umbilical points, such that (dip, dip) = E du2 + G dv2 (dtp, -dN) = RE du2 + X2G dv2 where the principal curvature A2 7^ R- Then, the structure equations are given by Vw = | | ^u - | § fa - REN - Exl> 1

Ev

Gu

Ipvv = ~ 7 7 ^ V>u + r ^

Nu = -R V>„ Nv — - A 2 ipv

i>v - A 2 G i V -

Glp

22

and the Mainardi-Codazzi equations for the immersion rp are

(fl-A2)§+(fl-A2)„=0 Since A2 ^ R, the coefficient E does not depend on v, that is, E = E(u). we consider the new parameters

Jvw>

(u) du,

y=v

the structure equations become I/JXX = -RN

- ip

x y

~ 2T*

%v = - ^ ^ + § ^ - X*GN - G^ Nx = -R V'x Ny = - A 2 tpy and the Mainardi-Codazzi equation is (JR-A2)^ + (i?-A2)x=0, whence the Gauss equation results

%)X+{^)2 = RX^1-R^-R)+R2~1Thus, if we take 1 V

~ R - A2

we obtain from (2) and (3) that

*— Usg) +(§)*) *--«+(«"-!),

23

Let 7 q be the maximal integral curve passing through a point q = ip(x0,y0) £ U for the principal curvature R. Then, from (1) it follows that -yq(t) = ip(x0 + t, y0) satisfies

{lq)tt = {Nolq)t

-R(.Nolq)-lq =

-R(lq)t

so that 7 q is a geodesic curve, which is a solution of the differential equation (7,)« - (R2 - 1)7, = Rv0 for a constant vector v0 £ L 4 . Therefore, j

q

is given by

7, = c o s h ^ i ? 2 - 1 t)vi + sinh(\/.R 2 - 1 i)v 2 - - ^ xl

r w0

(5)

— 1

when R > 1, 1 2 7 9 = wi + tv2 + - * i>„

(6)

when R = 1 and 7, = cos(Vl - R2 t)vi + sin(\/l - .R2 *)t>2 - - ^ - j -

w

(7)

°

when 0 < i? < 1, for suitable vectors iProm (4), the principal curvature A2 can be calculated on 7 g as R - A2 = Ucosh(V'-R 2 - 1 t) + bsmh(y/R2

R

-lt)+

_

J

(8)

when R > 1,

i? - A2 = U + bt - ]- t2 J

(9)

when i? = 1 and fl - A2 = U c o s ( V l ~R2t)

+ bsin(\/l - R2 t) +

2^_

J

(10)

when 0 < R < 1, for real constant a, b. Hence, if 7 g (*i) is the first umbilical point on 7,, we obtain from (8), (9), (10) and the continuity of A2 that 0 = R-

A 2 (7,(*i)) - lim R - A2(7,(*)) ? 0 t—>ti

which is a contradiction. Therefore, there is not any umbilical point on 7,. Moreover, since M is complete it follows that the geodesic 7, is defined for

24

all t G R, so that from the compactness of M the cases (5) and (6) are not possible, that is, necessarily 0 < R < 1. Moreover R ^ 0, because in that case acos( V / l -R2

t) +6sin(-\/l - -R2 *) = °

for some £ G R, which contradicts the continuity of A2. Let U be the connected component of non umbilical points containing p. Note that U is an open set, and from the above reasoning, can be parametrized by (x,y) G (-00,00) x (a, (3) for certain a,P G R, a < (3. Let us suppose that there exists an umbilical point q G dU. Then there exists a sequence of points qn = ip(xn,yn) G U tending to q. Therefore the sequence of geodesies 7„ passing through qn associated to the principal curvature R converges to a geodesic 7— passing through q which is also a line of curvature for the eigenvalue R. Now, from the above argument, it is sufficient to prove that there exists a non umbilical point on 7—. In fact, from (10), we are able to choose a point Pn £ In such that \2{pn) = 1/R -fi R- Finally, since M is compact, there exists a subsequence {pk} of {pn} converging to a non umbilical point p G j ~ . Consequently M is umbilically free, which is not possible because M is a topological sphere. Therefore M must be a totally umbilical round sphere. • Remark 2 Observe that we have not assumed that the principal curvatures Ai,A2 are necessarily ordered, but (Ai - R)(X2

4

-R)=0.

Linear Weingarten Surfaces

The following Theorem generalizes the results of Ramanathan and Li about constant mean curvature and constant Gaussian curvature respectively: Theorem 3 The only compact linear Weingarten spacelike surfaces in Sf are the totally umbilical round spheres. Proof: Let ip : M2 —> Sj C L 4 be a compact linear Weingarten spacelike surface in the de Sitter space Sf. Then we can choose constants a,b, c G R such that -2aH + b{K - 1) = c,

(11)

25

being a and b not both zero. Let us consider the symmetric tensor on M a{X, Y) = a{X, Y) - b(AX, Y),

X, Y e

X{M).

2

It can be easily seen that d e t a = (a — be) det((,)) where (,} is the induced metric on M, thus a is non-degenerate if and only if a2 — bc^= 0. Moreover, it is not possible that a2 — be < 0, because M admits no Lorentzian metric since M is a topological sphere. Hence, we must distinguish the following two cases: 1) If a 2 — be > 0, a defines a Riemannian metric on M if a suitable timeorientation is chosen on M. Let us consider (u, v) local isothermal parameters for the induced metric (,), and let {dip,dip) = E{du2 + dv2) {dip, dN) = edu2 + 2 / dudv + g dv2 be the first and second fundamental forms. Then, the structure equations are given by Vw = ^ ^uv

=

^vv

=

ipu ~ -~ i>v + eN - Eip

2E~ ^u + 2E~ ^v + fN Eu Ev ~~2M ^ +-^ ->Pv + gN - Eip

f

e

Nu = — ipu + —ipv JV„ = ^ „ + - | V* and the Mainardi-Codazzi equations for the immersion ip are e

* ~ f; = HEv (12) gu- fv = HEU Then we can obtain from a simple computation that the Laplacian Aa with respect to the metric a of a smooth function h : M—>R is given by aE -be u u a _aE-bg ( ( a 2 - - bc)E) A h f^uv i 7-1 *I>W

+4

-'(((§). -d)



vJ

n particular ((a 2 - bc)E) (A° iM«>=

9)

2E

Eu

bf +

F

-d).:

•-"((I).•

26

and ((a2 - bc)E) (A'N, <M = aE

h9 E

(eu - ~{eEu - fEv)

+2b-t(ev-1^{eEv + fEu, +^(fv-±(fEv

+

9Eu)

-(((D.-a).)-((i).

i) i/

Then, using (11) and the Codazzi-Mainardi equations (12), it can be shown that

and analogously

Hence, if M is considered as a Riemann surface with the conformal structure induced by a, we get that

for any conformal parameter z on M, that is, the 2-forms {ipz,tpz)dz2 and (Nz,ipz)dz2 are holomorphic and consequently they vanish identically. Therefore, the first and second fundamental forms of M are conformal, namely, M is a totally umbilical round sphere. 2) If a2 — be = 0, then the principal curvatures Ai,A2 associated to A verify that (a-6Ai)(o-6A2) = 0 on M. Then, from Theorem 1 and Remark 2, M must be a totally umbilical round sphere. • As a consequence we have: Theorem 4 The only compact linear Weingarten spacelike surfaces in the 3-dimensional hyperbolic space H 3 with non-degenerate second fundamental form are the totally umbilical round spheres. Proof: If xj) : M—>H3 C L 4 is a surface in the hyperbolic space H 3 with nondegenerate second fundamental form, then its Gauss map ./V : M—>S3 is a spacelike surface in the de Sitter space. Furthermore, the shape operators A^

27

and

of the immersions ip and N, respectively, satisfy A^QAN = I2, so that is a linear Weingarten surface in H 3 if and only if TV : M—>S3 is a linear Weingarten surface in S 3 . Hence, our assertion is a consequence of Theorem 3. • AN

Acknowledgments The second author is partially supported by DGICYT Grant No. BFM20013318 and Junta de Andalucia CEC: FQM0203. VT2 References 1. J.A. Aledo and J.A. Galvez, Remarks on Compact Linear Weingarten Surfaces in Space Forms, preprint. 2. J.A. Aledo and J.A. Galvez, Complete Surfaces in the Hyperbolic Space with a Constant Principal Curvature, preprint. 3. J.A. Aledo and A. Romero, Compact Spacelike Surfaces in the 3dimensional de Sitter Space with non-Degenerate Second Fundamental Form, preprint. 4. L.J. Alias, A Congruence Theorem for Compact Spacelike Surfaces in de Sitter Space, Tokyo J. Math. 24, (2001), 107-112. 5. L. Bianchi, Sulle Superficie a Curvatura Nulla in Geometria Ellittica, Ann. Mat. Pura Appl. 24 (1896), 93-129. 6. H. Li, Global Rigidity Theorems of Hypersurfaces, Ark. Mat. 35 (1997), 327-351. 7. S. Montiel, An Integral Inequality for Compact Spacelike Hypersurfaces in de Sitter Space and Applications to the Case of Constant Mean Curvature, Indiana Univ. Math. J. 37 (1988), 909-917. 8. J. Ramanathan, Complete Spacelike Hypersurfaces of Constant Mean Curvature in de Sitter Space, Indiana Univ. Math. J. 36 (1987), 349359. 9. K. Shiohama and R. Takagi, A Characterization of a Standard Torus in E 3 , J. Diff. Geometry 4 (1970), 477-485. 10. M. Umehara and K. Yamada, A Deformation of Tori with Constant Mean Curvature in R 3 to Those in other Space Forms, Trans. Amer. Math. Soc. 330 (1992), 845-857. 11. H.C. Wente, Counterexample to a Conjecture of H. Hopf, Pacific J. Math. 121 (1986), 193-243.

HYPERSURFACES WITH CONSTANT HIGHER ORDER M E A N CURVATURE IN EUCLIDEAN SPACE

LUIS J. ALIAS Departamento de Matemdticas, Universidad de Murcia, E-30100 Espinardo, Murcia, Spain E-mail: [email protected]

Departamento

J. MIGUEL MALACARNE de Matemdtica, Universidade Federal do Espirito 29060-900 Vitoria-ES, Brazil E-mail: [email protected]

Santo,

Dedicated to Professor Antonio M. Naveira on his 60th birthday The study of constant mean curvature (and, more generally, constant higher order mean curvature) hypersurfaces in Euclidean space is a classical topic in Differential Geometry. In this paper we review on recent advances on the study of such hypersurfaces, including some recent progress by the authors, trying to make the topic comprehensible for a general audience.

1

Introduction

In this paper we review on recent advances on the study of hypersurfaces

with constant higher order mean curvature in Euclidean space, including some recent progress by the authors. One of our main objectives on writing this paper has been to make the topic comprehensible for a general audience, trying to be as self-contained as possible. For that reason, we start in Section 2 with a classical seminal result, the Alexandrov theorem, which characterizes round spheres as the only closed (compact and without boundary) hypersurfaces with constant mean curvature which are embedded in Euclidean space. The Alexandrov theorem has become a basic result which can be taught (at least its two-dimensional version) at an elementary course on differential geometry of surfaces in R 3 , as a nice example of a geometric-analytic global theorem (see for instance [37] and [38]). In Section 4 we exhibit Alexandrov's original proof of that result *, which is based on the Hopf maximum principle for elliptic equations (see Section 3), and gave rise to the so called Alexandrov's reflection method. Later on, Reilly40 found a different and easier proof of the Alexandrov theorem. In Section 5 we develop Reilly's approach, which is based on the combination of certain integral formulae. More recently, using Reilly's method, the Alexan-

28

29

drov theorem was extended by Ros to the case of constant scalar curvature 42 , and more generally to the case of hypersurfaces with constant higher order mean curvature 43 , showing that round spheres are the only closed embedded hypersurfaces with constant r-mean curvature in R n + 1 . In Sections 6 and 7 we exhibit Ros' extension. In [31] Korevaar gave another proof of the Alexandrov theorem for higher order mean curvatures which uses the original ideas of Alexandrov. That approach is presented in Section 8. On the other hand, when one considers the corresponding problem to the Alexandrov theorem for the case of non-empty boundary, it is natural to ask whether a compact hypersurface in R™+1 with non-zero constant r-mean curvature and round spherical boundary is necessarily a round spherical cap. However, the general answer to this question is still unknown even in the simplest case of compact constant mean curvature surfaces in R 3 bounded by a circle. In the remaining of the paper we introduce some recent advances on this problem. In particular, we introduce a result due to Rosenberg [44] which states that this is true if the hypersurface is transverse along its boundary to the hyperplane where the spherical boundary is contained (Theorem 12). Let us remark that the transversality condition was first introduced by Brito, Sa Earp, Meeks and Rosenberg 15 for constant mean curvature hypersurfaces. Rosenberg's theorem is actually more general and has to do with the study of conditions which guarantee that the symmetries of the boundary are inherited by the whole hypersurface. This new point of view was introduced by Koiso 30 , and has also be considered by other authors 15 (see Section 9). In order to exhibit Rosenberg's result, we need to introduce some additional material and other related facts, like the Newton transformations (Section 10) and a flux formula (Section 12). Besides, in Section 11 we study the relationship between the geometry of a hypersurface and the geometry of its boundary. In particular, at boundary points, we are able to relate the r-mean curvatures of the hypersurface to the r-mean curvatures of its boundary, which allows us to state an essential auxiliary result (Lemma 14). After those preliminaries, in Section 13 we present Rosenberg's proof of Theorem 12, which is essentially contained in [15] and [44]. Finally, as an application of Theorem 12, we have recently proved that the only compact embedded hypersurfaces in Euclidean space Rn+1 with constant r-mean curvature Hr (with 2 < r < n) and spherical boundary are the hyperplanar round balls (with Hr = 0) and the spherical caps (with Hr a nonzero constant) [3] (Theorem 13). Our proof essentially consists on relating the transversality of the hypersurface along its boundary to the ellipticity of the Newton transformations. In Section 14 we exhibit the proof of our result.

30

2

A classical result: the Alexandrov theorem

One of the simplest and most important global results about the geometry of constant mean curvature hypersurfaces in Euclidean space E n + 1 is the Alexandrov theorem, which characterizes round spheres as the only compact hypersurfaces in Rn+1 with constant mean curvature. Specifically, Alexandrov f1] showed the following uniqueness result. Theorem 1 The only closed hypersurfaces with constant mean curvature which are embedded in Euclidean space are the round spheres. Here by closed we mean compact and without boundary. In the immersed case, Hopf 24 established that any constant mean curvature immersion of a topological 2-sphere in M3 must be a round sphere, and asked whether the same is true for every constant mean curvature immersion of a closed surface. Hsiang, Teng and Yu 26 were able to construct higher dimensional examples of non-spherical closed hypersurfaces with constant mean curvature, giving a negative answer to Hopf's question and showing that the hypothesis for the hypersurface to be embedded is essential in the Alexandrov theorem. Wente 48 settled the Hopf problem also in the negative for the 2-dimensional case, by constructing infinitely many constant mean curvature tori immersed in R 3 . More recently, Kapouleas has constructed new examples of constant mean curvature compact surfaces immersed in M3 with higher genus 28,29 . 3

Elliptic equations and maximum principles

Original Alexandrov's proof of Theorem 1 is essentially based on the honorable Hopf maximum principle for elliptic equations [25], creating the so called Alexandrov's reflection method which has been widely used by many other authors in several related problems [46,2i,45,i7,30,32,i4,i5,6,7,44,9,io,i8]_ I n o r d e r to state Hopf's maximum principle, we need to introduce some material from the theory of elliptic partial differential equations (for a more general treatment, we refer the reader to the excellent book [22]; here we will follow the notes by Leite [33]). Let fi C 1 " be a domain, u € C 2 (0), and let Du= (m,... ,u n ) € R n be its Euclidean gradient. By definition, a linear second order partial differential equation operator L has the form n L

M(x)

= ^ i,j = l

n

ay(x)uy(x) + 5^6 fc (x)« fc (x),

xefl,

fe=l

where the coefficients a^ = aji and bk are continuous functions. A linear

31

operator L is said to be elliptic at x e fi if the symmetric matrix [a^-(x)] is positive definite; it is elliptic in Q, if it is elliptic at each point of Q; it is uniformly elliptic in Q, if the function A/A is bounded in Q, where A(x) > 0 and A(x) > 0 denote, respectively, the maximum and minimum eigenvalues of the positive matrix [ a ^ x ) ] . The most important example of a linear elliptic operator is the Laplacian A[u] = J2iua- Obviously it is uniformly elliptic. It is well known that harmonic functions, which are defined as solutions of A[w] = 0, do not have interior points of maximum, unless they are constant. A fundamental generalization of this property is the famous Hopf's maximum, principle 25 , stated below (for a proof, see Theorem 3.5 in [22]). Theorem 2 (Hopf's maximum principle) a) (Interior point) Suppose that u satisfies the inequality L[u] > 0, with L uniformly elliptic in fi. If u achieves its maximum at an interior point of VL, then u is constant in Q.. b) (Boundary point) Let u satisfy L[u] > 0 with L uniformly elliptic in a domain Q, with smooth boundary dfl. Ifu achieves its maximum at a boundary point where Du exists, then any outward directional derivative of u at this point is positive, unless u is constant in Vi. Another type of operator which is very important is a second order quasilinear partial differential operator Q, which has the form n

Q[u] = ^

aij(Du)uij + b(Du),

where the coefficients a^ = aji and b are functions in C 1 (R"). Observe that the action of Q on D2u is linear, while it may be nonlinear on Du. The quasilinear operator Q is said to be elliptic with respect to a function u at x e Cl if the symmetric matrix [OJJ(DU)(X)] is positive definite; it is uniformly elliptic with respect to u if A/A is bounded in Q,, where A(x) and A(x) are the minimum and maximum eigenvalues of [aij(Du)(x)}. Before giving an example of such an operator, let us introduce some terminology. Let M be an orientable connected hypersurface in Euclidean space M.n+1 (eventually with non-empty smooth boundary dM). Since M is orientable, we may choose along M a globally defined unit normal vector field N , and we may assume that M is oriented by N. Let us denote by A : X(M) —> X(M) the shape operator (or second fundamental form) of the hypersurface with respect to the normal N. As is well known, for each p e M, A is a self-adjoint linear operator on TPM, and its eigenvalues K\ (p),... , Kn(p) are the principal curvatures of the hypersurface at p. The mean curvature H

32

of the hypersurface is then defined by nH(p) = tr(A) = J2iKi(p)> an s}, M and M* are two hypersurfaces with boundary with a common tangency point p £ dM n dM*, and dM and dM* are also tangent at p. The maximum principle for hypersurfaces with boundary gives that M and M* coincide in a neighborhood of p, because M and M* have the same orientation at p and the same constant mean curvature, and M is above M* locally near p. In any of the two cases above, let us consider S the connected component of M* which contains the point p, and let S be the part of Ms from which S is reflected, that is, S = S*. We put A — {q £ S : M and S coincide in a neighborhood of q}. It is easy to show that A is open, closed and non-empty. From the connectedness of S it follows that S c M. Then S U S is a compact, connected hypersurface contained in M, and therefore it coincides with M. Hence IIS is a symmetry hyperplane of M in the given direction, which finishes Alexandrov's proof. 5

Reilly's method

Later on, Reilly 40 found a different and easier proof of the Alexandrov theorem by combining certain integral formulae. To see it, let ip : Mn —> K" + 1 be an immersed hypersurface in R n + 1 . Then, by direct computation we obtain that

A ( ^ ) = 2 n ( l + ffMN)),

35

where A denotes the Laplacian of the induced metric on M. Therefore, if M is assumed to be closed (compact without boundary), then from the divergence theorem we have / (l + H{il>,*i))dM = A(M)+ JM

[ F(V>,N)dM = 0,

(2)

JM

where A(M) is the area (n-dimensional volume) of M and AM denotes the n-dimensional volume element of M with respect to the induced metric and the chosen orientation. On the other hand, when the hypersurface is assumed to be also embedded, then M is the boundary of a compact regular domain Q, C R™+1, d£l = M. In that case, we may choose N to be the interior unit normal on M. If x denotes the position vector in K n + 1 , then A(x, x) = 2(n + 1), where A denotes the Euclidean Laplacian on R n + 1 . Therefore, by the divergence theorem one gets that (n + l)V(Q)+

[ (V.N)dM = 0,

(3)

JM

where V(Cl) is the (n + l)-dimensional volume of Q,. The integral formulae (2) and (3) are one of the essential ingredients in Reilly's proof. The other one is a new integral formula established by Reilly in [40], which for the case of Euclidean hypersurfaces states as follows. Theorem 4 (Reilly's formula) Let M a compact hypersurface embedded in R™+1, bounding a compact domain ft, dil = M. Let N be the interior unit normal on M. For a given / e C°°(f2), it holds [ ((A/)2 Jn ^

- | V 2 / | 2 ) dV= '

[

(~2(Az)u

+ nHu2 + (A(Vs), V z » dM, (4)

JM o

where z = J\M and u = df/dN. Here V / is the Hessian of f in R n + 1 , and Vz and Az are the gradient and the Laplacian of z in M. It is worth pointing out that Reilly's formula (4) is essentially the classical Bochner formula for the case of non-empty boundary. In fact, it is known from Bochner formula [13] that for every vector field X on a Riemannian manifold M, it holds Div (VXX

- (TAvX)X)

= mcpf.-X") + \VX\2 - (Div(X)) 2 ,

where Div stands for the divergence operator on M, V denotes here the LeviCivita connection of M, and Ric is its Ricci curvature. In particular, if X is chosen to be V / , the gradient of a smooth function / 6 C°°(M), then one

36

gets Div ( V ^ V / - ( A / ) V / ) = R ^ ( V / , V / ) + | V 2 / | 2 - ( A / ) 2 , 2

-

where V / and A / are, respectively, the Hessian and the Laplacian of / in M. Therefore, if we apply this general expression to the case where M = fi is a compact domain in R" + 1 with smooth boundary M = dfi., then Ric = 0 and divergence theorem implies that

J ((A/)2 - |V2/|2) dV = J (V2/(V/, N) - A/(V/, N>) AM, (5) where N is the interior unit normal on M. Set z — / | M and u — df/dN = (V/, N). Taking into account that V / = Vz + uN, where Vz is the gradient of z in M, it is a standard calculation to see that V2f(X,

Y) = \72z(X, Y) - u{AX, Y),

and V2f(X,N)

= (Vu,X)

+

(A(Vz),X),

for every tangent vector fields X, Y e X(M). It then follows that A / = Az - nHu + V 2 / ( N , N), and V 2 / ( W , N) = (Vu, Vz) + (A(Vz), Vz) + u V 2 / ( N , N) = div(uVz) - uAz + (A{Wz), Vz) + u V 2 / ( N , N ) , where div stands for the divergence operator on M. Finally, this allows us to rewrite formula (5) as (4). What is really ingenious in Reilly's argument is the way to use (4) in order to study the geometry of the boundary. To see it, let us consider / € C°°(f2) the solution of the following Dirichlet problem A / = 1 in fl

and

/ = 0 on M = 89..

Therefore, combining the Schwarz inequality ( A / ) 2 < (n + 1)|V / | 2 and Reilly's formula (4), we obtain that V(Q) \> 1

I Hu2dM, JM

(6)

37 2

with equality if and only if V / i s proportional to the metric, that is,

By explicit integration of (7), that means that equality holds in (6) if and only if Q c R n + 1 is an Euclidean ball and M = dCl is a round sphere. On the other hand, from the divergence theorem we also have that

V(fi) = I AfdV = - f udM.

(8)

Jo. JM Therefore, by Schwarz inequality for the function u we obtain that

V{Q.f =ff

2

udM J < A(M) f f u2dM J ,

that is

2 /

v(n)2 S

,M" W

(9)

Let us assume from now on that the mean curvature H is constant. Since M is compact, M has an elliptic point, that is, a point where all the principal curvatures with respect to the interior unit normal are positive. In particular, if is a positive constant and from (9) and (6) we conclude that H

-(n

+ i)v(ny

(10)

with equality if and only if M is a round sphere. On the other hand, since H is constant, it follows from (2) and (3) that Hr-

A{M)

(n + l)V(Q)'

which gives the equality in (10) and finishes Reilly's proof of Theorem 1. 6

The Alexandrov theorem for higher order mean curvatures

Using Reilly's method, Ros 42 was able to extend the Alexandrov theorem to the case of hypersurfaces with constant scalar curvature (see also [41] for an expository approach), solving a problem proposed by Yau 49 . More generally, Ros 4 3 was able to extend it to the case of hypersurfaces with constant higher order mean curvature, stating the following result (see also [31] for another proof of the same result given simultaneous and independently by Korevaar).

38

Theorem 5 The only closed hypersurfaces with constant higher order mean curvature which are embedded in Euclidean space are the round spheres. Let us recall that the higher order mean curvatures of a hypersurface M in R n + 1 are the natural generalization of its mean and scalar curvatures. In fact, let ar : R n —> R be the r-th elementary symmetric function, denned by oy(a:i,... ,xn) =

^

x^.-.x^,

1 < r < n.

(11)

»l b / ( 0 ) ] , which implies that k(p) - k'(p) G T, where T = {x e R" : Xi > 0, for 1 < i < n) is the open positive cone of R™. We claim that k(p) = k'(p). To see it, let Yr be the connected component of {x e 1 " : ov(x) > 0} which contains the point ( 1 , . . . , 1), where ay is the r-th elementary symmetric function as defined in (11). Note that T C T r , and as showed by Garding 20 , Tr is also an open convex cone of K n . Garding 20 also established an inequality from which it is

43

possible to prove (see [16], Proposition 1.1) that • ^ • ( x ) > 0,

for every x G T r , 1 < i < n, and 1 < j < r.

(20)

By hypothesis, there exists some point qo G M' such that k'(q0) € T, hence k'(go) £ r r . By continuity of the ordered eigenvalues of M' and the fact that M' is connected we conclude that k'(g) G r r , for every q G M'. On the other hand, define a(t) = k'(p) + t(k{p) - k'(p)), for t > 0. We will see that a(t) G IV, for t > 0 (see Lemma 4.1 in [19]). In fact, if it does not hold, then there exists t0 > 0 such that ar(a(t)) > 0 in 0 < t < to and ar(a(t0)) — 0. This implies that ji^r{oL(t))\t=tl < 0 for some 0 < t\ < t0. But this is impossible, because of

jt 0,

by (20). In particular, we have that a ( l ) = k(p) G r r , and by the same reason as before k(q) G Tr, for every q G M. Besides, from the convexity of r r , it follows that the segment from k(p) to k'(p) is contained in IV. Now, the mean value theorem and (20) implies that n

ar(k(p)) - ar(k'(p))

„

= Y, ^ ( ( 1 " s)k(p) + sk'toXmip) «=i

- «;(p)) > 0

OXi

for some 0 < s < 1. But, the left hand side of this equality is equal to (™)(i7r — H'r) < 0. Thus k(p) = k'(p), which proves our claim. Besides, we also conclude that D2u(0) = D2u'(0) = A, where A is the second fundamental form of M at p (and Ap = A'p). Now we explain the linearization procedure. The hypothesis Hr < H'r implies that ?i r [u] < Tir[u'\. Write Uj = (1 — i)u + tu', a segment from u to u'. Let w = u' — u and observe that -^ut = w, -^Dut — Dw and that jkD2ut = D2w, with Hessian matrix [wy]. Then

0 < Hr[u'] -Hr[u) = J* jtHr[ut}dt = ^Y^WtHj

+ E ^ N ^ H

The above inequality may be written as 0 < L[w] = Y i,j

c w

ij ij + Y

bkWk

'

k

where the coefficients Cjj(x) = /Q | ^ - [ u t ] (x)dt and 6fc(x) = / 0 §^[u t ](x)di of the linear operator L are clearly continuous.

44

For all t e [0,1], one has that u t (0) = 0, Dut(0) = 0, and D2ut(0) = A. This fact, together with (19), implies that [f^[ut]](0) is a matrix that only depends on A, and thus it is independent of t. Therefore [cy(0)] = [f^ n M](0) is a matrix that depends only on A, and we claim that it has all eigenvalues positive. To see it, we first note that from (19) it follows that Wr[u](0) = J2\j\=r det(Aj), where \J\ = r means that J = {j\ < • • • < jr} is a set of r indices such that 1 < j \ < • • • < j r < n, and Aj denotes the r x r principal submatrix of A with rows and columns indexed by J. Let P be an orthogonal matrix such that P~lAP = diag(fti(p),... ,nn{p)). Then, from the above expression for Hr[u}(0) we conclude that Hr[u)(0) = o>(k(p)). Since KS — J2i,jPisuHPjs> implies that

the chain rule

dHr dun

M(o)

S i v e s l ^ j M C 0 ) = Y,s^(Hp))PisPjs,

P = diag(g«P))

which

g(k(p,,).

Therefore, the eigenvalues of the matrix [cy(0)] are | ^ ( k ( p ) ) , 1 < % < n, which are positive due to k(p) 6 Vr and (20). Therefore L is elliptic at 0, and we may assume it is uniformly elliptic in a neighborhood U of 0, since its coefficients are continuous. The hypothesis w = u' — u < 0 near the origin implies that w achieves its maximum value 0 at the point 0. The Hopf maximum principle applied to L[w] > 0 implies that w = 0 in U, whether 0 is an interior or boundary point. Therefore, the hypersurfaces coincide locally. Once we have the maximum principle, Alexandrov reflection method can be applied without change as in the case of the mean curvature to prove Theorem 5. As a result of the above proof of maximum principle, we also get the following interesting consequence. Corollary 9 Let M be an oriented connected hypersurface in Wn+lwith constant r-mean curvature Hr. Assume that M has an interior point where all principal curvatures are positive. Then —-i-(k(p)) > 0

for every p e M,

1 < i < n,

1 < j < r.

OXi

9

The case of non-empty boundary

When one considers the corresponding problem to the Alexandrov theorem for the case of non-empty boundary, it is natural to ask whether a compact hypersurface in R n + 1 with non-zero constant mean curvature and round spherical boundary is necessarily a round spherical cap. However, the general answer

45

to this question is still unknown even in the simplest case of compact constant mean curvature surfaces in R 3 bounded by a circle. In [28], Kapouleas showed that there exist examples of higher genus compact, non-spherical immersed surfaces in R 3 with constant mean curvature and circular boundary. However the original question remains open if one requires in addition that the surface has genus zero or that it is embedded, and one has the following conjectures

[ 1 5 ]\ Conjecture 10 An immersed disc in R with non-zero constant mean curvature and circular boundary is a spherical cap. Conjecture 11 An embedded surface in R 3 with non-zero constant mean curvature and circular boundary is a spherical cap. In recent years, different authors have considered this problem obtaining several partial results [2,5,6,7,11,14,15,34,35^ j ^ t ^ o r j g m a i conjectures remain open. In [30], Koiso gave a new interpretation of this problem by studying under what conditions the symmetries of the boundary of a non-zero constant mean curvature hypersurface M embedded into R n + 1 are inherited by the whole hypersurface. In particular, she showed that when the boundary E is a round (n — l)-sphere contained in a hyperplane II of R" + 1 , and M does not intersect the outside of E in II, then M is symmetric with respect to every hyperplane which contains the center of E and is perpendicular to II, and hence M must be a round spherical cap. Related to Koiso's symmetry theorem, Brito, Sa Earp, Meeks and Rosenberg 15 also showed the following symmetry result. Let E be a strictly convex (n — l)-dimensional submanifold contained in a hyperplane II of R™+1, and let M be a compact embedded hypersurface in M™+1 with non-zero constant mean curvature and bounded by E. If M is transverse to II along the boundary DM, then M is contained in one of the half-spaces of R n + 1 determined by II and M has all the symmetries of E. In particular, if E is a round sphere, then M must be a round spherical cap. Here, transversality means that the hypersurface M is never tangent to the hyperplane n along its boundary. More recently, using the ideas in [15], Rosenberg [44] extended this result to the case of the higher order r-mean curvatures as follows. Theorem 12 Let E be a strictly convex (n —1)-dimensional submanifold contained in a hyperplane n o / R n + 1 , and let M be a compact embedded hypersurface in R n + 1 with non-zero constant r-mean curvature Hr and bounded by E. If M is transverse to H along the boundary dM, then M is contained in one of the half-spaces o / R n + 1 determined by U and M has all the symmetries o / E . In particular, if E is a round sphere, then M must be a round spherical cap. Moreover, ifr = n then one does not need to assume that M is transverse to

46

II along its boundary. As an application of Theorem 12, we have recently proved that the corresponding n-dimensional version of Conjecture 11 is true for the case of the scalar curvature and, more generally, for the case of the higher order r-mean curvatures, when r > 2. Specifically, we have shown the following result [3]. Theorem 13 The only compact embedded hypersurfaces in Euclidean space M.n+1 with constant r-mean curvature Hr (with 2 < r < n) and spherical boundary are the hyperplanar round balls (with Hr = 0) and the spherical caps (with Hr a non-zero constant). Our objective in the rest of the paper is to exhibit both Rosenberg's proof of Theorem 12 and our proof of Theorem 13, trying to be as self-contained as possible. In order to do that, we need to introduce some additional terminology and other related facts. 10

T h e N e w t o n transformations

The classical Newton transformations Tr : X(M) —> X(M) are defined inductively from the shape operator A by T0 = I

and Tr = SrI - A o r

M

,

1 < r < n,

(21)

where / denotes the identity in X(M), and Sr = (™)Hr, or equivalently by Tr = SrI - Sr_xA + ••• + ( - l ) r - 1 S 1 y r - 1 + ( - l ) M r . Note that by the Cayley-Hamilton theorem, we have Tn = 0. Let us recall that each TT is also a self-adjoint linear operator which commutes with A. Indeed, A and Tr can be simultaneously diagonalized; for a fixed point p £ M, if {e\,... , e n } C TpM are eigenvectors of A corresponding to the eigenvalues K\(p),... , Kn(p), respectively, then they are also eigenvectors of Tr corresponding to the eigenvalues of Tr, and Tr(ei) = /Xj)r(p)ej with Hi,r(p) =-^-(niip),...

,Kn(p)),

l,N)N denotes the vector field on M given by the tangential component of the position vector field. Observe that for any tangent vector field Z G -£(M), using that V z T r - i is self-adjoint, it follows that div(T r _iZ) = (div Tr-i,Z)

+ tr(T r _i o VZ) = tr(T r _i o VZ),

(24)

where V Z : I M V X 2 for every X € X(M). In particular, if Z = xpT then V x ^ T = X + {ifj,~N)AX and, using formula (23), expression (24) simplifies to div(T r _i^ T ) = tr(T r _i) + tr(r r _i o A) = c r _!(iJ r _i + ( ^ , N ) F r ) . Thus, divergence theorem implies now (13). 11

A geometric configuration

In what follows, we will consider the following geometric configuration in Euclidean space R™+1. Let II C Rn+1 beahyperplaneinK n + 1 , and let S " " 1 c II be an orientable (n — l)-dimensional closed submanifold contained in II. Let ip : Mn —> R™+1 be an orientable compact hypersurface immersed into R" + 1 with smooth boundary dM. As usual, M is said to be a hypersurface with boundary S if the immersion xp restricted to the boundary dM is a diffeomorphism onto S. The following question naturally arises from this geometric

48

configuration: How is the geometry of M along its boundary related to the geometry ofE as a hypersurface ofU? We will consider this configuration oriented by the following fields: i) N , the unit normal vector field globally defined on M; ii) v, the outward pointing unit conormal vector field along dM; iii) rj, the unitary vector field normal to E in II which points outward with respect to the domain in II bounded by S; and iv) a, the unique unitary vector field normal to II which is compatible with 77 and with the orientation of E. With this orientation, given a point p € E, a basis {vi,... , f n - i } for T P E is positively oriented if and only if {r](p),Vi,... ,i>„_i} is a positively oriented basis for II. Let As denote the shape operator of E C II with respect to the unit normal vector field n. At each point p G dM, let {e\,... e„_i} be eigenvectors of AY. , and let us denote its corresponding eigenvalues by T\ ,... , T„_ 1, so that A^ei — Tjej, 1 < % < n — 1. After a simple computation, it follows that the matrix of A in the orthonormal basis { e i , . . . e„_i, 1^} of TPM is given by /-n(a,i/) 0 ••• 0 -T2(a,zv

0 0

(Av,ei) {Av, e2)

\

A =

(25) 0 \ (Ai/,ei)

0 {Au,e2)

-rn_x(a.,u) (Av,en-i)

{Au,en-r) {Av,v) J

Computing from this expression, one can conclude that the r-mean cur vature Hr of the hypersurface M at a boundary point p € dM is given by nHi — Si = -si{a.,v)

+ {Av,v}, n-l

\H2 = S2 = s 2 (a,z/) 2 - si{a,v){Av,v)

-^(Au^i

(26)

i=l

Hr = Sr = ( - l f s r ( a , v)r + ( - l y - V - i f c , v)r-'{Au,

+(-l)'- 1 M

v)

(27)

n-l

sr-2(Ti){Ais,ei 2=1

3 < r < n,

49

where sr (resp. s r (fj)) denotes the r-th elementary symmetric function of the principal curvatures n , . . . ,r„_i (resp. n , . . . , f j , . . . ,r„_i) of E as a hypersurface of II (and sn = 0 by definition). Observe that this expression provides us with a partial answer to our initial question, since it relates the geometry of the hypersurface M along its boundary (given by the r-mean curvatures Hr) to the geometry of E C II (given by the r-mean curvatures hr defined by (™~ )hr = sr). On the other hand, this expression, and the relationship between A and A-£ given by (25), is the key for the following essential auxiliary result, which can be found in [3]. Lemma 14 Let E be an orientable (n — 1)-dimensional compact submanifold contained in a hyperplane II = a-1 of M n + 1 . Let ip : M™ —> R n + 1 be an orientable hypersurface with boundary E = tp(dM), and let u stands for the outward pointing unit conormal vector field along dM c M. Then, along the boundary dM and for every 1 < r < n — 1, it holds (Trv,v) = (-l)rsr{a,v)r,

(28)

where sr is the r-th symmetric function of the principal curvatures of E c II with respect to the outward pointing unitary normal. 12

A flux formula

An essential ingredient in the proof of Theorem 12 is the following flux formula, which was first stated by Rosenberg in [44]. Lemma 15 (Flux formula) Let E be a closed (n -1)-dimensional submanifold contained in a hyperplane II C Rn+1, and let ip : Mn —> R n + 1 be a compact hypersurface with boundary E and constant r-mean curvature Hr, 1 < r < n. Let Dn be a compact orientable hypersurface in E n + 1 with 3D = dM such that M U D is an oriented n-cycle of R n + 1 , with D oriented by the unit normal field njg. Then for every constant vector field Y on Rn+1, we have K n + 1 be an immersed compact hypersurface with boundary T, and constant r-mean curvature Hr, 1 < r < n. Then 0 0 and we can assume that M is oriented by the mean curvature vector. Let us begin reasoning as in [15]. We suppose that II = {x e R™+1 : xn+i = 0}. As M is transverse to II along S we may assume that, in a neighborhood of dM, M is contained in the closed half-space II + = {x e Rn+1 : xn+\ > 0}, and that M is globally transverse to II. In this situation we claim that M C I I + . We will see it by showing that M is disjoint from Ext(D) and Int(£>), where D is the domain in II bounded by E, Ext(£>) is the set II — D, and Int(D) is the interior of D. We first observe that M n Int(D) ^ 0 and M n Ext(D) = 0 leads to a contradiction. If that is the case, let M\ be the connected component of M fl I l + that contains E. M\ together with a proper submanifold D of D bound an embedded (n + l)-manifold W\ C I I + . We apply the flux formula in Lemma 15 to the n-cycle M\ U D, so that - / ( r r _ i ^ , Y)ds = r(n)H r JdMi \rJ

f Jb

= r()HrVol{D)

(Y,nB)dD ) = 0 and M D Ext(-D) = 0, which means that M is contained in the closed half-space n + determined by the hyperplane II. The Alexandrov reflection f1] with vertical hyperplanes immediately proves that M inherits the symmetries of E. In the case r = n, as Hn is a positive constant and there is at least an elliptic point of M, it follows that all the principal curvatures are positive at every point. Hence M is strictly convex at each point. Now come towards n from above with horizontal hyperplanes II t . Since M D lit is strictly convex for all t, hence M is contained in one half-space determined by II, and it is topologically a disc. 14

Transversality versus ellipticity. Proof of Theorem 13

Lemma 14 implies a very strong relationship between the transversality of M with respect to n along the boundary dM, and the ellipticity on M of the Newton transformation Tr, when r > 1 (recall that To = I). That relationship between transversality and ellipticity is actually the key of our proof of Theorem 13. In fact, saying that M is not transverse to II along its boundary means that there exists a point p £ dM such that (a, v)(p) = 0, which implies from (28) that (Tru, v)(p) = 0, r > 1. Therefore we can conclude that if the Newton transformation Tr is positive definite on M for some 1 < r < n — 1, then the hypersurface M is transverse to n along its boundary. On the other hand,

55

in the case where Hn does not vanish on M and n > 3, transversality easily follows from (27). Indeed, in that case we have along the boundary dM n-1 i=i

In particular, if there exists a point p € dM where (a, v)(p) = 0, then Hn(p) = 0. In the same way, if we assume that H2 (or, equivalently, the scalar curvature of the hypersurface) is positive everywhere on M, then (26) also implies that M is transverse to II along the boundary. Now we are ready to proof Theorem 13. Let us assume that M is not a hyperplanar round ball. Then the constant r-mean curvature Hr must be necessarily non-zero because we know from the proof of Theorem 12 that there exists at least one interior elliptic point of M, and under the appropriate orientation of M we may assume that Hr is a positive constant. Besides, we also proved that the Newton transformation T r _i is positive definite on M. Since r — 1 > 1, from the relationship between transversality and ellipticity above, that means that M is transverse to II along the boundary. Our result then is a direct consequence of Theorem 12. Acknowledgments L.J. Alias was partially supported by DGICYT, MECD, Spain (Grant N BFM2001-2871-C04-02) and by Fundacion Seneca, CARM, Spain (Grant N PI-3/00854/FS/01). J.M. Malacarne was partially supported by PICD/CAPES, Brazil. References 1. A.D. Alexandrov, Uniqueness theorems for surfaces in the large V, Vestnik Leningrad Univ. Math. 13, 5-8 (1958); English translation: AMS Transl. 21, 412-416 (1962). 2. L.J. Alias, R. Lopez and B. Palmer, Stable constant mean curvature surfaces with circular boundary, Proc. Amer. Math. Soc. 127, 11951200 (1999). 3. L.J. Alias and J.M. Malacarne, Constant scalar curvature hypersurfaces with spherical boundary in Euclidean space, to appear in Rev. Mat. Iberoamericana (2002). 4. L.J. Alias and J.M. Malacarne, Constant higher order mean curvature hypersurfaces in Riemannian spaces, preprint, 2002.

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5. L.J. Alias and B. Palmer, On the area of constant mean curvature discs and annuli with circular boundaries, Math. Z. 237, 585-599 (2001). 6. J.L. Barbosa, Constant mean curvature surfaces bounded by a planar curve, Mat. Contemp. 1, 3-15 (1991). 7. J.L. Barbosa, Hypersurfaces of constant mean curvature in R" + 1 bounded by an Euclidean sphere, in Geometry and topology of submanifolds, II (Avignon, 1988), 1-9, (World Sci. Publishing, Teaneck, NJ, 1990). 8. J.L. Barbosa and A.G. Colares, Stability of hypersurfaces with constant r-mean curvature, Ann. Global Anal. Geom. 15, 277-297 (1997). 9. J.L. Barbosa and R. Sa Earp, Prescribed mean curvature hypersurfaces in EP +1 (—1) with convex planar boundary I, Geom. Dedicata 71, 61-74 (1998). 10. J.L. Barbosa and R. Sa Earp, Prescribed mean curvature hypersurfaces in H" +1 (—1) with convex planar boundary II, in Seminaire de Theorie Spectrale et Geometrie, Vol. 16, Annee 1997-1998, 43-79, (Univ. Grenoble I, Saint-Martin-d'Hres, 1998). 11. J.L. Barbosa and L.P. Jorge, Stable if-surfaces spanning S 1 (l), An. Acad. Brasil. Cienc. 6 1 , 259-263 (1994). 12. I. Bivens, Integral formulas and hyperspheres in a simply connected space form, Proc. Amer. Math. Soc. 88, 113-118 (1983). 13. S. Bochner, Vector fields and Ricci curvature, Bull. Amer. Math. Soc. 52, 776-797 (1946). 14. F. Brito and R. Sa Earp, Geometric configurations of constant mean curvature surfaces with planar boundary, An. Acad. Brasil. Cienc. 63, 5-19 (1991). 15. F. Brito, R. Sa Earp, W. Meeks and H. Rosenberg, Structure theorems for constant mean curvature surfaces bounded by a planar curve, Indiana Univ. Math. J. 40, 333-343 (1991). 16. L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations, III: Functions of the eigenvalues of the Hessian, Acta Math. 155, 261-301 (1985). 17. M.P. do Carmo and H.B. Lawson Jr., On Alexandrov-Bernstein theorems in hyperbolic space, Duke Math. J. 50, 995-1003 (1983). 18. R. Sa Earp and E. Toubiana, Variants on Alexandrov reflection principle and other applications of maximum principle, in Seminaire de Theorie Spectrale et Geometrie, Vol. 19, Annee 2000-2001, 93-121, (Univ. Grenoble I, Saint-Martin-d'Hres, 2001). 19. F. Fontenele and S.L. Silva, A tangency principle and applications, Illinois J. Math. 45, 213-228 (2001) 20. L. Garding, An inequality for hyperbolic polynomials, J. Math. Mech.

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8, 957-965 (1959). 21. B. Gidas, W.M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Coram. Math. Phys. 68, 209-243 (1979). 22. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der mathematischen Wissenschaften 224 Springer-Verlag, Berlin, 1983. 23. E. Heintze and H. Karcher, A general comparison theorem with applications to volume estimates for submanifolds, Ann. Sci. Ecole Norm. Sup. 11, 451-470 (1978). 24. H. Hopf, Differential geometry in the large, Lecture Notes in Mathematics, 1000 (Springer-Verlag, Berlin, 1989). 25. H. Hopf, Elementare Bemerkunge iiber die Losungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus, Sitz. Ber. Preuss. Akad. Wissensch. Berlin. Math.-Phys. Kl. 19, 147 (1927). 26. W.Y. Hsiang, Z.H. Teng and W.C.Yu, New examples of constant mean curvature immersions of (2k — l)-spheres into Euclidean 2fc-space. Ann. of Math. 117, 609-625 (1983). 27. C.C. Hsiung, Some integral formulas for closed hypersurfaces, Math. Scand. 2, 286-294 (1954). 28. N. Kapouleas, Compact constant mean curvature surfaces in Euclidean three-space, J. Differential Geom. 33, 683-715 (1991). 29. N. Kapouleas, Constant mean curvature surfaces constructed by fusing Wente tori, Invent. Math. 119, 443-518 (1995). 30. M. Koiso, Symmetry of hypersurfaces of constant mean curvature with symmetric boundary, Math. Z. 191, 567-574 (1986). 31. N.J. Korevaar, Sphere theorems via Alexandrov for constant Weingarten curvature hypersurfaces: Appendix to a note of A. Ros, J. Differential Geom. 27, 221-223 (1988). 32. N.J. Korevaar, R. Kusner and B. Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30, 465-503 (1989). 33. M.L. Leite, The tangency principle for hypersurfaces with a null intermediate curvature, XI Escola de Geometria Diferencial, Brazil (2000). 34. R. Lopez and S. Montiel, Constant mean curvature discs with bounded area, Proc. Amer. Math. Soc. 123, 1555-1558 (1995). 35. R. Lopez and S. Montiel, Constant mean curvature surfaces with planar boundary, Duke Math. J. 85, 583-604 (1996). 36. S. Montiel and A. Ros, Compact hypersurfaces: the Alexandrov theorem for higher order mean curvatures, in Differential geometry, Pitman Monogr. Surveys Pure Appl. Math., 52, 279-296, (Longman Sci. Tech.,

58

Harlow, 1991). 37. J. Oprea, Differential geometry and its applications, Prentice Hall, New Jersey, 1997. 38. R. Osserman, Curvature in the eighties, Amer. Math. Monthly 97, 731756 (1990). 39. R.C. Reilly, Variational properties of functions of the mean curvature for hypersurfaces in space forms, J. Differential Geom. 8, 465-477 (1973). 40. R.C. Reilly, Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J. 26, 459-472 (1977). 41. A. Ros, Theoremes globales pour les hypersurfaces, in Conferenze del Seminario di Matematica dell'Universita di Ban, 223 (Bari, 1987) (Gius. Laterza & Figli S.p.A., Bari). 42. A. Ros, Compact hypersurfaces with constant scalar curvature and a congruence theorem, J. Differential Geom. 27, 215-220 (1988). 43. A. Ros, Compact hypersurfaces with constant higher order mean curvatures, Rev. Mat. IberoamericanaZ, 447-453 (1987). 44. H. Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sc. Math. 117, 211-239 (1993). 45. R.M. Schoen, Uniqueness, symmetry, and embeddedness of minimal surfaces, J. Differential Geom. 18, 791-809 (1983). 46. J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43, 304-318 (1971). 47. M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol IV, 2nd edition, Publish or Perish Inc. (1979). 48. H.C. Wente, Counterexample to a conjecture of H. Hopf, Pacific J. Math. 121, 193-243 (1986). 49. S.T. Yau, Problem section, in Seminar on Differential Geometry, Annals Math. Studies No. 102 (Princeton University Press, Princeton NJ, 1982).

U N I Q U E N E S S OF SPACELIKE H Y P E R S U R F A C E S W I T H C O N S T A N T M E A N CURVATURE IN GENERALIZED ROBERTSON-WALKER SPACETIMES LUIS J. ALIAS Departamento de Matemdticas, Universidad de Murcia, E-30100 Espinardo, Murcia, Spain E-mail: [email protected]

Departamento

SEBASTIAN MONTIEL de Geometria y Topologia, Universidad de Granada, E-18071 Granada , Spain E-mail: smontiel&ugr. es

Dedicated to Professor Antonio M. Naveira on the occasion of his 60th birthday In a recent paper 2 , the second author stated some uniqueness results for compact spacelike hypersurfaces with constant mean curvature in generalized RobertsonWalker spacetimes. Our objective here is to correct the proof of one of those uniqueness results, as well as to derive other new related results. In particular, we extend some of those uniqueness statements to the case of complete (non-compact) spacelike hypersurfaces.

1

Introduction

Spacelike hypersurfaces with constant mean curvature in Lorentzian manifolds have been objects of increasing interest in recent years, both from physical and mathematical points of view. A basic question on this topic is the problem of uniqueness for this type of hypersurfaces. In [*], the first author together with Romero and Sanchez studied the uniqueness of spacelike hypersurfaces with constant mean curvature in a wide family of spacetimes, the family of the so called generalized Robertson- Walker (GRW) spacetimes, which are Lorentzian warped products with 1-dimensional negative definite base and Riemannian fiber (for the details, see Section 2). In particular, it was shown in f1] that in a GRW spacetime obeying the timelike convergence condition, every compact spacelike hypersurface with constant mean curvature must be umbilical. Recall that a spacetime is said to obey the timelike convergence condition if the Ricci curvature is non negative on timelike directions. In particular, for a GRW spacetime with Riemannian fiber Fn and warping function / , the

59

60

timelike convergence condition is equivalent to the conditions RicF>(n-l)sup(//"-/'2)(,)F and / " < 0, F

where Ric is the Ricci curvature of the Riemannian manifold Fn. More recently, the second author 2 observed that, although the timelike convergence condition is physically reasonable, it is not so good as a hypothesis to deduce uniqueness results. Specifically, he observed that any of the two conditions above implies separately the required uniqueness. In particular, the sole hypothesis / " < 0 suffices to guarantee uniqueness, without any other restriction on the Ricci curvature of the fiber. In fact, a more general condition on the warping function / is sufficient in order to obtain uniqueness, as stated in the following result. Theorem 1. (Theorem 7 in [2]) Let f : I —> R be a positive smooth function defined on an open interval, such that ff" — / ' < 0, that is, such that - log / is convex. Then, the only compact spacelike hypersurfaces immersed into a generalized Robertson-Walker spacetime —I Xj Fn with constant mean curvature are the slices {t} x F, for any (necessarily compact) Riemannian manifold F. Unfortunately, the proof of this result given in [2] is not totally right. Actually, that proof uses the fact that every subharmonic or superharmonic function on a compact Riemannian manifold must be constant, and it applies that fact to the function n : M —> I obtained by projecting the spacelike hypersurface M on the interval / . However, there is a mistake in the computation of the Laplacian of IT in [2], which makes the argument not work. Our objective here is to correct the proof of Theorem 1, as well as to obtain some other new related results. In particular, we extend Theorem 1 to the case of complete (non-compact) spacelike hypersurfaces (see Theorem 10). 2

Preliminaries

Let Fn be an n-dimensional Riemannian manifold, and let J be a 1dimensional manifold (either a circle or open interval of R). Throughout this paper, —IXfFn will denote the (n + l)-dimensional product manifold I x F endowed with the Lorentzian metric R is a positive smooth function, 717 and np are the projections from I x F onto each factor, and (,)F is the Riemannian metric on F. That is, —I Xf Fn is a Lorentzian warped product with base (I, —dt2), fiber (Fn, (,) F ), and warping function / . For simplicty we will write {,) = -dt2 +

f2(t)(,)F.

Following I1], we will refer to —Ix-f Fn as a generalized Robertson-Walker (GRW) spacetime. In the case where Fn is a Riemannian space form with constant sectional curvature, —IXfFn is classically called a Robertson-Walker spacetime, and it is a spatially homogeneous spacetime. Observe that spatial homogeneity, which is reasonable as a first approximation of the large scale structure of the universe, may not be realistic when one considers a more accurate scale. For that reason, GRW spacetimes could be suitable spacetimes to model universes with inhomogeneous spacelike geometry [5]. Besides, small deformations of the metric on the fiber of Robertson-Walker spacetimes fit into the class of GRW spacetimes. A smooth immersion ip : M —> — / Xf Fn of a an n-dimensional connected manifold M is said to be a spacelike hypersurface if the induced metric via tp is a Riemannian metric on M, which, as usual, is also denoted by (,). Since dt = (d/dt)(t,g),

t€l,q£F,

is a unitary timelike vector field globally defined on —I Xf Fn, it determines a time-orientation on —I Xf Fn. Thus, for a given spacelike hypersurface M, there exists a unique timelike unit normal field N globally defined on M which is in the same time-orientation as dt, so that (dt,N) X(M) defines the shape operator of M with respect to N. The mean curvature of M is then defined as H — —(\/n)tv{A). The choice of the sign — in our definition of H is motivated by the fact that in that case the mean curvature vector is given by H — HN. Therefore, H(p) > 0

62

at a point p £ M if and only if H(p) is in the same time-orientation as N, and hence as dt. We will need the following remarkable properties of a spacelike hypersurface in a GRW spacetime, which can be found in f1 ]. Lemma 2. • If a GRW spacetime —I Xf Fn admits a compact spacelike hypersurface, then the fiber F is necessarily compact. • If the universal covering of the fiber of a GRW spacetime —IXfFn is compact, then every complete spacelike hypersurface on which /(TTJ) is bounded is necessarily compact. 3

Compact spacelike hypersurfaces

In this section we will consider the case of compact spacelike hypersurfaces in a GRW spacetime, necessarily with compact Riemannian fiber because of Lemma 2. Lemma 3. Let f : I —> ]R be a positive smooth function defined on an open interval, such that ff" — f < 0, that is, such that — log / is convex, and let —IXfFn be a GRW spacetime with (necessarily compact) Riemannian fiber F. Then the mean curvature of every compact spacelike hypersurface immersed into —IxjFn satisfies min H < (log / ) ' ( W ) < (log /)'(t m in) < max H,

(4)

where tm[n and i m a x denote, respectively, the minimum and the maximum values of TTJ\M = 717 o ip on M. Proof. Let us write for simplicity 7r = 7T/|M = iri ° ip, which is a smooth function on M. It is not difficult to see that the gradient of TV is —dj, where dj G X{M) denotes the tangential component of dt, that is VTT = - 5 7 = -dt-(dt,N)N.

(5)

From the general relationship between the Levi-Civita connection of a warped product and the Levi-Civita connections of its base and fiber (see Chapter 7, Proposition 35 in [3]), it follows that V z ^ = (log/)'(7rj)Z F for every vector field Z on —IXfFn, on the fiber, that is,

(6)

where ZF denotes the projection of Z

ZF = Z + (Z,dt)dt.

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Therefore, by taking covariant derivative in (5) and using (2) and (3), we obtain from (6) that VXVTT

= -VxdJ

= -(log/)'(*) (X + (X,dt)dJ) + (dt,N)AX

for every tangent vector field X e X(M). given by

In particular, the Laplacian of n is

Ayr = -n ((log f)'(ir)+H(dt,N))

- (log/)'(TT)|VTT| 2 .

(7)

Since M is compact, there exist points pm\n,Pma.x 6 M where the function TT attains its minimum and its maximum, respectively, that is, Tr(Pmin) = min7r(p) = fmin < £ max = Tr(Pmax) = max7r(p). pGM

p£M

In particular, p m i n and p m a x are critical points of IT, and from (5) we have N(pmin) = (dt)pmin and N(pmax) = (dt)pmax- Using this in (7), we also obtain that A7r(pmin) = -n((log/) / (< m i n ) - H(pmin))

> 0,

and A7r(p max ) = - n ( ( l o g / ) ' ( t m a x ) - ff(pmax)) < 0, that is, min H < H(pmax)

< (log / ) ' ( t m a x )

(8)

maxF>ff(pmin)>(log/)'(tmin).

(9)

and

Finally, since —log/ is convex, then (log/)' is non increasing and so, (log/)'(*max) < (log/)'(*min), which jointly with (8) and (9) yields (4). This finishes the proof of Lemma 3. • In the proof above, the hypothesis on the warping function has been used just to assure that (log/)'(t m a x ) < (log/)'(i m j„). Therefore, we can state in general the following. Corollary 4. Let —IXfFn be a GRW spacetime with (necessarily compact) Riemannian fiber F. Then the mean curvature of every compact spacelike hypersurface immersed into —IXfFn satisfies mini? < (log/)'(i m a x )

and

maxff > (log/)'(* mi „),

where tm\n and t m a x denote, respectively, the minimum and the maximum values of I:I\M = T R is the increasing function given by

9(t) = I f(s)ds,

(12)

Jto

for a fixed arbitrary to G / . Then, it follows from (10) and (11) that Au = g'(7r)A7r + 5"(7r)|V7r|2 = -nHf(w) = -nHf(ir)(l

(1 + (dt, N)) + +

(/'(TT)

- Hf(ir))

|VTT|2

(dt,N)).

Besides, recall that 1 + (dt,N) < 0 by (1). Hence u is either subharmonic or superharmonic on M, which is compact. Prom this, it follows that u is constant on the hypersurface, and since g(t) is increasing that means that 7r itself is constant on M, that is, it is a slice. This finishes the proof of Theorem 1. On the other hand, let us recall that a GRW spacetime is said to be static when the warping function is constant (/ — 1 = constant without loss of generality). As a nice consequence of Corollay 4 we also obtain the following. Corollary 6. Let I/J : M —> —M x Fn be a compact spacelike hypersurface immersed into a static GRW spacetime. Then its mean curvature H satisfies min H < 0 < max H.

(13)

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In particular, it follows from (13) that every compact spacelike hypersurface immersed into a static GRW spacetime with constant mean curvature H must be maximal, that is, H = 0. Hence, by (7) we get Air = 0 and IT must be constant on M. That is, M must be a slice. 4

Complete spacelike hypersurfaces

In this section we will extend to the case of complete spacelike hypersurfaces in GRW spacetimes some of the previous results. In order to do that, we will make use of the following generalized maximum principle for Riemannian manifolds due to Omori [4] and Yau [6]. Lemma 7 (A generalized maximum principle). Let M be a complete Riemannian manifold whose Ricci curvature is bounded away from — oo and let u : M —> R be a smooth function on M. a) If u is bounded from above on M, then for each e > 0 there exists a point pe £ M such that |Vu(p e )| < £,

Au(pe) < e,

sup it - £ < u(p£) < supu;

(14)

b) If u is bounded from below on M, then for each e > 0 there exists a point qe e M such that \Vu(q£)\ < e,

Au(qe) > —e,

infu < u(qe) < infw + e.

(15)

Here Vu and Au denote, respectively, the gradient and the Laplacian of u on M. Using this generalized maximum principle, Corollary 4 can be extended as follows. Lemma 8. Let -I Xf Fn be a GRW spacetime with Riemannian fiber F, and let tp : M —> —I Xf Fn be a complete spacelike hypersurface immersed into —IxfFn whose Ricci curvature is bounded away from — oo. If KI\M is bounded on M with ijnf,iSup S / , then its mean curvature H satisfies infff —/ Xf Fn be a complete spacelike hypersurface immersed into —IXfFn whose Ricci curvature is bounded away from —oo. If TTI\M is bounded on M with t-mf,tsup € I, then its mean curvature H satisfies inf H < (log/)'(t BUp ) < (log/)'(t i n f ) < supff, where i;nf and tsup denote, respectively, the infimum and the supremum of — TTI ° V' on M. As a consequence of this, we obtain the following uniqueness result for the complete case. Theorem 10. Let f : I —> R be a positive smooth function defined on an open interval, such that ff" — f < 0 and equality holds only at isolated points of I. Then, the only complete spacelike hypersurfaces immersed into a generalized Robertson-Walker spacetime —I Xf Fn which are contained in a timelike bounded region TT/|M

n(«i,* 2 ) = [h,h] x f " = {(t,q) e-IxfFn:h