DOCSLIB.ORG

Existence and Non-Existence for a Mean Curvature Equation in Hyperbolic Space

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Existence and Non-Existence for a Mean Curvature Equation in Hyperbolic Space COMMUNICATIONS ON Website: http://AIMsciences.org PURE AND APPLIED ANALYSIS Volume 4, Number 3, September 2005 pp. 549{568 EXISTENCE AND NON-EXISTENCE FOR A MEAN CURVATURE EQUATION IN HYPERBOLIC SPACE Elias M. Guio Departamento de Matem¶atica Universidade Federal do Espirito Santo, Vit¶oria29060-900 ES, Brazil Ricardo Sa Earp Departamento de Matem¶atica Pontif¶³ciaUniversidade Cat¶olicado Rio de Janeiro Rio de Janeiro 22453-900 RJ, Brazil (Communicated by Gigliola Sta±lani ) Abstract. There exists a well-known criterion for the solvability of the Dirich- let Problem for the constant mean curvature equation in bounded smooth do- mains in Euclidean space. This classical result was established by Serrin in 1969. Focusing the Dirichlet Problem for radial vertical graphs P.-A. Nitsche has established an existence and non-existence results on account of a criterion based on the notion of a hyperbolic cylinder. In this work we carry out a similar but distinct result in hyperbolic space considering a di®erent Dirichlet Problem based on another system of coordinates. We consider a non standard cylin- der generated by horocycles cutting orthogonally a geodesic plane P along the boundary of a domain ­ ½ P: We prove that a non strict inequality between 0 the mean curvature HC (y) of this cylinder along @­ and the prescribed mean 0 curvature H(y); i.e HC (y) > jH(y)j; 8 y 2 @­ yields existence of our Dirichlet Problem. Thus we obtain existence of surfaces whose graphs have prescribed mean curvature H(x) in hyperbolic space taking a smooth prescribed bound- ary data ': This result is sharp because if our condition fails at a point y a non-existence result can be inferred. 1. Introduction. A classical result in Partial Di®erential Equations and Di®er- ential Geometry due to Serrin (1969) is the following: Given a constant a, there exists a condition on the boundary of the domain ­ such that the Dirichlet Prob- lem for the mean equation H = a is solvable. In fact, if the mean curvature of the Euclidean cylinder over @­ is bigger than a; then some a priori gradient estimates ensure a posteriori existence for the Dirichlet Problem. This inequality is sharp in the following sense: if it does not hold at a point then it can be inferred non- existence for a certain smooth boundary data. In this article we carry out a similar program in hyperbolic space, taking into account a certain coordinates system and a certain geometric boundary condition in hyperbolic space based on a suitable but not standard notion of \cylinder". A novelty value in our paper is the deduction of a solution of the related Dirichlet Problem for prescribed (not necessarily constant) mean curvature H(x): Moreover, the sharpness of the result is assured by a non- 2000 Mathematics Subject Classi¯cation. 35J25, 53A10. Key words and phrases. Dirichlet problem, mean curvature equation, hyperbolic space. 549 550 ELIAS M. GUIO AND RICARDO SA EARP existence result if our criterion in hyperbolic space \fails at a point". In fact, let P be a n-dimensional totally geodesic plane in hyperbolic space Hn+1 and let ­ be a domain in P with C2 boundary @­: We shall focus the upper halfspace model of hyperbolic space, i.e Hn+1 := f(x0; x1; : : : ; xn); xn > 0gequipped with the metric 2 1 2 2 2 ds = 2 (dx0 + dx1 + ¢ ¢ ¢ + dxn). For sake of simplicity, we ¯x a hyperplane xn n+1 n+1 P := fx0 = 0g. Notice that the asymptotic boundary of H denoted @1H is n+1 de¯ned by @1H := fxn = 0g [ f1g: We represent by @­ a (n ¡ 1)-dimensional closed connected embedded smooth submanifold of P. We then de¯ne C by the n- dimensional cylinder generated by horocycles with asymptotic boundary 1 cutting orthogonally P: That is, C := f(t; x1; ¢ ¢ ¢ ; xn); t 2 R; (0; x1; ¢ ¢ ¢ ; xn) 2 @­g: Notice that in this model P is a vertical hyperplane and our cylinder C is an \Euclidean horizontal cylinder". We shall consider throughout this paper the fol- lowing de¯nition of horizontal graph of a function in hyperbolic space: Let ­ ½ P be the domain whose boundary is @­. Given a function u : ­ ! R, the graph of u is de¯ned as the set G(u) = f(u(x1; : : : ; xn); x1; : : : ; xn); (0; x1; : : : ; xn) 2 ­g: Indeed, this is the natural notion of graph in our system of coordinates given by horocycles cutting orthogonally P: We shall commence to describe the equation on which we are going to focus: Given a C1 function H : ­¹ ½ P ! R, and a C2 function ' : ­ ! R; we shall consider the following Dirichlet Problem for the prescribed horizontal mean curvature equation, say Problem (P ): µ ¶ µ ¶ Du n D u div = H + n in ­ (1.1) W (u) xn W (u) u = ' on @­ p where Du means the Euclidean gradient of u in P, W (u) = 1 + jDuj2 @u n Diu = ; i = 1; : : : ; n and the symbol \div" denotes the divergence in R = @xi f(x1; : : : ; xn)g: We shall explain with details some historical backgrounds and mo- tivations. A main uniqueness result obtained by Lucas Barbosa and the second author (see [5], [6]) gives a strong motivation for the study of the above Dirichlet Problem. In fact, under some assumptions about the relation of the size of H with the geometry of @­ , uniqueness of the above Dirichlet Problem (horizontal graphs) can be inferred in the class of immersed compact hypersurfaces with same mean curvature and same boundary. Particularly, if H is constant with H2 6 1; and if the principal curvatures ¸i of the boundary @­ satisfy an inequality ¸i > h0 > 1; then any smooth compact connected immersed n-dimensional manifold with bound- ary @­ is an horizontal graph. Lucas Barbosa and the second author have focused attention exclusively on the case of zero boundary conditions. Horizontal graphs satisfying the assumptions above were constructed to be used as barriers to obtain the cited main uniqueness theorem. Before proceeding with the discussion of our problem in hyperbolic space, we pause to make precise the criterion for the solvability of the Dirichlet Problem in Euclidean space, and for the equation of constant mean curvature equation in a MEAN CURVATURE EQUATION IN HYPERBOLIC SPACE 551 0 bounded C2 domain ­: Let H³ be the´ mean curvature of @­. Then the constant Du mean curvature equation div W (u) = nH in ­ , u = ' on @­ is solvable for 0 constant H and arbitrary boundary continuous data ' if and only if (n ¡ 1)H > n jHj everywhere on @­. This theorem for smooth boundary data was established by J. Serrin in [26]. Notice that convexity of ­ is the sharp criterion of solvability for arbitrary continuous boundary data of the Dirichlet Problem on a bounded domain for the minimal surface equation in two variables; this was proved by R. Finn [10]. For more than two variables H. Jenkins and J. Serrin in [13] showed that the minimal hypersurface equation on a bounded C2;® domain, say ­, 0 < ® < 1; 0 is solvable for arbitrary ' 2 C2;®(­)¹ if and only if the mean curvature H of @­ is non-negative at every point of @­. Notice that the mean curvature equation in Euclidean space is now classical in the literature (see, for instance [7], [3], [9], [12]). Notice that there exist some recent interesting papers on minimal graphs over unbounded domains (see [29], [15]). We should mention the remarkable work of R. Schoen on the mean curvature equation in Euclidean space and its application to Geometry (see [27]). L. Bers showed the classical minimal surface equation cannot have an isolated singularity (see [2]). The same result for the constant mean curvature equation was established by R. Finn (see [11]). The removable singularity theorem for prescribed mean curvature equations in Rn+1 is due to the H. Rosenberg in a joint work with the second author ( see [22]). Returning to hyperbolic space, we observe that a removable singularity type theorem for the horizontal mean curvature equation in hyperbolic space was done by B. Nelli and the second author (see [20]). Now we shall write down the geometric condition (say, condition (¤)) that plays a crucial role in the whole study of our Dirichlet Problem (P ) for the prescribed mean curvature H(x): 0 HC (y) > jH(y)j; 8 y 2 @­ (¤) 0 where HC (y) is the mean curvature of the cylinder generated by horocycles at a boundary point y: We can express the above inequality in the following equivalent analytic form 0 (n ¡ 1)H (y) + xn(y)Dnd(y) > njH(y)j; 8 y 2 @­ (¤) 0 where H (y); is the mean curvature of the boundary of ­ and xn(y)Dnd(y) is the nth vertical component of the Euclidean unit interior normal to @­ at a boundary point y (d(x) is the hyperbolic distance from x 2 ­ to the boundary @­). We feel that it is interesting to now give a geometric insight about the second term that appears on the left of the above inequality : We recall that Serrin's condition in Euclidean space ensures a priori boundary gradient estimates that are essential for the related Dirichlet Problem. On the other hand, looking to the simplest situation in hyperbolic space, it is not hard to see that there exist \spherical caps" of geodesic (minimal) planes cutting orthogonally the plan of the domain ­ at some point y; whose projections are convex curves (Euclidean ellipse) in the hyperbolic plane P: Thus, Serrin inequality holds but we have no control of the gradient at y: However, condition (¤) allows us to infer a priori boundary gradient estimates for the smooth solutions of Dirichlet Problem (P ): We infer these estimates in Section 1.
Load more

Recommended publications
  • Marcel Berger Remembered
    Marcel Berger Remembered Claude LeBrun, Editor and Translator Gérard Besson EDITOR’S NOTE. Claude LeBrun has kindly assembled this memorial for Marcel Berger. This month also Marcel Berger,1 one of the world’s leading differential marks the conference “Riemannian Geometry Past, geometers and a corresponding member of the French Present and Future: An Homage to Marcel Berger” at Academy of Sciences for half a century, passed away on IHÉS. October 15, 2016, at the age of eighty-nine. Marcel Berger’s contributions to geometry were both broad and deep. The classification of Riemannian holo- nomy groups provided by his thesis has had a lasting impact on areas ranging from theoretical physics to alge- braic geometry. His 1960 proof that a complete oriented even-dimensional manifold with strictly quarter-pinched positive curvature must be a topological sphere is the Claude LeBrun is professor of mathematics at Stony Brook Univer- sity. His email address is [email protected]. Gérard Besson is CNRS Research Director at Institut Fourier, Uni- For permission to reprint this article, please contact: versité Grenoble, France. His email address is g.besson@ujf [email protected]. -grenoble.fr. DOI: http://dx.doi.org/10.1090/noti1605 1Pronounced bare-ZHAY. December 2017 Notices of the AMS 1285 research director by the CNRS. He served as president of the French Mathematical Society (SMF) during the period of 1979–80 and in this capacity helped oversee the foundation of the International Center for Mathematical Workshops (CIRM) in Luminy. In 1985 he then became director of the Institute for Higher Scientific Studies (IHÉS) in Bures-sur-Yvette, a position in which he served until 1993, when he was succeeded by his former student Jean-Pierre Bourguignon.
    [Show full text]
  • Robert B. Kusner Department of Mathematics and Statistics
    Robert B. Kusner Department of Mathematics and Statistics University of Massachusetts at Amherst Amherst, MA 01003 USA 413-545-6022, FAX 413-545-1801 (but please call my cell during the COVID19 pandemic) [email protected], [email protected] www.gang.umass.edu/∼kusner www.math.umass.edu/directory/faculty/rob-kusner Publications There are many more projects underway, but those listed here are ordered by when the project came into substantial form with specific methods and results outlined in a draft preprint. This order reveals a feature of doing mathematics research that is unfortunately lost in more public-facing lists (like the Selected Publications list on Rob Kusner's Department/CNS homepage): some projects take considerably longer to cultivate and curate | to draft, to polish, to post, to revise, to submit, to publish | than others. Since a variety of other order- ings, including those based on auto-updated citation records, are offered by Google Scholar and by the American Mathematical Society's MathSciNet, the reviewer is invited to peruse those too (as of early 2021, for example, Google Scholar records over 2700 citations to Rob's professional publications, with well over 100 citations per year for each of the past 7 years). 64. On the Canham Problem [in preparation] (with P. McGrath). 63. Gordian Unlinks [in preparation] (with W. Kusner). 62. On the Willmore Problem for surfaces with symmetry, arXiv:2103.09432 (with P. Wang). 61. On Steklov Eigenspaces for Free Boundary Minimal Surfaces in the Unit Ball [submitted] (with P. McGrath). 60. The geometry of complete minimal surfaces at infinity and the Willmore Morse index of their inversions [in preparation] (with J.
    [Show full text]
  • Simons' Equation and Minimal Hypersurfaces in Space Forms
    SIMONS' EQUATION AND MINIMAL HYPERSURFACES IN SPACE FORMS BIAO WANG n Abstract. Let n > 3 be an integer, and let Σ be a non totally geodesic complete minimal hypersurface immersed in the (n + 1)-dimensional space form M n+1(c), where the constant c denotes the sectional curva- ture of the space form. If Σn satisfies the Simons' equation (3.9), then n n either (1) Σ is a catenoid if c 6 0, or (2) Σ is a Clifford minimal hypersurface or a compact Ostuki minimal hypersurface if c > 0. This paper is motivated by Tam and Zhou [17]. 1. Introduction In 1968 Simons [14] (see also [2, 7 or 9] and [18, 1.6]) showed that the second fundamental form of an immersedx x minimal hypersurfacex in the sphere or in the Euclidean space satisfies a second order elliptic partial differential equation, which can imply the famous Simons' inequality. The Simons' inequality enabled him to prove a gap phenomenon for minimal n submanifolds in the sphere and the Bernstein's problem in R for n 6 7. Since then the Simons' inequality has been used by various authors to study minimal immersions. In this paper, we shall classify all complete minimal hypersurfaces im- mersed in the space form M n+1(c) which satisfy the Simons' equation (3.9), where n > 3. Roughly speaking, a catenoid is a minimal rotation hypersurface im- mersed in M n+1(c). In the case when c = 0, Tam and Zhou [17, Theorem 3.1] proved that if a non-flat complete minimal hypersurface Σn immersed n+1 in R satisfies the Simons' equation (3.9) on all nonvanishing points of A , then Σn must be a catenoid.
    [Show full text]
  • Remembering Shoshichi Kobayashi
    Remembering Shoshichi Kobayashi Gary R. Jensen, Coordinating Editor hoshichi Kobayashi was born January 4, 1932, in Kofu City, Japan. He grew up amidst the colossal devastation of World War II. After completing a BS degree Sin 1953 at the University of Tokyo, he spent the academic year 1953–1954 on a French scholarship at the University of Paris and the University of Strasbourg. From there he went to the University of Washington in Seattle, where he received a Ph.D. in 1956 for the thesis “Theory of connections,” (Annali di Mat. (1957), 119–194), written under the direction of Carl B. Allendoerfer. There followed two years as a member of the Photo from the Oberwolfachcourtesy Photo of Collection, the Archives of the MFO. Institute for Advanced Study in Princeton and then Shoshichi Kobayashi two years as a research associate at MIT. In 1960 he became an assistant professor at the University of British Columbia, which he left in 1962 to go to many of Shoshichi’s mathematical achievements. the University of California at Berkeley, where he Several of his students relate their experiences of remained for the rest of his life. working under Professor Kobayashi’s supervision. Shoshichi published 134 papers, widely admired Some Berkeley colleagues write of their personal not only for their mathematical content but for the experiences with Shoshichi. elegance of his writing. Among his thirteen books, His brother, Hisashi, recounts some details of Hyperbolic Manifolds and Holomorphic Mappings Shoshichi’s early education. Shoshichi’s wife and and the Kobayashi metric in complex manifolds daughters tell a fascinating story of his childhood, created a field, while Foundations of Differential his studies abroad, and major life experiences Geometry, I and II, coauthored with Nomizu, are preceding his move to Berkeley.
    [Show full text]
  • IMPA and the Dissemination of Mathematical Economics in Brazil Matheus Assaf FEA - Universidade De São Paulo
    Applying pure mathematics: IMPA and the dissemination of mathematical economics in Brazil Matheus Assaf FEA - Universidade de São Paulo Resumo: O objetivo do artigo é contribuir para o entendimento histórico da internacionalização e disseminação da economia matemática no Brasil. Em foco, o papel central do Instituto de Matemática Pura e Aplicada (IMPA) para o desenvolvimento da economia matemática. Avalia-se o porquê da economia tornar-se a disciplina representante da matemática aplicada em um instituto influenciado pelo purismo matemático do grupo Bourbaki. O artigo também explora a prática da teoria econômica dentro de um departamento de matemática, analisando a relação entre estas duas disciplinas. Palavras-chave: Pensamento Econômico Brasileiro, Economia Matemática, Bourbaki, IMPA, História do Pensamento Econômico Área: História Econômica, do Pensamento Econômico e Demografia Histórica O autor agradece à FAPESP pelo financiamento da pesquisa necessária para a realização deste trabalho. Introduction Unity and purity are two main features of the mathematics advocated by Nicolas Bourbaki, a fictitious scholar created by a group of mathematicians1 in 1935. Bourbaki (1950) takes up on the fragmentation of mathematics that followed the increase in the number of works devoted to the discipline. Facing the question of whether this growth would develop a “strongly constructed organism, acquiring ever greater cohesion and unity with its new growths, or whether it is the external manifestation of a tendency towards a progressive splintering” (221) Bourbaki advocates for the existence of a single mathématique, based on the axiomatic method and the study of mathematical structures. On the other hand, this single mathématique had a defined internal pecking order where pure mathematics – the foundation of the axiomatic method that provided unity to mathematics – held the highest rank.
    [Show full text]
  • Front Matter
    John M. Lee Introduction to Riemannian Manifolds Second Edition 123 John M. Lee Department of Mathematics University of Washington Seattle, WA, USA ISSN 0072-5285 ISSN 2197-5612 (electronic) Graduate Texts in Mathematics ISBN 978-3-319-91754-2 ISBN 978-3-319-91755-9 (eBook) https://doi.org/10.1007/978-3-319-91755-9 Library of Congress Control Number: 2018943719 Mathematics Subject Classification (2010): 53-01, 53C20, 53B20 Originally published with title “Riemannian Manifolds: An Introduction to Curvature” 1st edition: © Springer-Verlag New York, Inc. 1997 2nd edition: © Springer International Publishing AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.
    [Show full text]
  • Prizes and Awards
    SEATTLE JAN 6-9, 2016 January 2016 Prizes and Awards 4:25 p.m., Thursday, January 7, 2016 PROGRAM OPENING REMARKS Francis Su, Mathematical Association of America CHEVALLEY PRIZE IN LIE THEORY American Mathematical Society OSWALD VEBLEN PRIZE IN GEOMETRY American Mathematical Society DAVID P. R OBBINS PRIZE American Mathematical Society LEVI L. CONANT PRIZE American Mathematical Society AWARD FOR DISTINGUISHED PUBLIC SERVICE American Mathematical Society E. H. MOORE RESEARCH ARTICLE PRIZE American Mathematical Society LEROY P. S TEELE PRIZE FOR MATHEMATICAL EXPOSITION American Mathematical Society LEROY P. S TEELE PRIZE FOR SEMINAL CONTRIBUTION TO RESEARCH American Mathematical Society LEROY P. S TEELE PRIZE FOR LIFETIME ACHIEVEMENT American Mathematical Society AMS-SIAM NORBERT WIENER PRIZE IN APPLIED MATHEMATICS American Mathematical Society Society for Industrial and Applied Mathematics FRANK AND BRENNIE MORGAN PRIZE FOR OUTSTANDING RESEARCH IN MATHEMATICS BY AN UNDERGRADUATE STUDENT American Mathematical Society Mathematical Association of America Society for Industrial and Applied Mathematics COMMUNICATIONS AWARD Joint Policy Board for Mathematics LOUISE HAY AWARD FOR CONTRIBUTION TO MATHEMATICS EDUCATION Association for Women in Mathematics M. GWENETH HUMPHREYS AWARD FOR MENTORSHIP OF UNDERGRADUATE WOMEN IN MATHEMATICS Association for Women in Mathematics ALICE T. SCHAFER PRIZE FOR EXCELLENCE IN MATHEMATICS BY AN UNDERGRADUATE WOMAN Association for Women in Mathematics CHAUVENET PRIZE Mathematical Association of America EULER BOOK PRIZE Mathematical Association of America THE DEBORAH AND FRANKLIN TEPPER HAIMO AWARDS FOR DISTINGUISHED COLLEGE OR UNIVERSITY TEACHING OF MATHEMATICS Mathematical Association of America YUEH-GIN GUNG AND DR.CHARLES Y. HU AWARD FOR DISTINGUISHED SERVICE TO MATHEMATICS Mathematical Association of America CLOSING REMARKS Robert L.
    [Show full text]
  • 266 O Percurso Profissional De Manfredo Perdigão Do Carmo E a Geometria Diferencial No Brasil Antonio José Melo De Queiroz
    266 O percurso profissional de Manfredo Perdigão do Carmo e a Geometria Diferencial no Brasil Antonio José Melo de Queiroz O PERCURSO PROFISSIONAL DE MANFREDO PERDIGÃO DO CARMO E A GEOMETRIA DIFERENCIAL NO BRASIL THE PROFESSIONAL CAREER OF MANFREDO PERDIGÃO DO CARMO AND THE DIFFERENTIAL GEOMETRY IN BRAZIL Antonio José Melo de Queiroz1 Universidade Estadual do Ceará Resumo Este trabalho elaborou uma visão panorâmica sobre o início da carreira do professor e pesquisador brasileiro Manfredo Perdigão do Carmo (1928 – 2018). Será discutida de forma breve sua formação em Engenharia Civil, a pós-graduação em Geometria Diferencial e suas atividades iniciais de pesquisa e ensino. No texto será abordada a produção científica deste eminente geômetra, uma breve exposição dos primeiros artigos científicos, formação de parcerias de pesquisa, orientação de diversos estudantes em nível de doutorado e o constante esforço para colaborar com a produção de conhecimento matemático no Brasil. Diversos artigos publicados pelo professor Manfredo foram elaborados em parcerias com matemáticos brasileiros ou estrangeiros. Essas colaborações se ampliavam à medida que Manfredo do Carmo orientava vários estudantes de pós-graduação em Matemática no Brasil, principalmente, no Instituto de Matemática Pura e Aplicada (IMPA), além disso, estas e outras atividades de ensino colaboraram para a formação de uma fecunda escola de Geometria Diferencial no país. Manfredo do Carmo também escreveu uma série de livros, a maioria deles dedicados à Geometria Diferencial, algumas destas obras são referências de prestígio nacional e internacional. Além das atividades de ensino e pesquisa, o matemático esteve envolvido com ações de gerência institucional, foi presidente da Sociedade Brasileira de Matemática (SBM) e membro da Academia Brasileira de Ciências (ABC).
    [Show full text]
  • The Willmore Conjecture: a Celebration of Mathematics by Áurea Quintino*
    The Willmore Conjecture: a Celebration of Mathematics by Áurea Quintino* Cum enim Mundi universi fabrica sit perfectissima, atque a Creatore sapientissimo absoluta, nihil omnino in mundo contingit, in quo non maximi minimive ratio quaepiam eluceat[1] — Leonhard Euler 1 Introduction 2 Willmore energy and the Willmore Conjecture Established in 1961, the Oswald Veblen Prize in Geometry is an award granted by the American Mathematical Society in A central theme in Mathematics is the search for the op- recognition of a notable research memoir in geometry and timal representative within a certain class of objects, of- topology. Presented every three years, this year’s edition of ten driven by the minimization of some energy, reflecting the prize distinguished the joint work of the Brazilian math- what occurs in many physical processes. From the early ematician Fernando Codá Marques (Princeton University) 1960s, Thomas Willmore devoted particular attention to and the Portuguese mathematician André Neves (Imperial the quest for the optimal immersion of a compact surface College London), for their landmark achievement and ma- in Euclidean 3-space regarding the minimization of some jor contribution to the use of variational methods in differ- natural energy motivated by questions on the elasticity of ential geometry, with a special highlight for the proof of the membranes and the energetic cost associated with mem- long-standing Willmore Conjecture. brane bending deformations. Proposed in 1965 by the English geometer Thomas J. We can characterize how much a membrane is bent at Willmore, the Willmore Conjecture concerned the quest for a particular point on the membrane by means of the curva- the torus with the lowest bending energy of all and pre- ture of the osculating circles of the planar curves obtained dicted the equilibrium state of such curved surfaces.
    [Show full text]
  • A Survey on Hiperbolicity of Projective Hypersurfaces
    A Survey on Hiperbolicity of Projective Hypersurfaces Publicações Matemáticas A Survey on Hiperbolicity of Projective Hypersurfaces Simone Diverio Université Pierre et Marie Curie Erwan Rousseau Université de Strasbourg et CNRS impa Copyright 2011 by Simone Diverio e Erwan Rousseau Impresso no Brasil / Printed in Brazil Capa: Noni Geiger / Sérgio R. Vaz Publicações Matemáticas Introdução à Topologia Diferencial – Elon Lages Lima Criptografia, Números Primos e Algoritmos – Manoel Lemos Introdução à Economia Dinâmica e Mercados Incompletos – Aloísio Araújo Conjuntos de Cantor, Dinâmica e Aritmética – Carlos Gustavo Moreira Geometria Hiperbólica – João Lucas Marques Barbosa Introdução à Economia Matemática – Aloísio Araújo Superfícies Mínimas – Manfredo Perdigão do Carmo The Index Formula for Dirac Operators: an Introduction – Levi Lopes de Lima Introduction to Symplectic and Hamiltonian Geometry – Ana Cannas da Silva Primos de Mersenne (e outros primos muito grandes) – Carlos Gustavo T. A. Moreira e Nicolau Saldanha The Contact Process on Graphs – Márcia Salzano Canonical Metrics on Compact almost Complex Manifolds – Santiago R. Simanca Introduction to Toric Varieties – Jean-Paul Brasselet Birational Geometry of Foliations – Marco Brunella Introdução à Teoria das Probabilidades – Pedro J. Fernandez Teoria dos Corpos – Otto Endler Introdução à Dinâmica de Aplicações do Tipo Twist – Clodoaldo G. Ragazzo, Mário J. Dias Carneiro e Salvador Addas Zanata Elementos de Estatística Computacional usando Plataformas de Software
    [Show full text]
  • Conformally Flat Metrics, Constant Mean Curvature Surfaces in Product
    CONFORMALLY FLAT METRICS, CONSTANT MEAN CURVATURE SURFACES IN PRODUCT SPACES AND R-STABILITY OF HYPERSURFACES Tese Apresentada `aBanca Examinadora do Instituto Nacional de Matem´aticaPura e Aplicada como Requerimento Parcial para Obten¸c˜aodo Grau de Doutor em Ciˆencias por Marcos Petr´uciode Almeida Cavalcante 6 Janeiro de 2006 . O conhecimento incha; o Amor ´eque constr´oi. (I Cor 8, 1b) ii . Tutus tuus ego sum, o Maria, et omnia mea tua sunt... Aos meus pais Francisco Petr´ucio e Maria Das Neves e aos meus irm˜aosFrancisco Jr., Amanda e Marcius pelo Amor incondicional Ao Professor Edmilson Pontes (in memorian) por mostrar-me a beleza da matem´atica iii Agradecimentos Ao Prof. Manfredo do Carmo, pela orienta¸c˜ao do meu mestrado, do meu doutorado, culminando na orienta¸c˜ao desta tese. Agrade¸cotamb´empor todas as agrad´aveis conversas e pelo constante incentivo nestes anos todos. Ao Prof. Fernando Cod´aMarques, pela co-orienta¸c˜ao, por ter-me proposto o problema que resultou no primeiro cap´ıtulo desta tese, pelas excelentes aulas do curso de T´opicos em Geometria Riemanniana, pelas esclarecedoras conversas sobre matem´atica e por ter sido um verdadeiro amigo durante todos estes anos. Ao Prof. Hil´ario Alencar, por sempre ter-me apoiado e incentivado, desde os tempos em que fui seu aluno na Ufal. Aos Professores Jo˜ao Lucas Barbosa, Marcos Dajczer e Henrique Bursztyn, mem- bros da banca, pelos comet´arios e sugest˜oespara a esta tese. Aos Professores L´ucioRodrigues e Luis Florit do grupo de geometria diferencial do Impa.
    [Show full text]