Augsburg, Univ., Diss., 2007.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Baseados nos trabalhos De Gerhardt e Urbas [12], [36], provamos um resultado de convergÃncia global e determinamos precisamente o comportamento assintÃtico de soluÃÃes de um fluxo geomÃtrico que descreve a evoluÃÃo de hipersuperfÃcies estreladas e k-convexas por funÃÃes das curvaturas principais. Como aplicaÃÃo, e seguindo o argumento de Guan e Li [16], utilizamos um caso particular deste resultado de convergÃncia para generalizar a clÃssica desigualdade de Alexandrov-Fenchel para domÃnios estrelados e k-convexos. / Based on the work of Gerhardt and Urbasa [12], [36], we prove a global convergence result and precisely determine the asymptotic behavior of solutions of a geometric flow describing the evolution of starshaped, k-convex hypersurfaces according to certain functions of the principal curvatures. As an application, and following the argument of Guan and Li [16], we use a special case of this convergence result to generalize the classical Alexandrov-Fenchel inequality for domains starry and k-convex.

resumo não disponível.

resumo não disponível.

resumo não disponível.

Un polyèdre fuchsien de l'espace hyperbolique est une surface polyédrale invariante sous l'action d'un groupe fuchsien d'isométries (c.a.d. un groupe d'isométries qui laissent globalement invariante une surface totalement géodésique et sur laquelle il agit de manière cocompacte). La métrique induite sur un polyèdre fuchsien convexe est isométrique à une métrique hyperbolique avec des singularités coniques de courbure singulière positive sur une surface compacte de genre $>1$. On démontre que ces métriques sont en fait réalisées par un unique polyèdre fuchsien convexe (modulo les isométries globales). Ce résultat étend un théorème célèbre de A.D. Alexandrov. <br />On montre aussi que chaque métrique à courbure constante avec des courbures singulières négatives sur une surface compacte de genre $>1$ peut-être réalisée par un unique polyèdre ``fuchsien'' convexe dans un espace modèle lorentzien.<br />Finalement on présente des extensions possibles de ces résultats, ce qui amène à des énoncés généraux sur la réalisation de métriques sur les surfaces.

Submitted by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2018-02-20T14:22:00Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Dissertação_Cícero K. M. Pereira.pdf: 13494805 bytes, checksum: dec78fc54d9f514a09ab374d40872d52 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2018-02-20T14:22:12Z (GMT) No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Dissertação_Cícero K. M. Pereira.pdf: 13494805 bytes, checksum: dec78fc54d9f514a09ab374d40872d52 (MD5) / Made available in DSpace on 2018-02-20T14:22:12Z (GMT). No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Dissertação_Cícero K. M. Pereira.pdf: 13494805 bytes, checksum: dec78fc54d9f514a09ab374d40872d52 (MD5) Previous issue date: 2017-10-20 / FAPEAM - Fundação de Amparo à Pesquisa do Estado do Amazonas / We will present some height estimates for compact surfaces with positive constant extrinsic curvature (𝐾−surfaces) in ℳ2 × R, where ℳ2 is a surface with constant Gauss curvature. We will initially show a vertical height estimate for compact 𝐾−graphs in ℳ2 × R, with boundary in a slice and later horizontal height estimate for compact, embedded 𝐾−surfaces in H2 × R with boundary on a vertical plane. Such results have been proven by josé Espinar, José Galvez and Harold Rosenberg in the article entitled "Complete surfaces with positive extrinsic curvature in product spaces". The tools used to demonstrate these estimates are based on the Hopf Maximum Principle and the Alexandrov Reflection Method. / Neste trabalho apresentamos algumas estimativas de altura para superfícies compactas com curvatura extrínseca constante positiva (𝐾−superfícies) em ℳ2 × R, em que ℳ2 denota uma superfície com curvatura de Gauss constante. Mostraremos inicialmente uma estimativa de altura vertical para 𝐾−gráficos compactos em ℳ2 × R, com bordo em um plano horizontal e posteriormente uma estimativa de altura horizontal para 𝐾−superfícies compactas mergulhadas em H2 × R com bordo em um plano vertical. Tais resultados foram provados por José Espinar, José Galvez e Harold Rosenberg no artigo intitulado "Complete surfaces with positive extrinsic curvature in product spaces". As ferramentas utilizadas para demonstrar estas estimativas se baseiam no princípio do máximo de Hopf e no Método de Reflexão de Alexandrov.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22974号 / 理博第4651号 / 新制||理||1668(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 山口 孝男, 教授 藤原 耕二, 教授 入谷 寛 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

The goal of this dissertation is to present a R. Reilly's demonstration of the theorem of Alexandrov . The theorem states that The only compact hypersurfaces, conected, of constant mean curvature, immersed in Euclidean space are spheres. The theorem of Alexandrov was proved by A. D. Alexandrov in the article Uniqueness Theorems for Surfaces in the Large V, published in 1958 by Vestnik Leningrad University, volume 13, number 19, pages 5 to 8. In his demonstration, Alexandrov used the famous Principle of tangency, introduced by him in that article. In the year 1962, M. Obata shown in Certain Conditions for a Riemannian Manifold to be isometric With the Sphere, published by the Journal of Mathematical Society of Japan, volume 14, pages 333 to 340, that a Riemannian Manifold M, compact, connected and without boundary, is isometric to a sphere, since the Ricci curvature of M satisfies certain lower bound. This theorem solves the problem of finding manifolds that reach equality in the estimate of Lichnerowicz for the first eigenvalue. In 1977, R. Reilly, in the article Applications of the Hessian operator in a Riemannian Manifold, published in Indianna University Mathematical Journal, volume 23, pages 459 to 452, showed a generalization of the Obata theorem for compact manifolds with boundary. As an example of the technique developed in this demonstration, he presents a new demonstration of the theorem of Alexandrov. This demonstration, as well as the techniques involved are the object of study of this work. / Conselho Nacional de Desenvolvimento Científico e Tecnológico / O objetivo desta dissertação é apresentar uma demonstração de R. Reilly para o Teorema de Alexandrov. O teorema estabelece que As únicas hipersuperfícies compactas, conexas, de curvatura média constante, mergulhadas no espaço Euclidiano são as esferas. O teorema de Alexandrov foi provado por A. D. Alexandrov no artigo Uniqueness Theorems for Surfaces in the Large V, publicado em 1958 pela Vestnik Leningrad University, volume 13, número 19, páginas 5 a 8. Em sua demonstração, Alexandrov usou o famoso Princípio de Tangência, introduzido por ele no citado artigo. No ano de 1962, M. Obata demonstrou em Certain Conditions for a Riemannian Manifold to be Isometric With a Sphere, publicado pelo Journal of Mathematical Society of Japan, volume 14, páginas 333 a 340, que uma variedade Riemanniana M, compacta, conexa e sem bordo, é isométrica a uma esfera, desde que a curvatura de Ricci de M satisfaça determinada limitação inferior. Este teorema resolve o problema de encontrar as variedades que atingem a igualdade na estimativa de Lichnerowicz para o primeiro autovalor. Em 1977, R. Reilly, no artigo Applications of the Hessian Operator in a Riemannian Manifold, publicado no Indianna University Mathematical Journal, volume 23, páginas 459 a 452, demonstrou uma generalização do Teorema de Obata para variedades compactas com bordo. Como exemplo da técnica desenvolvida nesta demonstração, ele apresenta uma nova demonstração do Teorema de Alexandrov. Esta demonstração, bem como as técnicas envolvidas, são o objeto de estudo deste trabalho.

This thesis is broadly divided in two parts. In the first part we give a survey of various distances between metric spaces, namely the uniform distance, Lipschitz distance, Hausdor distance and the Gramoz Hausdor distance. Here we talk about only the most basic of their properties and give a few illustrative examples. As we wish to study collections of metric measure spaces, which are triples (X; d; m) consisting of a complete separable metric space (X; d) and a Boral probability measure m on X, there are discussions about some distances between them. Among the three that we discuss, the transportation and distortion distances were introduced by Sturm. The later, denoted by 2, on the space X2 of all metric measure spaces having finite L2-size is the focus of the second part of this thesis. The second part is an exposition based on the work done by Sturm. Here we prove a number of results on the analytic and geometric properties of (X2; 2). Beginning by noting that (X2; 2) is a non-complete space, we try to understand its completion. Towards this end, the notion of a gauged measure space is useful. These are triples (X; f; m) where X is a Polish space, m a Boral probability measure on X and f a function, also called a gauge, on X X that is symmetric and square integral with respect to the product measure m2. We show that, Theorem 1. The completion of (X2; 2) consists of all gauged measure spaces where the gauges satisfy triangle inequality almost everywhere. We denote the space of all gauged measure spaces by Y. The space X2 can be embedded in Y and the transportation distance 2 extends easily from X2 to Y. These two spaces turn out to have similar geometric properties. On both these spaces 2 is a strictly intrinsic metric; i.e. any two members in them can be joined by a shortest path. But more importantly, using a description of the geodesics in these spaces, the following result is proved. Theorem 2. Both (X2; 2) and (Y; 2) have non-negative curvature in the sense of Alexandrov.