Apresentaremos alguns resultados de rigidez e de bifurcação para soluções do problema de Yamabe em variedades produto com bordo. / We will discuss some rigidity and bifurcation results for solutions of the Yamabe problem in product manifolds with boundary.

Made available in DSpace on 2015-04-22T22:16:05Z (GMT). No. of bitstreams: 1 Marcos Aurelio de Alcantara.pdf: 720745 bytes, checksum: 08ea1f8125881d86a363610223d345ea (MD5) Previous issue date: 2013-04-03 / CNPq - Conselho Nacional de Desenvolvimento Científico e Tecnológico / This paper presents main result of a theorem in rigidity of convex three dimensional homogeneous spaces, which was proved by Hosenberg and Tribuzy in 2011. More precisely, we prove that given smooth family of isometric immersions strictly convex f(t) : M 􀀀! N, with f(0) = f, Ke(ft(x)) = Ke(f(x)) for x 2 M and for all t, and H(ft(x)) = H(f(x)) in three distinct points x of M. Then there are isometries h(t) : N 􀀀! N such that h(t)f(t) = f. / Este trabalho apresenta como principal resultado um teorema de rigidez de superfícies convexas em espaços homogêneos tridimensionais, que foi provado por Hosenberg e Tribuzy em 2011. Mais precisamente, provaremos que dada uma família suave de imersões isométricas estritamente convexa f(t) : M 􀀀! N, com f(0) = f, Ke(ft(x)) = Ke(f(x)) para x 2 M e todo t, e H(ft(x)) = H(f(x)) em três pontos distintos x de M. Então existem isometrias h(t) : N 􀀀! N tal que h(t)f(t) = f.

Submitted by Alison Vanceto (alison-vanceto@hotmail.com) on 2016-10-05T11:50:42Z No. of bitstreams: 1 TeseSCC.pdf: 2428184 bytes, checksum: e1ac9bcc617f51c6e101914c2bf485ad (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2016-10-05T18:25:42Z (GMT) No. of bitstreams: 1 TeseSCC.pdf: 2428184 bytes, checksum: e1ac9bcc617f51c6e101914c2bf485ad (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2016-10-05T18:25:51Z (GMT) No. of bitstreams: 1 TeseSCC.pdf: 2428184 bytes, checksum: e1ac9bcc617f51c6e101914c2bf485ad (MD5) / Made available in DSpace on 2016-10-05T18:37:11Z (GMT). No. of bitstreams: 1 TeseSCC.pdf: 2428184 bytes, checksum: e1ac9bcc617f51c6e101914c2bf485ad (MD5) Previous issue date: 2015-06-08 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / In this thesis we classify the isometric immersions f : Mm ^ Sm+p x R with m > 3 P < m — ^d c < 1, where Mm denotes a Riemannian manifold with constant sectional curvature equal to c. We obtain partial results on the classification of isometric immersions f : Mm ^ Hm+p x R with m > 3 P < m — ^d c < 0, We also characterize the hvpersurfaces f : M3 ^ Q4(c) for which there exists another isometric immersion f : M3 ^ L4, where Q4(c^d L4 denote a 4-dimensional space form of constant sectional curvature c and the 4-dimensional Lorentz space, respectively. / Nesta tese classificamos as imersões isométrieas f : Mm ^ gm+p x r com m > 3 P < m — 3 e c < 1, em que Mm denota uma variedade Riemanniana com curvatura seccional constante igual a c. Obtemos resultados parciais sobre a classificação das imersões isométrieas f : Mm ^ Hm+P x R com m > 3 P < m — 3 e c< 0, Caracterizamos ainda as hipersuperfíeies f : M3 ^ Q4(c) para as quais existe outra imersão isométrica f : M3 ^ L4, em que Q4(c) e L4 denotam, respectivamente, uma forma espacial Riemanniana com curvatura constante igual a c e o espaço de Lorentz de dimensão 4.

Un problema clásico en geometría lorentziana es la descripción de las inmersiones isométricas entre los espacios lorentzianos de curvatura constante. En este trabajo nos centramos en la clasificación de las inmersiones isométricas del plano lorentziano en el espacio anti-de Sitter tridimensional. Damos una fórmula de representación de estas inmersiones en términos de pares de curvas (con posibles singularidades) en el plano hiperbólico. Esto nos permite resolver los problemas propuestos por Dajczer y Nomizu en 1981. De entre todas las inmersiones isométricas del plano lorentziano en el espacio anti-de Sitter, algunas de ellas corresponden a toros lorentzianos (los ejemplos más sencillos son los toros de Hopf). Como aplicación de nuestra anterior descripción, probamos que todos estos toros pueden obtenerse a partir de dos curvas cerradas en el espacio hiperbólico. Finalmente, demostramos que los toros de Hopf son los únicos toros llanos lorentzianos inmersos en una amplia familia de sumersiones de Killing lorentzianas tridimensionales. / A classical problem in Lorentzian geometry is the description of the isometric immersions between Lorentzian spaces of constant curvature. We investigate the problem of classifying the isometric immersion from the Lorentz plane into the three-dimensional anti-de Sitter space, providing a representation formula of these isometric immersions in terms of pairs of curves (possibly with singularities) in the hyperbolic plane. We then give an answer to the open problems proposed by Dajczer and Nomizu in 1981. Among all isometric immersions of the Lorentz plane into the anti-de Sitter space, some of them are actually Lorentzian tori (the basic examples are the Hopf tori). As an application of our previous description, we prove that any such torus can be recovered from two closed curves in the hyperbolic plane. Finally, we prove that Lorentzian Hopf tori are the only immersed Lorentzian flat tori in a wide family of Lorentzian three-dimensional Killing submersions.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-08-09T17:38:25Z No. of bitstreams: 1 JAMILLY LOURÊDO ROCHA - DISSERTAÇÃO PPGMAT 2014..pdf: 5707925 bytes, checksum: 8010cd451ac64c8a7fccc36a2f8313f6 (MD5) / Made available in DSpace on 2018-08-09T17:38:25Z (GMT). No. of bitstreams: 1 JAMILLY LOURÊDO ROCHA - DISSERTAÇÃO PPGMAT 2014..pdf: 5707925 bytes, checksum: 8010cd451ac64c8a7fccc36a2f8313f6 (MD5) Previous issue date: 2014-08 / Capes / Com uma aplicação adequada do conhecido princípio do máximo generalizado de Omori-Yau, obtemos resultados de rigidez com relação a hipersuperfícies imersas completascomcurvaturamédiadelimitadanoespaçohiperbólicoHn+1 (n+1)-dimensional. Em nossa abordagem exploramos a existência de uma dualidade natural entreHn+1 e a metade Hn+1 do espaço de SitterSn+11 , cujo modelo é chamado de steady state space. / As a suitable application of the well known generalized maximum principle of Omori-Yau, we obtain rigidity results concerning to a complete hypersurface immersed with bounded mean curvature in the (n+1)-dimensional hyperbolic spaceHn+1. In our approach, we explore the existence of a natural duality betweenHn+1 and the half Hn+1 of the de Sitter spaceSn+11 , which models the so-called steady state space.