Alguns resultados tipo-Bernstein em variedades semi-riemannianas

• Ano de defesa: 2011
• Autor(a) principal: Parente, Ulisses Lima
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: por
• Palavras-chave em Português: Hipersuperfícies; Curvatura; Geometria diferencial
• Link de acesso: http://www.repositorio.ufc.br/handle/riufc/846

In this thesis, we study complete space-like hypersurfaces immersed in semi-Riemannian manifolds, satisfying some conditions on their higher-order mean curvatures in order to get Bernstein-type results. Analytical tools we use are some versions of the maximum principle. When the ambient space is a generalized Robertson-Walker spacetime which is supposed to obey the strong null convergence condition, we establish new characterizations of totally geodesic spacelike hypersurfaces. Furthermore, we obtain a lower estimate the minimum index of relative nullity when the space-like hypersurface is r-maximal, or when there are two consecutive higher-order mean curvatures that do not change sin. We also obtain rigidity results and new characterizations of totally umbilical hypersurfaces, assuming they have some constant higher-order mean curvature, and that the ambient space is a spacetime Robertson-Walker obeying the null convergence condition. These results are applied to the de Sitter and anti-de Sitter spaces. Finally, we prove a Bernstein-type theorem for constant mean curvature complete hypersurfaces immersed in a riemannian product.