Sobre a geometria de imersÃes isomÃtricas em variedades de Lorentz conformemente estacionÃrias / On the geometry of varieties of isometric immersions in Lorents stationary conformally
AUTOR(ES)
Marco Antonio LÃzaro VelÃsquez
FONTE
IBICT - Instituto Brasileiro de Informação em Ciência e Tecnologia
DATA DE PUBLICAÇÃO
03/12/2010
RESUMO
In this thesis we study several aspects of the geometry of conformally stationary Lorentz manifolds and, more particularly, of generalized Robertson-Walker spaces, under the presence of a closed conformal vector field. We initiate by focusing our study on the r-stability and on the strong r-stability of closed spacelike hypersurfaces of conformally stationary ambient spaces of constant sectional curvature; more precisely, we obtain a characterization of the r-stable ones by means of the first eigenvalue of a suitable elliptic operator naturally associated to its r-th mean curvature, as well classify the strongly r-stable ones by means of an appropriate condition on the conformal factor of the conformal vector field on the ambient space. Following,we establish general Bernstein-type theorems for spacelike hypersurfaces of conformally stationary Lorentz manifolds, one of which does not require the hypersurface to be of constant mean curvature. We end by extending, to conformally stationary Lorentz manifolds, a result of J. Simons on the minimality of certain cones in Euclidean spaces, and apply this result to build complete, non-compact minimal submanifolds in the de Sitter space and in the anti-de Sitter space.
ASSUNTO(S)
geometria diferencial campos conformes fechados r-estabilidade de hipersuperfÃcies resultados tipo-bernstein conformal field closed r-stability of hypersurfaces bernstein-type results hipersuperfÃcies geometria de subvariedades hypersurfaces
