Sobre subvariedades com segunda forma fundamental dominada em espaços de Hadamard.

Abstract

Our work has as object of study submanifolds in Hadamard spaces and in N×R product spaces where N is a complete Riemannian manifold with pole and upper bounded radial sectional curvatures. In Hadamard spaces we show that immersions with second norm dominated fundamental form are proper and have finite topology. The submanifolds of Euclidean space with this condition on the second fundamental form have null fundamental tone. On the N × R product space, we initially study immersed hypersurfaces contained in a vertical cylinder. We observe that, after a certain limitation in the mean curvature vector, these hypersurfaces are non-parabolic. Another result obtained is the non-existence of complete hypersurfaces immersed in Hn × R with Ricci curvature with super quadratic decay.