On the geometry of Riemannian hypersurfaces: uniqueness, nonexistence, stability and bifurcation.

Abstract

In this work we study some problems related to the geometry of Riemannian hypersurfaces immersed in semi-Riemannian manifolds (with index zero or one) equipped with a density function and that can be modeled by a certain class of warped products. Initially, assuming reasonable conditions in the weighted mean curvature of such hypersurfaces and considering certain restrictions in the ambient space, we establish some results of uniqueness and non-existence. We also establish results of stability, bifurcation and local rigidity associated with variational problems involving the functional 1-area and the functional weighted area of such a hypersurface.