Rigidity, uniqueness and nonexistence results in certain warped product spaces.

Abstract

This thesis presents the study of hypersurfaces immersed in Lorentzian and warped Riemannian products ambient. In the first part, we analyze hypersurfaces that satisfy conditions on the mean curvature, obtaining rigidity and non-existence results for solitons of the mean curvature flow in GRW spacetimes and standard static spaces. We demonstrate applications of these results in ambient such as Einstein-de Sitter Spacetime, Steady State Type Spacetimes, LorentzMinkowski space, and more. We obtain Calabi-Bernstein type results and highlight stability results of hypersurfaces. In the second part, we study two-sided hypersurfaces immersed in warped Riemannian products, establishing results on existence, rigidity, and non-existence of solitons of the mean curvature flow, subject to conditions on the mean curvature and warping function of the ambient. We demonstrate applications of these results in ambient such as Real projective space, pseudo-hyperbolic spaces, Schwarzchild space, and Reissner-Nordstr¨om space. We also dedicate part of the study to submanifolds immersed in weighted ambient.