𝑝-parabolic submanifolds in certain spacetimes: rigidity, uniqueness and non-existence results.

Abstract

In this work we present rigidity and uniqueness results for parabolic and stable constant mean curvature hypersurfaces immersed in Generalized Robertson-Walker and Satandard Static spacetimes. We obtained some conditions under which a hypersurface in these ambiences must be parabolic, as well as stable. In order to achieve uniqueness results, we used some cut-off functions coming from the parabolicity jointly with the stability operator. Also, we introduced the concept of totally trapped submanifold and obtained some uniqueness and non-existence results when the submanifold is 𝑝-parabolic. We also presented a lemma of Nishikawa in order to obtain Calabi-Berstein type results for surfaces in Robertson-Walker Generalized spacetimes.