Subvariedades aprisionadas em modelos do espaço-tempo

Abstract

In this work, we present some geometric applications of a Liouville-type result for the infinity Laplacian operator. More specifically, we prove rigidity and nonexistence results for complete spacelike submanifolds immersed in two particular types of Lorentzian warped products: the generalized Robertson-Walker spacetimes and the standard static spacetimes. Furthermore, as a way to motivate and complement our study, we also present, for the case of generalized Robertson-Walker spacetimes, analogous results for parabolic spacelike submanifolds, through a Liouville-type property satisfied by the Laplacian operator in such submanifolds. We also prove the validity of an Omori-Yau type principle for infinity Laplacian. Among the submanifolds for which the results obtained are valid are the well-known weakly trapped submanifolds and the totally trapped submanifolds, which emerged in the literature from works related to gravitational collapse and the existence of singularities in spacetimes of General Relativity.