Parabolicity criteria and characterization results for submanifolds of bounded mean curvature in model manifolds with weights. (March 2020) - Hurtado, A.; Palmer, V.; Rosales, C.
Abstract: Let P be a submanifold properly immersed in a rotationally symmetric manifold having a pole and endowed with a weight e h . The aim of this paper is twofold. First, by assuming certain control on the h -mean curvature of P, we establish comparisons for the h -capacity of extrinsic balls in P, from which we deduce criteria ensuring the h -parabolicity or h -hyperbolicity of P . Second, we employ functions with geometric meaning to describe submanifolds of bounded h -mean curvature which are confined into some regions of the ambient manifold. As a consequence, we derive half-space and Bernstein-type theorems generalizing previous ones. Our results apply for some relevant h -minimal submanifolds appearing in the singularity theory of the mean curvature flow.