Parabolicity criteria and characterization results for submanifolds of bounded mean curvature in model manifolds with weights
Let be a submanifold properly immersed in a rotationally symmetric manifold having a pole and endowed with a weight . The aim of this paper is twofold. First, by assuming certain control on the -mean curvature of , we establish comparisons for the -capacity of extrinsic balls in , from which we deduce criteria ensuring the -parabolicity or -hyperbolicity of . Second, we employ functions with geometric meaning to describe submanifolds of bounded -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 -minimal submanifolds appearing in the singularity theory of the mean curvature flow.