Parabolicity criteria and characterization results for submanifolds of bounded mean curvature in model manifolds with weights

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.