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Let $P$ be a submanifold properly immersed in a rotationally symmetricmanifold having a pole and endowed with a weight $e^h$. The aim of this paperis 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 todescribe submanifolds of bounded $h$-mean curvature which are confined intosome regions of the ambient manifold. As a consequence, we derive half-spaceand Bernstein-type theorems generalizing previous ones. Our results apply forsome relevant $h$-minimal submanifolds appearing in the singularity theory ofthe mean curvature flow.

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