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

TL;DR

This paper establishes criteria for the parabolicity or hyperbolicity of submanifolds with bounded mean curvature in weighted, rotationally symmetric manifolds, and characterizes such submanifolds using geometric functions, with applications to mean curvature flow.

Contribution

It provides new comparison theorems for $h$-capacity and introduces geometric characterizations of bounded $h$-mean curvature submanifolds, extending classical results.

Findings

Criteria for $h$-parabolicity and $h$-hyperbolicity based on $h$-mean curvature control.

Characterization of submanifolds confined in regions using geometric functions.

Generalizations of half-space and Bernstein-type theorems.

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.