On the characterization of parabolicity and hyperbolicity of submanifolds

TL;DR

This paper establishes conditions for determining when submanifolds with controlled mean curvature are parabolic or hyperbolic within Riemannian manifolds that have a pole and bounded sectional curvatures.

Contribution

It provides a comprehensive set of necessary and sufficient conditions for parabolicity and hyperbolicity of submanifolds under curvature constraints.

Findings

Characterization of parabolicity and hyperbolicity based on curvature bounds.

Conditions involving mean curvature control for submanifolds.

Applicable to Riemannian manifolds with a pole and bounded sectional curvatures.

Abstract

We give a set of sufficient and necessary conditions for parabolicity and hyperbolicity of a submanifold with controlled mean curvature in a Riemannian manifold with a pole and with sectional curvatures bounded from above or from below.