# 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.

## Key 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.