Comparison principle, stochastic completeness and half-space theorems

TL;DR

This paper establishes a criterion for stochastic completeness of submanifolds using Hessian comparison, and applies it to derive half-space theorems in various Riemannian settings.

Contribution

It introduces a new criterion for stochastic completeness based on distance to hypersurfaces and applies comparison principles to derive geometric half-space theorems.

Findings

Criterion for stochastic completeness via Hessian comparison.

Mean curvature estimates for submanifolds in product and wedge spaces.

Horizontal and vertical half-space theorems in hyperbolic and Euclidean products.

Abstract

We present a criterion for the stochastic completeness of a submanifold in terms of its distance to a hypersurface in the ambient space. This relies in a suitable version of the Hessian comparison theorem. In the sequel we apply a comparison principle with geometric barriers for establishing mean curvature estimates for stochastically complete submanifolds in Riemannian products, Riemannian submersions and wedges. These estimates are applied for obtaining both horizontal and vertical half-space theorems for submanifolds in $\mathbb{H}^n \times \mathbb{R}^\ell$.