On stochastically complete submanifolds

TL;DR

This paper proves that certain complete submanifolds are stochastically complete using advanced criteria, and applies this to derive curvature estimates for specific types of submanifolds.

Contribution

It extends stochastic completeness results to properly immersed submanifolds using a deep criterion and derives geometric curvature bounds.

Findings

Properly immersed submanifolds of stochastically complete manifolds are stochastically complete.

The weak Omori-Yau maximum principle holds on these submanifolds.

Sectional curvature estimates are obtained for cylindrically bounded submanifolds.

Abstract

Using a deep criteria due to Pigola, Rigoli and Setti, we prove that a geodesically complete, properly immersed submanifold M of a stochastically complete Riemannian manifold N is stochastically complete. This implies that the weak Omori-Yau maximum principle holds on M. As geometric application, we prove sectional curvature estimates for properly immersed cilindrically bounded submanifolds.