Mean exit times from submanifolds with bounded mean curvature

TL;DR

This paper investigates conditions under which submanifolds with bounded mean curvature cannot be minimally immersed into certain product space regions, providing new non-immersibility results and moment estimates.

Contribution

It introduces a novel approach that does not rely on the weak maximum principle at infinity to establish non-immersibility of certain submanifolds.

Findings

Submanifolds with infinite mean exit time cannot be minimally immersed into cylinders, horocylinders, cones, and wedges.

Provides estimates for the moments of submanifolds with small mean curvature in cylinders.

Generalizes previous non-immersibility results using a new approach.

Abstract

We show that submanifolds with infinite mean exit time can not be isometrically and minimally immersed into cylinders, horocylinders, cones, and wedges of some product spaces. Our approach is not based on the weak maximum principle at infinity, and thus it permits us to generalize previous results concerning non-immersibility of stochastically complete submanifolds. We also produce estimates for the complete tower of moments for submanifolds with small mean curvature immersed into cylinders.