Mean Curvature of Hypersurfaces in Killing Submersions with Bounded Shadow

TL;DR

This paper establishes a lower bound for the mean curvature of hypersurfaces immersed in manifolds with Killing submersions, under conditions of bounded projection and stochastic completeness or properness.

Contribution

It provides a new lower bound for the mean curvature of hypersurfaces in Killing submersions with bounded shadow, under specific geometric and completeness assumptions.

Findings

Lower bound for mean curvature vector norm established

Conditions on ambient manifold and hypersurface properties specified

Results applicable to stochastically complete or proper immersions

Abstract

Given a complete hypersurface isometrically immersed in an ambient manifold, in this paper we provide a lower bound for the norm of the mean curvature vector field of the immersion assuming that: 1) The ambient manifold admits a Killing submersion with unit-length Killing vector field. 2)The projection of the image of the immersion is bounded in the base manifold. 3)The hypersurface is stochastically complete, or the immersion is proper.