Bakry-\'Emery Ricci Curvature Bounds on Manifolds with Boundary

TL;DR

This paper extends classical geometric theorems to manifolds with boundary under Bakry-Émery curvature bounds, including intersection and splitting results, without requiring the vector field to be a gradient.

Contribution

It generalizes key theorems to Bakry-Émery curvature bounds on manifolds with boundary, broadening their applicability in geometric analysis.

Findings

Proved a Bakry-Émery version of Petersen and Wilhelm's theorem.

Established splitting theorems for manifolds with boundary under curvature bounds.

Extended results without assuming the vector field is of gradient type.

Abstract

We prove a Bakry-\'Emery generalization of a theorem of Petersen and Wilhelm, itself a generalization of a theorem of Frankel, that closed minimal hypersurfaces in a complete manifold with a suitable curvature bound must intersect. We then prove splitting theorems of Croke-Kleiner type for manifolds bounded by hypersurfaces obeying Bakry-\'Emery curvature bounds. Motivated in part by the near-horizon geometry programme of general relativity, we do not assume that the Bakry-\'Emery vector field is of gradient type.