Bakry–Émery Ricci curvature, X-minimal hypersurfaces, and near horizon geometries

Abstract

Motivated by the extreme black hole near horizon geometry equation and the Ellis–Ehlers equation of mathematical cosmology, we prove a Bakry–Émery generalization of a theorem of Frankel that closed minimal hypersurfaces in a complete manifold with a suitable curvature bound must intersect. We do not assume that the Bakry–Émery vector field is of gradient type. We also present splitting theorems of warped product type for manifolds bounded by hypersurfaces obeying Bakry–Émery curvature bounds.