A Bakry-\'Emery Almost Splitting Result With Applications to the Topology of Black Holes

TL;DR

This paper extends the almost splitting theorem to manifolds with generalized Bakry-Émery Ricci curvature, deriving topological constraints and applying these results to the topology of higher-dimensional black hole horizons.

Contribution

It generalizes the almost splitting theorem to non-gradient Bakry-Émery Ricci curvature and applies it to analyze the topology of black hole horizons in higher dimensions.

Findings

Fundamental group is almost abelian under certain bounds.

Topological restrictions on black hole horizons at low temperature.

Extended classical results to generalized Bakry-Émery Ricci curvature.

Abstract

The almost splitting theorem of Cheeger-Colding is established in the setting of almost nonnegative generalized $m$-Bakry-\'{E}mery Ricci curvature, in which $m$ is positive and the associated vector field is not necessarily required to be the gradient of a function. In this context it is shown that with a diameter upper bound and volume lower bound the fundamental group of such manifolds is almost abelian. Furthermore, extensions of well-known results concerning Ricci curvature lower bounds are given for generalized $m$-Bakry-\'{E}mery Ricci curvature. These include: the first Betti number bound of Gromov and Gallot, Anderson's finiteness of fundamental group isomorphism types, volume comparison, the Abresch-Gromoll inequality, and a Cheng-Yau gradient estimate. Finally, this analysis is applied to stationary vacuum black holes in higher dimensions to find that low temperature horizons must have limited topology, similar to the restrictions exhibited by (extreme) horizons of zero temperature.