A generalization of Hawking's black hole topology theorem to higher dimensions

TL;DR

This paper generalizes Hawking's black hole topology theorem to higher dimensions, showing that horizon cross sections are of positive Yamabe type, which constrains their possible topologies.

Contribution

It extends Hawking's theorem to higher dimensions by proving that black hole horizons admit metrics of positive scalar curvature, broadening understanding of their topological restrictions.

Findings

Horizon cross sections in higher dimensions are of positive Yamabe type.

The results are consistent with known higher-dimensional black hole topologies.

The proof builds on Schoen and Yau's work without directly using the Jang equation.

Abstract

Hawking's theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. This conclusion extends to outer apparent horizons in spacetimes that are not necessarily stationary. In this paper we obtain a natural generalization of Hawking's results to higher dimensions by showing that cross sections of the event horizon (in the stationary case) and outer apparent horizons (in the general case) are of positive Yamabe type, i.e., admit metrics of positive scalar curvature. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology $S^2 \times S^1$. The proof is inspired by previous work of Schoen and Yau on the existence of solutions to the Jang equation (but does not make direct use of that equation).