Extrinsic black hole uniqueness in pure Lovelock gravity

TL;DR

This paper introduces a new notion of extrinsic black holes in pure Lovelock gravity, proving a global uniqueness theorem and a local rigidity result for Lovelock-Schwarzschild solutions under certain conditions.

Contribution

It defines extrinsic black holes in pure Lovelock gravity and proves a global uniqueness theorem and a local rigidity result for these solutions.

Findings

Proved a global uniqueness theorem for extrinsic black holes in pure Lovelock gravity.

Established a local rigidity result for Lovelock-Schwarzschild solutions.

Connected the results to the Penrose inequality for graphs.

Abstract

We define a notion of extrinsic black hole in pure Lovelock gravity of degree $k$ which captures the essential features of the so-called Lovelock-Schwarzschild solutions, viewed as rotationally invariant hypersurfaces with null $2k$-mean curvature in Euclidean space $\mathbb R^{n+1}$, $2\leq 2k\leq n-1$. We then combine a regularity argument with a rigidity result by Ara\'ujo-Leite to prove, under a natural ellipticity condition, a global uniqueness theorem for this class of black holes. As a consequence we obtain, in the context of the corresponding Penrose inequality for graphs established by Ge-Wang-Wu, a local rigidity result for the Lovelock-Schwarzschild solutions.