On the geometry of the level sets of bounded static potentials

TL;DR

This paper introduces a new approach to analyze the geometry of level sets of bounded static potentials in general relativity, leading to monotonic quantities, symmetry detection, and simplified proofs of black hole uniqueness.

Contribution

It develops a novel method involving monotone quantities along level sets to identify symmetry and prove black hole uniqueness in static solutions.

Findings

Monotone quantities are established along level set flows.

The approach simplifies the proof of the 3D Black Hole Uniqueness Theorem.

Conditions for higher-dimensional black hole uniqueness are discussed.

Abstract

In this paper we present a new approach to the study of asymptotically flat static metrics arising in general relativity. In the case where the static potential is bounded, we introduce new quantities which are proven to be monotone along the level set flow of the potential function. We then show how to use these properties to detect the rotational symmetry of the static solutions, deriving a number of sharp inequalities. As a consequence of our analysis, a simple proof of the classical $3$-dimensional Black Hole Uniqueness Theorem is recovered and some geometric conditions are discussed under which the same statement holds in higher dimensions.