Monotonicity formulas for static metrics with non-zero cosmological constant

TL;DR

This paper develops new monotonicity formulas for static space-times with non-zero cosmological constant, leading to geometric inequalities and uniqueness results for de Sitter and anti-de Sitter solutions.

Contribution

It introduces novel integral quantities and monotonicity properties to characterize static solutions with positive or negative cosmological constant.

Findings

Proves monotonicity of new integral quantities along level set flow.

Derives sharp geometric and analytic inequalities for static solutions.

Establishes uniqueness of de Sitter and anti-de Sitter metrics under certain conditions.

Abstract

In this paper we study non-singular vacuum static space-times with non-zero cosmological constant. We introduce new integral quantities, and under suitable assumptions we prove their monotonicity along the level set flow of the static potential. We then show how to use these properties to derive a number of sharp geometric and analytic inequalities, whose equality case can be used to characterize the rotational symmetry of the underlying static solutions. As a consequence, we are able to prove some new uniqueness statements for the de Sitter and the anti-de Sitter metrics. In particular, we show that the de Sitter solution has the least possible surface gravity among three-dimensional static metrics with connected boundary and positive cosmological constant.