Monotonicity formulas for static metrics with non-zero cosmological constant

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.

43 pages. arXiv admin note: substantial text overlap with arXiv:1504.04563

References in corpus (4)

• On the mass of static metrics with positive cosmological constant - I
• Uniqueness of the de Sitter spacetime among static vacua with positive cosmological constant
• On static three-manifolds with positive scalar curvature
• Some Sphere Theorems in Linear Potential Theory

Cited by in corpus (4)

• On the mass of static metrics with positive cosmological constant - I
• Quantum reverse hypercontractivity: its tensorization and application to strong converses
• On the mass of static metrics with positive cosmological constant -- II
• Black Hole and Equipotential Photon Surface Uniqueness in 4-dimensional Asymptotically Flat Electrostatic Electro-Vacuum Spacetimes