Calculus of Variations and Geometric Measure Theory
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S. Borghini - L. Mazzieri

Monotonicity formulas for static metrics with non-zero cosmological constant

created by borghini on 07 Apr 2018

[BibTeX]

preprint

Inserted: 7 apr 2018

Year: 2016

ArXiv: 1605.04578 PDF

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.

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