On the geometry of the level sets of bounded static potentials

2015/04/17 by Virginia Agostiniani, Agostiniani, Virginia, Lorenzo Mazzieri +1 · 1 citation
Mathematics · #35B06 #35N25 #53C21 #83C57 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #math.AP #math.DG #msc:35B06 #msc:35N25 #msc:53C21 #msc:83C57

paper · pdf · doi:10.48550/arxiv.1504.04563

arxiv created 2015/04/17 · arxiv updated 2015/04/20

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.

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