Monotonicity Formulas for Static Metrics with Non-zero Cosmological Constant

  • Stefano Borghini
  • Lorenzo MazzieriEmail author
Part of the Springer INdAM Series book series (SINDAMS, volume 33)


In this paper we adopt the approach presented in Agostiniani and Mazzieri (J Math Pures Appl 104:561–586, 2015; Commun Math Phys 355:261–301, 2017) to 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.


Static metrics Splitting theorem (Anti)-de Sitter solution Overdetermined boundary value problems 

MSC (2010)

35B06 53C21 83C57 35N25 



The authors would like to thank P. T. Chruściel for his interest in our work and for stimulating discussions during the preparation of the manuscript. The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and are partially founded by the GNAMPA Project “Principi di fattorizzazione, formule di monotonia e disuguaglianze geometriche”.


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Authors and Affiliations

  1. 1.Università degli Studi di TrentoPovoItaly

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