# Monotonicity Formulas for Static Metrics with Non-zero Cosmological Constant

## Abstract

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.

## Keywords

Static metrics Splitting theorem (Anti)-de Sitter solution Overdetermined boundary value problems## **MSC (2010)**

35B06 53C21 83C57 35N25 ## Notes

### Acknowledgements

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”.

## References

- 1.V. Agostiniani, L. Mazzieri, Riemannian aspects of potential theory. J. Math. Pures Appl.
**104**(3), 561–586 (2015)MathSciNetCrossRefGoogle Scholar - 2.V. Agostiniani, L. Mazzieri, Monotonicity formulas in potential theory (2016). https://arxiv.org/abs/1606.02489.
- 3.V. Agostiniani, L. Mazzieri, Comparing monotonicity formulas for electrostatic potentials and static metrics. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.
**28**(1), 7–20 (2017)MathSciNetCrossRefGoogle Scholar - 4.V. Agostiniani, L. Mazzieri, On the geometry of the level sets of bounded static potentials. Commun. Math. Phys.
**355**(1), 261–301 (2017)MathSciNetCrossRefGoogle Scholar - 5.V. Agostiniani, S. Borghini, L. Mazzieri, On the torsion problem for domains with multiple boundary components (in preparation)Google Scholar
- 6.V. Agostiniani, M. Fogagnolo, L. Mazzieri, Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature. arXiv preprint arXiv:1812.05022 (2018)Google Scholar
- 7.L. Ambrosio, G. Da Prato, A. Mennucci, Introduction to measure theory and integration, in
*Appunti. Scuola Normale Superiore di Pisa (Nuova Serie)*[Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], vol. 10 (Edizioni della Normale, Pisa 2011)Google Scholar - 8.L. Ambrozio, On static three-manifolds with positive scalar curvature. J. Differ. Geom.
**107**(1), 1–45 (2017)MathSciNetCrossRefGoogle Scholar - 9.S. Borghini, L. Mazzieri, On the mass of static metrics with positive cosmological constant-II. 2017. ArXiv Preprint Server https://arxiv.org/abs/1711.07024
- 10.S. Borghini, L. Mazzieri, On the mass of static metrics with positive cosmological constant: I. Classical and Quantum Gravity
**35**(12), 125001 (2018)Google Scholar - 11.S. Borghini, G. Mascellani, L. Mazzieri, Some sphere theorems in linear potential theory. Trans. Am. Math. Soc. (2019). https://doi.org/10.1030/tran/7637
- 12.W. Boucher, G.W. Gibbons, G.T. Horowitz, Uniqueness theorem for anti-de Sitter spacetime. Phys. Rev. D (3)
**30**(12), 2447–2451 (1984)MathSciNetCrossRefGoogle Scholar - 13.P.T. Chruściel, On analyticity of static vacuum metrics at non-degenerate horizons. Acta Phys. Polon. B
**36**(1), 17–26 (2005)MathSciNetzbMATHGoogle Scholar - 14.P.T. Chruściel, Remarks on rigidity of the de sitter metric. http://homepage.univie.ac.at/piotr.chrusciel/papers/deSitter/deSitter2.pdf
- 15.P.T. Chruściel, M. Herzlich, The mass of asymptotically hyperbolic Riemannian manifolds. Pac. J. Math.
**212**(2), 231–264 (2003)MathSciNetCrossRefGoogle Scholar - 16.E. De Giorgi, Complementi alla teoria della misura (
*n*− 1)-dimensionale in uno spazio*n*-dimensionale, in*Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960–1961*(Editrice Tecnico Scientifica, Pisa, 1961)Google Scholar - 17.E. De Giorgi,
*Frontiere Orientate di Misura Minima*(Editr. Tecnico scientifica, 1961)Google Scholar - 18.W. De Sitter, On the curvature of space. Proc. Kon. Ned. Akad. Wet
**20**, 229–243 (1917)Google Scholar - 19.H. Federer, The Gauss–Green theorem. Trans. Am. Math. Soc.
**58**, 44–76 (1945)MathSciNetCrossRefGoogle Scholar - 20.H. Federer, A note on the Gauss-Green theorem. Proc. Am. Math. Soc.
**9**, 447–451 (1958)MathSciNetCrossRefGoogle Scholar - 21.H. Federer, Geometric measure theory, in
*Die Grundlehren der mathematischen Wissenschaften, Band 153*(Springer, New York, 1969)Google Scholar - 22.M. Fogagnolo, L. Mazzieri, A. Pinamonti, Geometric aspects of p-capacitary potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire. https://doi.org/10.1016/j.anihpc.2018.11.005
- 23.G.W. Gibbons, S.A. Hartnoll, C.N. Pope, Bohm and Einstein-Sasaki metrics, black holes, and cosmological event horizons. Phys. Rev. D (3)
**67**(8), 084024 (2003)Google Scholar - 24.O. Hijazi, S. Montiel, Uniqueness of the AdS spacetime among static vacua with prescribed null infinity. Adv. Theor. Math. Phys.
**18**(1), 177–203 (2014)MathSciNetCrossRefGoogle Scholar - 25.O. Hijazi, S. Montiel, S. Raulot, Uniqueness of the de Sitter spacetime among static vacua with positive cosmological constant. Ann. Glob. Anal. Geom.
**47**(2), 167–178 (2015)MathSciNetCrossRefGoogle Scholar - 26.F. Kottler, Über die physikalischen grundlagen der Einsteinschen gravitationstheorie. Ann. Phys. (Berlin)
**361**(14), 401–462 (1918)Google Scholar - 27.S.G. Krantz, H.R. Parks, A primer of real analytic functions, in
*Birkhäuser Advanced Texts: Basler Lehrbücher*, 2nd edn. [Birkhäuser Advanced Texts: Basel Textbooks] (Birkhäuser, Boston, 2002)Google Scholar - 28.J. Lafontaine, Sur la géométrie d’une généralisation de l’équation différentielle d’Obata. J. Math. Pures Appl. (9)
**62**(1), 63–72 (1983)Google Scholar - 29.L. Lindblom, Static uniform-density stars must be spherical in general relativity. J. Math. Phys.
**29**(2), 436–439 (1988)MathSciNetCrossRefGoogle Scholar - 30.S. Łojasiewicz,
*Introduction to Complex Analytic Geometry*(Birkhäuser, Basel, 1991). Translated from the Polish by Maciej KlimekGoogle Scholar - 31.H. Nariai, On a new cosmological solution of Einstein’s fieldequations of gravitation. Sci. Rep. Tohoku Univ. Ser. I
**35**(1), 62–67 (1951)MathSciNetzbMATHGoogle Scholar - 32.M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Jpn.
**14**, 333–340 (1962)MathSciNetCrossRefGoogle Scholar - 33.R. Penrose, Asymptotic properties of fields and space-times. Phys. Rev. Lett.
**10**, 66–68 (1963)MathSciNetCrossRefGoogle Scholar - 34.J. Qing, On the uniqueness of AdS space-time in higher dimensions. Ann. Henri Poincaré
**5**(2), 245–260 (2004)MathSciNetCrossRefGoogle Scholar - 35.K. Schwarzschild, On the gravitational field of a mass point according to Einstein’s theory. Gen. Relativ. Gravit.
**35**(5), 951–959 (2003). Translated from the original German article [Sitzungsber. Königl. Preussich. Akad. Wiss. Berlin Phys. Math. Kl.**1**916, 189–196] by S. Antoci and A. LoingerGoogle Scholar - 36.J. Serrin, Isolated singularities of solutions of quasi-linear equations. Acta Math.
**113**, 219–240 (1965)MathSciNetCrossRefGoogle Scholar - 37.J. Souček, V. Souček, Morse-Sard theorem for real-analytic functions. Comment. Math. Univ. Carol.
**13**, 45–51 (1972)MathSciNetzbMATHGoogle Scholar - 38.X. Wang, The mass of asymptotically hyperbolic manifolds. J. Differ. Geom.
**57**(2), 273–299 (2001)MathSciNetCrossRefGoogle Scholar - 39.X. Wang, On the uniqueness of the AdS spacetime. Acta Math. Sin. (Engl. Ser.)
**21**(4), 917–922 (2005)Google Scholar - 40.H.F. Weinberger, Remark on the preceding paper of Serrin. Arch. Ration. Mech. Anal.
**43**, 319–320 (1971)MathSciNetCrossRefGoogle Scholar - 41.H.M. Zum Hagen, On the analyticity of static vacuum solutions of Einstein’s equations. Proc. Camb. Philos. Soc.
**67**, 415–421 (1970)MathSciNetCrossRefGoogle Scholar