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On the mass of static metrics with positive cosmological constant: II. (English) Zbl 1443.83019

Summary: This is the second of two works, in which we discuss the definition of an appropriate notion of mass for static metrics, in the case where the cosmological constant is positive and the model solutions are compact. In the first part [Classical Quantum Gravity 35, No. 12, Article ID 125001, 43 p. (2018; Zbl 1391.83037)], we have established a positive mass statement, characterising the de Sitter solution as the only static vacuum metric with zero mass. In this second part, we prove optimal area bounds for horizons of black hole type and of cosmological type, corresponding to Riemannian Penrose inequalities and to cosmological area bounds à la Boucher-Gibbons-Horowitz, respectively. Building on the related rigidity statements, we also deduce a uniqueness result for the Schwarzschild-de Sitter spacetime.

MSC:

83C40 Gravitational energy and conservation laws; groups of motions
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C57 Black holes
53Z05 Applications of differential geometry to physics

Citations:

Zbl 1391.83037
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References:

[1] Abbott, LF; Deser, S., Stability of gravity with a cosmological constant, Nucl. Phys. B, 195, 1, 76-96 (1982) · Zbl 0900.53033
[2] Agostiniani, V.; Mazzieri, L., Riemannian aspects of potential theory, J. Math. Pures Appl. (9), 104, 3, 561-586 (2015) · Zbl 1326.35216
[3] Agostiniani, V.; Mazzieri, L., On the geometry of the level sets of bounded static potentials, Commun. Math. Phys., 355, 1, 261-301 (2017) · Zbl 1375.53090
[4] Agostiniani, V.; Mazzieri, L., Monotonicity formulas in potential theory, Calc. Var. Partial. Differ. Equ., 59, 1, 6 (2020) · Zbl 1428.35008
[5] Ambrozio, L., On static three-manifolds with positive scalar curvature, J. Differ. Geom., 107, 1, 1-45 (2017) · Zbl 1385.53020
[6] Anninos, Dionysios, de Sitter musings, Int. J. Mod. Phys. A, 27, 13, 1230013 (2012) · Zbl 1247.83068
[7] Ashtekar, A.; Bonga, B.; Kesavan, A., Asymptotics with a positive cosmological constant: I. Basic framework, Class. Quantum Gravity, 32, 2, 025004 (2014) · Zbl 1307.83011
[8] Balasubramanian, V.; De Boer, J.; Minic, D., Mass, entropy, and holography in asymptotically de sitter spaces, Phys. Rev. D, 65, 12, 123508 (2002)
[9] Baltazar, H.; Ribeiro, E., Remarks on critical metrics of the scalar curvature and volume functionals on compact manifolds with boundary, Pac. J. Math., 297, 1, 29-45 (2018) · Zbl 1400.53028
[10] Beig, R.; Simon, W., On the uniqueness of static perfect-fluid solutions in general relativity, Commun. Math. Phys., 144, 2, 373-390 (1992) · Zbl 0760.53043
[11] Besse, A.L.: Einstein Manifolds. Classics in Mathematics. Springer, Berlin (2008). Reprint of the 1987 edition
[12] Böhm, C., Non-compact cohomogeneity one Einstein manifolds, Bull. Soc. Math. Fr., 127, 1, 135-177 (1999) · Zbl 0935.53021
[13] Borghini, S., Chruściel, P.T., Mazzieri, L.: On the uniqueness of Schwarzschild-de Sitter spacetime. arXiv Preprint Server arXiv:1909.05941 (2019)
[14] Borghini, S.; Mazzieri, L., On the mass of static metrics with positive cosmological constant: I, Class. Quantum Gravity, 35, 12, 125001 (2018) · Zbl 1391.83037
[15] Borghini, S.; Mazzieri, L.; Dipierro, S., Monotonicity formulas for static metrics with non-zero cosmological constant, Contemporary Research in Elliptic PDEs and Related Topics, 129-202 (2019), Berlin: Springer, Berlin · Zbl 1444.35013
[16] Boucher, W.; Gibbons, GW; Horowitz, GT, Uniqueness theorem for anti-de Sitter spacetime, Phys. Rev. D (3), 30, 12, 2447-2451 (1984)
[17] Bousso, R.: Adventures in de Sitter space, pp. 539-569 (2003) · Zbl 1186.83079
[18] Bousso, R.; Hawking, SW, Pair creation of black holes during inflation, Phys. Rev. D, 54, 6312-6322 (1996)
[19] Bray, HL, Proof of the Riemannian Penrose inequality using the positive mass theorem, J. Differ. Geom., 59, 2, 177-267 (2001) · Zbl 1039.53034
[20] Bunting, GL; Masood-ul-Alam, AKM, Nonexistence of multiple black holes in asymptotically Euclidean static vacuum space-time, Gen. Relativ. Gravit., 19, 2, 147-154 (1987) · Zbl 0615.53055
[21] Cardoso, V.; Dias, ÓJ; Lemos, JP, Nariai, Bertotti-Robinson, and anti-Nariai solutions in higher dimensions, Phys. Rev. D, 70, 2, 024002 (2004)
[22] Chruściel, P.T.: Remarks on rigidity of the de sitter metric. http://homepage.univie.ac.at/piotr.chrusciel/papers/deSitter/deSitter2.pdf
[23] Chruściel, PT, On analyticity of static vacuum metrics at non-degenerate horizons, Acta Phys. Pol. B, 36, 1, 17-26 (2005) · Zbl 1066.83007
[24] Chruściel, PT; Jezierski, J.; Kijowski, J., Hamiltonian mass of asymptotically Schwarzschild-de sitter space-times, Phys. Rev. D, 87, 12, 124015 (2013)
[25] Chruściel, PT; Simon, W., Towards the classification of static vacuum spacetimes with negative cosmological constant, J. Math. Phys., 42, 4, 1779-1817 (2001) · Zbl 1009.83009
[26] da Silva, A., Baltazar, H.: On static manifolds and related critical spaces with cyclic parallel Ricci tensor. arXiv Preprint Server arXiv:1811.04447 (2018)
[27] De Sitter, W., On the curvature of space, Proc. Kon. Ned. Akad. Wet, 20, 229-243 (1917)
[28] Fang, Y.; Yuan, W., Brown-york mass and positive scalar curvature II: Besse’s conjecture and related problems, Ann. Glob. Anal. Geom., 56, 1-15 (2018) · Zbl 1418.53052
[29] Gibbons, GW; Hartnoll, SA; Pope, CN, Bohm and Einstein-Sasaki metrics, black holes, and cosmological event horizons, Phys. Rev. D (3), 67, 8, 084024 (2003)
[30] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, volume 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd edn. Springer, Berlin (1983) · Zbl 0562.35001
[31] Ginsparg, P.; Perry, MJ, Semiclassical perdurance of de Sitter space, Nucl. Phys. B, 222, 245-268 (1983)
[32] Hijazi, O.; Montiel, S.; Raulot, S., Uniqueness of the de Sitter spacetime among static vacua with positive cosmological constant, Ann. Glob. Anal. Geom., 47, 2, 167-178 (2015) · Zbl 1318.53054
[33] Huisken, G.; Ilmanen, T., The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differ. Geom., 59, 3, 353-437 (2001) · Zbl 1055.53052
[34] Huisken, G., Polden, A.: Geometric evolution equations for hypersurfaces. In: Calculus of Variations and Geometric Evolution Problems (Cetraro, 1996). Lecture Notes in Mathematics, vol. 1713, pp. 45-84. Springer, Berlin (1999) · Zbl 0942.35047
[35] Israel, W., Event horizons in static vacuum space-times, Phys. Rev., 164, 5, 1776-1779 (1967)
[36] Kastor, D.; Traschen, J., A positive energy theorem for asymptotically de sitter spacetimes, Class. Quantum Gravity, 19, 23, 5901 (2002) · Zbl 1019.83007
[37] Kobayashi, O., A differential equation arising from scalar curvature function, J. Math. Soc. Jpn., 34, 4, 665-675 (1982) · Zbl 0486.53034
[38] Kottler, F., Über die physikalischen grundlagen der Einsteinschen gravitationstheorie, Ann. Phys. (Berlin), 361, 14, 401-462 (1918) · JFM 46.1306.01
[39] Krantz, SG; Parks, HR, Distance to \(C^k\) hypersurfaces, J. Differ. Equ., 40, 1, 116-120 (1981) · Zbl 0431.57009
[40] Krantz, S.G., Parks, H.R.: A Primer of Real Analytic Functions. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], 2nd edn. Birkhäuser Boston, Inc., Boston (2002)
[41] Lafontaine, J., Sur la géométrie d’une généralisation de l’équation différentielle d’Obata, J. Math. Pures Appl. (9), 62, 1, 63-72 (1983) · Zbl 0513.53046
[42] Lee, DA; Neves, A., The Penrose inequality for asymptotically locally hyperbolic spaces with nonpositive mass, Commun. Math. Phys., 339, 2, 327-352 (2015) · Zbl 1322.53038
[43] Łojasiewicz, S.: Une propriété topologique des sous-ensembles analytiques réels. In: Les Équations aux Dérivées Partielles (Paris, 1962), pp. 87-89. Éditions du Centre National de la Recherche Scientifique, Paris (1963)
[44] Łojasiewicz, S.: Introduction to Complex Analytic Geometry. Birkhäuser, Basel (1991). Translated from the Polish by Maciej Klimek · Zbl 0747.32001
[45] Luo, M.; Xie, N.; Zhang, X., Positive mass theorems for asymptotically de Sitter spacetimes, Nucl. Phys. B, 825, 1-2, 98-118 (2010) · Zbl 1196.83017
[46] Nariai, H., On a new cosmological solution of Einstein’s fieldequations of gravitation, Sci. Rep. Tohoku Univ. Ser. I Phys. Chem. Astron., 35, 1, 62-67 (1951) · Zbl 0045.13202
[47] Qing, J.; Yuan, W., A note on static spaces and related problems, J. Geom. Phys., 74, 18-27 (2013) · Zbl 1287.83016
[48] Robinson, DC, A simple proof of the generalization of Israel’s theorem, Gen. Relativ. Gravit., 8, 8, 695-698 (1977) · Zbl 0429.53043
[49] Schoen, R.; Yau, ST, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys., 65, 1, 45-76 (1979) · Zbl 0405.53045
[50] Schoen, R., Yau, S.T.: Positive scalar curvature and minimal hypersurface singularities. 2017. arXiv Preprint Server arXiv:1704.05490
[51] Schwarzschild, K.: On the gravitational field of a sphere of incompressible fluid according to Einstein’s theory. arXiv Preprint Server https://arxiv.org/abs/physics/9912033 (1999) · Zbl 1210.83012
[52] Shiromizu, T., Positivity of gravitational mass in asymptotically de Sitter space-times, Phys. Rev. D, 49, 10, 5026 (1994)
[53] Shiromizu, T.; Ida, D.; Torii, T., Gravitational energy, dS/CFT correspondence and cosmic no-hair, J. High Energy Phys., 2001, 11, 010 (2001)
[54] Souček, J.; Souček, V., Morse-Sard theorem for real-analytic functions, Comment. Math. Univ. Carol., 13, 45-51 (1972) · Zbl 0235.26012
[55] Witten, E., A new proof of the positive energy theorem, Commun. Math. Phys., 80, 3, 381-402 (1981) · Zbl 1051.83532
[56] Witten, E.: Quantum gravity in de sitter space. Technical report (2001) · Zbl 1054.83013
[57] Yuan, W.: Brown-york mass and positive scalar curvature I-first eigenvalue problem and its applications. arXiv Preprint Server arXiv:1806.07798 (2018)
[58] Zum Hagen, HM, On the analyticity of static vacuum solutions of Einstein’s equations, Proc. Camb. Philos. Soc., 67, 415-421 (1970) · Zbl 0191.52505
[59] Zum Hagen, HM; Robinson, DC; Seifert, HJ, Black holes in static vacuum space-times, Gen. Relativ. Gravit., 4, 1, 53-78 (1973)
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