## On the uniqueness of the ADS spacetime.(English)Zbl 1082.53040

The author proves that the $$n$$-dimensional anti-de Sitter space-time $$(M,g,V)$$ is the unique static solution to the vacuum Einstein equation with negative cosmological constant, provided: $$M$$ is a spin manifold; $$(M,g)$$ is conformally compact and the conformal boundary is the sphere $$S^{n-1}$$; $$V^{-2}g$$ restricted to $$S^{n-1}$$ is the standard metric. Moreover, it is proven that for $$n \leq 7$$, the spin assumption may be removed.

### MSC:

 53C24 Rigidity results 58Z05 Applications of global analysis to the sciences 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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### References:

 [1] Boucher, W., Gibbons, G. W., Horowitz, Gary, T.: Uniqueness theorem for anti–de Sitter spacetime. Phys. Rev., D(3), 30(12), 2447–2451 (1984) [2] Galloway, G. J., Surya, S., Woolgar, E.: On the geometry and mass of static, asymptotically ads spacetimes, and the uniqueness of the ads soliton: e–Print, hep–th/0204081 [3] Wang, X. D.: The mass of asymptotically hyperbolic manifolds. J. Differential Geom., 57(2), 273–299 (2001) · Zbl 1037.53017 [4] Piotr, T. Chruściel, Herzlich, M.: The mass of asymptotically hyperbolic riemannian manifolds, e–Print math.DG/0110035. · Zbl 1056.53025 [5] Robin Graham, C.: Volume and area renormalizations for conformally compact Einstein metrics. The Proceedings of the 19th Winter School ”Geometry and Physics”, (Srní, 1999), 63, 31–42 (2000) [6] Qing, J.: On the rigidity for conformally compact Einstein manifolds. Int. Math. Res. Not., 21, 1141–1153 (2003) · Zbl 1042.53031 [7] Bunting, Gary, L., Masood–ul Alam, A. K. M.: Nonexistence of multiple black holes in asymptotically Euclidean static vacuum space–time. Gen. Relativity Gravitation, 19(2), 147–154 (1987) · Zbl 0615.53055 [8] Masood-ul Alam, A. K. M.: Uniqueness of a static charged dilaton black hole. Classical Quantum Gravity, 10(12), 2649–2656 (1993) · Zbl 0787.53081 [9] Masood-ul Alam, A. K. M.: Uniqueness proof of static charged black holes revisited. Classical Quantum Gravity, 9(5), L53–L55 (1992) · Zbl 0991.83546 [10] Miao, P. Z.: Positive Mass Theorem on Manifolds admitting Corners along a Hypersurface, arXiv mathph/ 0212025 [11] Shi, Y., Tam, L.: Positive mass theorem and the boundary behavior of compact manifolds with nonegative scalar curvature. arXiv math. [12] Chruściel, Piotr, T. Simon, W.: Towards the classification of static vacuum spacetimes with negative cosmological constant. J. Math. Phys., 42(4), 1779–1817 (2001) · Zbl 1009.83009 [13] Wang, X. D.: On conformally compact Einstein manifolds. Math. Res. Lett., 8(5–6), 671–688 (2001) · Zbl 1053.53030
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