Uniqueness of Kottler spacetime and the Besse conjecture. (Unicité de l’espace-temps de Kottler et conjecture de Besse.) (English) Zbl 1202.83068

Summary: We establish a black hole uniqueness theorem for Schwarzschild-de Sitter spacetime, also called Kottler spacetime, which satisfies Einstein’s field equations of general relativity with positive cosmological constant. Our result concerns the class of static vacuum spacetimes with compact spacelike slices and regular maximal level set of the lapse function. We provide a characterization of the interior domain of communication of the Kottler spacetime, which surrounds an inner horizon and is surrounded by a cosmological horizon. The proof combines arguments from the theory of partial differential equations and differential geometry, and is centered on a detailed study of a possibly singular foliation. We also apply our technique in the Riemannian setting, and establish the validity of the so-called Besse conjecture.


83C57 Black holes
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C25 Approximation procedures, weak fields in general relativity and gravitational theory
35L15 Initial value problems for second-order hyperbolic equations
53Z05 Applications of differential geometry to physics
Full Text: DOI arXiv


[1] Beig, R.; Simon, W., On the uniqueness of static perfect-fluid solutions in general relativity, Comm. math. phys., 144, 373-390, (1992) · Zbl 0760.53043
[2] Besse, A.L., Einstein manifolds, Ergeb. math. grenzgeb., vol. 10, (1987), Springer Verlag · Zbl 0613.53001
[3] Hawking, S.W., Black holes in general relativity, Comm. math. phys., 25, 152-166, (1972)
[4] Israel, W., Event horizons in static vacuum spacetimes, Phys. rev., 164, 1776-1779, (1967)
[5] Kobayashi, O., Scalar curvature of a metric with unit volume, Math. ann., 279, 253-265, (1987) · Zbl 0611.53037
[6] Kottler, F., Uber die physikalischen grundlagen der einsteinschen gravitation theori, Ann. phys., 56, 401-402, (1918) · JFM 46.1306.01
[7] Lafontaine, J., Sur la géométrie d’une généralisation de l’équation différentielle d’obata, J. math. pures appl., 62, 63-72, (1983) · Zbl 0513.53046
[8] P.G. LeFloch, L. Rozoy, A uniqueness theorem for Schwarzschild-de Sitter spacetime, in preparation. · Zbl 1202.83068
[9] P.G. LeFloch, L. Rozoy, in preparation.
[10] Lindblom, L., Static uniform-density stars must be spherical in general relativity, J. math. phys., 29, 436-439, (1988) · Zbl 0641.76139
[11] Obata, M., Certain conditions for a Riemannian manifold to be isometric with a sphere, J. math. soc. Japan, 14, 333-340, (1962) · Zbl 0115.39302
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.