Communications in Mathematical Physics

, Volume 266, Issue 2, pp 571–576 | Cite as

A Generalization of Hawking’s Black Hole Topology Theorem to Higher Dimensions



Hawking’s theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. This conclusion extends to outer apparent horizons in spacetimes that are not necessarily stationary. In this paper we obtain a natural generalization of Hawking’s results to higher dimensions by showing that cross sections of the event horizon (in the stationary case) and outer apparent horizons (in the general case) are of positive Yamabe type, i.e., admit metrics of positive scalar curvature. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology S 2 × S 1. The proof is inspired by previous work of Schoen and Yau on the existence of solutions to the Jang equation (but does not make direct use of that equation).


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  1. 1.
    Ashtekar A., Galloway G.J. (2005) Uniqueness theorems for dynamical horizons. Adv. Theor. Math. Phys. 8, 1–30MathSciNetGoogle Scholar
  2. 2.
    Ashtekar A., Krishnan B. (2003) Dynamical horizons and their properties. Phys. Rev. D 68, 261101CrossRefMathSciNetGoogle Scholar
  3. 3.
    Andersson L., Mars M., Simon W. (2005) Local existence of dynamical and trapping horizons. Phys. Rev. Lett. 95, 111102CrossRefADSGoogle Scholar
  4. 4.
    Cai, M. Volume minimizing hypersurfaces in manifolds of nonnegative scalar curvature. In: Minimal Surfaces, Geometric Analysis, and Symplectic Geometry, Advanced Studies in Pure Mathematics, eds. Fukaya, K., Nishikawa, S., Spruck, J., 34, 1–7 (2002)Google Scholar
  5. 5.
    Cai M., Galloway G.J. (2000) Rigidity of area minimzing tori in 3-manifolds of nonnegative scalar curvature. Commun. Anal. Geom. 8, 565–573MATHMathSciNetGoogle Scholar
  6. 6.
    Cai M., Galloway G.J. (2001) On the topology and area of higher dimensional black holes. Class. Quant. Grav. 18, 2707–2718MATHCrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Emparan R., Reall H.S. (2002) A rotating black ring in five dimensions. Phys. Rev. Lett. 88, 101101CrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Gibbons G.W. (1972) The time symmetric initial value problem for black holes. Commun. Math. Phys. 27, 87–102CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Gromov M., Lawson B. (1980) Spin and scalar curvature in the presence of the fundamental group. Ann. of Math. 111, 209–230CrossRefMathSciNetGoogle Scholar
  10. 10.
    Gromov M., Lawson B. (1983) Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Publ. Math. IHES 58, 83–196MATHMathSciNetGoogle Scholar
  11. 11.
    Hawking S.W. (1972) Black holes in general relativity. Commun. Math. Phys. 25, 152–166CrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Hawking, S.W. The event horizon. In ‘Black Holes, Les Houches lectures’ (1972), edited by C. DeWitt, B. S. DeWitt Amsterdam: North Holland, 1972Google Scholar
  13. 13.
    Hawking S.W., Ellis G.F.R. (1973) The large scale structure of space-time. Cambridge University Press, CambridgeMATHGoogle Scholar
  14. 14.
    Helfgott, C., OZ, Y., Yanay, Y. On the topology of black hole event horizons in higher dimensions. JHEPO2 (2006) 024Google Scholar
  15. 15.
    Kazdan J., Warner F. (1975) Prescribing curvatures. Proc. Symp. in Pure Math. 27, 309–319MathSciNetGoogle Scholar
  16. 16.
    Lichnerowicz A. (1963) Spineurs harmoniques. Cr. Acd. Sci. Paris, Sér. A-B 257, 7–9MATHMathSciNetGoogle Scholar
  17. 17.
    Schoen R., Yau S.T. (1979) 1 Existence of incompressible minimal surfaces and the topology of three dimensional manifolds of non-negative scalar curvature. Ann. of Math. 110, 127–142CrossRefMathSciNetGoogle Scholar
  18. 18.
    Schoen R., Yau S.T. (1979) On the structure of manifolds with positive scalar curvature. Manuscripta Math. 28, 159–183MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Schoen R., Yau S.T. (1981) Proof of the positive of mass theorem. II. Commun. Math. Phys., 79, 231–260MATHCrossRefADSMathSciNetGoogle Scholar

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© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MiamiCoral GablesUSA
  2. 2.Department of MathematicsStanford UniversityStanfordUSA

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