Letters in Mathematical Physics

, Volume 109, Issue 3, pp 661–673 | Cite as

New restrictions on the topology of extreme black holes

  • Marcus Khuri
  • Eric WoolgarEmail author
  • William Wylie


We provide bounds on the first Betti number and structure results for the fundamental group of horizon cross sections for extreme stationary vacuum black holes in arbitrary dimension, without additional symmetry hypotheses. This is achieved by exploiting a correspondence between the associated near-horizon geometries and the mathematical notion of m-quasi-Einstein metrics, in addition to generalizations of the classical splitting theorem from Riemannian geometry. Consequences are analyzed and refined classifications are given for the possible topologies of these black holes.


Black holes Horizon topology m-Quasi-Einstein metric Splitting theorem 

Mathematics Subject Classification

83C57 53C20 



The authors would like to thank Hari Kunduri for comments, as well as Stefan Hollands and James Lucietti for independently pointing out that \(\mathbb {RP}^3 \#\mathbb {RP}^3\) can be ruled out in Corollary 7 when an extra U(1) symmetry is present. The first author would also like to thank Luca Di Cerbo for useful discussions.


  1. 1.
    Breunholder, V., Lucietti, J.: Moduli space of supersymmetric solitons and black holes in five dimensions. arXiv:1712.07092
  2. 2.
    Cheeger, J., Gromoll, D.: The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differ. Geom. 6, 119–128 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, Y., Teo, E.: A rotating black lens solution in five dimensions. Phys. Rev. D 78, 064062 (2008)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Chruściel, P., Reall, H., Tod, P.: On non-existence of static vacuum black holes with degenerate components of the event horizon. Class. Quantum Gravity 23, 549–554 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Donaldson, S., Kronheimer, P.: The Geometry of Four-Manifolds. Oxford Mathematical Monographs (1997)Google Scholar
  6. 6.
    Emparan, R., Reall, H.: A rotating black ring in five dimensions. Phys. Rev. Lett. 88, 101101 (2002)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Eschenburg, J., Heintze, E.: An elementary proof of the Cheeger–Gromoll splitting theorem. Ann. Glob. Anal. Geom. 2, 141–151 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fang, F., Li, X.-D., Zhang, Z.: Two generalizations of Cheeger–Gromoll splitting theorem via Bakry–Emery Ricci curvature. Ann. Inst. Fourier (Grenoble) 59(2), 563–573 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Freedman, M., Quinn, F.: Topology of \(4\)-Manifolds. Princeton University Press, Princeton (1990)zbMATHGoogle Scholar
  10. 10.
    Friedman, J., Schleich, K., Witt, D.: Topological censorship. Phys. Rev. Lett. 71(10), 1486–1489 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Galloway, G.: Rigidity of marginally trapped surfaces and the topology of black holes. Commun. Anal. Geom. 16(1), 217–229 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Galloway, G.: Constraints on the topology of higher dimensional black holes. In: Horowitz, G. (ed.) Black Holes in Higher Dimensions, pp. 159–179. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  13. 13.
    Galloway, G.: Rigidity of outermost MOTS—the initial data version, Gen. Relativity Gravitation. arXiv:1712.02247
  14. 14.
    Galloway, G., Schoen, R.: A generalization of Hawking’s black hole topology theorem to higher dimensions. Commun. Math. Phys. 266(2), 571–576 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gregory, R., Laflamme, R.: Black strings and p-branes are unstable. Phys. Rev. Lett. 70, 2837–2840 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gromov, M.: Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. 53, 53–73 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gromov, M., Lawson, H.B.: Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Inst. Hautes Études Sci. Publ. Math. (1983) 58, 83–196 (1984)CrossRefzbMATHGoogle Scholar
  18. 18.
    Hawking, S.: The event horizon, Black Holes. In: Dewitt, C., Dewitt, B. (eds.) Les Houches Lectures. pp. 1–55 (1973)Google Scholar
  19. 19.
    Hawking, S., Ellis, G.: The Large Scale Structure of Space–Time. Cambridge University Press, London (1973)CrossRefzbMATHGoogle Scholar
  20. 20.
    Helfgott, C., Oz, Y., Yanay, Y.: On the topology of stationary black hole event horizons in higher dimensions. J. High Energy Phys. 2, 025 (2006)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Hempel, J: 3-Manifolds. Annals of Mathematical Studies, No. 86. Princeton University Press, Princeton (1976)Google Scholar
  22. 22.
    Hollands, S., Holland, J., Ishibashi, A.: Further restrictions on the topology of stationary black holes in five dimensions. Ann. Henri Poincaré 12(2), 279–301 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hollands, S., Ishibashi, A.: On the ‘stationary implies axisymmetric’ theorem for extremal black holes in higher dimensions. Commun. Math. Phys. 291, 403–441 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hollands, S., Ishibashi, A.: Black hole uniqueness theorems in higher dimensional spacetimes. Class. Quantum Gravity 29(16), 163001 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hollands, S., Ishibashi, A., Wald, R.: A higher dimensional stationary rotating black hole must be axisymmetric. Commun. Math. Phys. 271(3), 699–722 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hollands, S., Yazadjiev, S.: A uniqueness theorem for stationary Kaluza–Klein black holes. Commun. Math. Phys. 302(3), 631–674 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Khuri, M., Weinstein, G., Yamada, S.: Stationary vacuum black holes in 5 dimensions, preprint (2018) arXiv:1711.05229
  28. 28.
    Khuri, M., Weinstein, G., Yamada, S.: Asymptotically locally Euclidean/Kaluza–Klein stationary vacuum black holes in 5 dimensions. Prog. Theor. Exp. Phys. 2018(5), 053E01 (2018)Google Scholar
  29. 29.
    Khuri, M., Woolgar, E.: Nonexistence of de Sitter black rings. Class. Quantum Gravity 34, 22LT01 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Khuri, M., Woolgar, E.: Nonexistence of degenerate horizons in static vacua and black hole uniqueness. Phys. Lett. B 777, 235–239 (2018)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Kim, D.S., Kim, Y.H.: Compact Einstein warped product spaces with nonpositive scalar curvature. Proc. Am. Math. Soc. 131, 2573–2576 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kleiner, B.: A new proof of Gromov’s theorem on groups of polynomial growth. J. Am. Math. Soc. 23(3), 815–829 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kreck, M., Lück, W., Teichner, P.: Stable prime decompositions of four-manifolds, Prospects in Topology (Princeton, NJ, 1994), pp. 251–269, Ann. of Math. Stud., vol. 138, Princeton University Press (1995)Google Scholar
  34. 34.
    Kunduri, H., Lucietti, J.: An infinite class of extremal horizons in higher dimensions. Commun. Math. Phys. 303, 31–71 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Kunduri, H., Lucietti, J.: Classification of near-horizon geometries of extremal black holes. Living Rev. Relat. 16, 8 (2013)ADSCrossRefzbMATHGoogle Scholar
  36. 36.
    Kunduri, H., Lucietti, J.: A supersymmetric black lens. Phys. Rev. Lett. 113, 211101 (2014)ADSCrossRefGoogle Scholar
  37. 37.
    Lucietti, J.: Two remarks on near-horizon geometries. Class. Quantum Gravity 29, 235014 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Mann, A.: The growth of free products. J. Algebra 326(1), 208–217 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Milnor, J.: A note on curvature and fundamental group. J. Differ. Geom. 2, 1–7 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Moncrief, V., Isenberg, J.: Symmetries of higher dimensional black holes. Class. Quantum Gravity 25(19), 195015 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Myers, R., Perry, M.: Black holes in higher dimensional space–times. Ann. Phys. 172(2), 304–347 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Petersen, P.: Riemannian Geometry, Graduate Texts in Mathematics 171. Springer, New York (1998)Google Scholar
  43. 43.
    Pomeransky, A., Sen’kov, R.: Black ring with two angular momenta. arXiv:hep-th/0612005
  44. 44.
    Tomizawa, S., Nozawa, M.: Supersymmetric black lenses in five dimensions. Phys. Rev. D 94, 044037 (2016)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    Villani, C.: Optimal Transport Old and New, Grundlehren der Mathematischen Wissenschaften 338. Springer, New York (2009)Google Scholar
  46. 46.
    Wolf, J.: Spaces of Constant Curvature. Publish or Perish, Wilmington (1984)zbMATHGoogle Scholar
  47. 47.
    Wylie, W.: A warped product version of the Cheeger–Gromoll splitting theorem. Trans. Am. Math. Soc. 369(9), 6661–6681 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsStony Brook UniversityStony BrookUSA
  2. 2.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada
  3. 3.Department of MathematicsSyracuse UniversitySyracuseUSA

Personalised recommendations