Skip to main content

Advertisement

SpringerLink
Log in

Brendle’s Inequality on Static Manifolds

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

We generalize Brendle’s geometric inequality considered in Brendle (Publ Math Inst Hautes Études Sci 117:247–269, 2013) to static manifolds. The inequality bounds the integral of inverse mean curvature of an embedded mean-convex hypersurface by geometric data of the horizon. As a consequence, we obtain a reverse Penrose inequality on static asymptotically locally hyperbolic manifolds in the spirit of Chruściel and Simon (J Math Phys 42(4):1779–1817, 2001).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ambrozio, L.: On static three-manifolds with positive scalar curvature, arXiv:1503.03803

  2. Brendle, S.: Constant mean curvature surfaces in warped product manifolds. Publ. Math. Inst. Hautes Études Sci. 117, 247–269 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boucher, W., Gibbons, G.W., Horowitz, G.T.: Uniqueness theorem for anti-de Sitter spacetime. Phys. Rev. D 30(12), 2447–2451 (1984)

    Article  MathSciNet  Google Scholar 

  4. Brendle, S., Hung, P.-K., Wang, M.-T.: A Minkowski inequality for hypersurfaces in the anti-de Sitter–Schwarzschild manifold. Commun. Pure Appl. Math. 69(1), 124–144 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bunting, G.L., Masood-ul-Alam, A.K.M.: Nonexistence of multiple black holes in asymptotically Euclidean static vacuum space-time. Gen. Relat. Gravit. 19(2), 147–154 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  6. Beig, R., Schoen, R.M.: On static \(n\)-body configurations in relativity. Class. Quantum Gravit. 26(7), 075014, 7 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chruściel, P.T.: The classification of static vacuum spacetimes containing an asymptotically flat spacelike hypersurface with compact interior. Class. Quantum Gravit. 16(3), 661–687 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cederbaum, C., Galloway, G.J.: Uniqueness of photon spheres via positive mass rigidity, arXiv:1504.05804.v1

  9. Chruściel, P.T., Simon, W.: Towards the classification of static vacuum spacetimes with negative cosmological constant. J. Math. Phys. 42(4), 1779–1817 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Galloway, G.J., Miao, P.: Variational and rigidity properties of static potentials, arXiv: 1412.1062v3

  11. Israel, W.: Event horizons in static vacuum space-times. Phys. Rev. 164(5), 1776–1779 (1967)

    Article  Google Scholar 

  12. Lee, D.A., Neves, A.: The Penrose inequality for asymptotically locally hyperbolic spaces with nonpositive mass. Commun. Math. Phys. 339(2), 327–352 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, J., Xia, C.: An integral formula and its applications on sub-static manifolds, arXiv: 1603.02201

  14. Miao, P., Tam, L.-F.: Static potentials on asymptotically flat manifolds. Ann. Henri Poincaré 16(10), 2239–2264 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Qing, J.: On the uniqueness of AdS space-time in higher dimensions. Ann. Henri Poincaré 5(2), 245–260 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Robinson, D.: A simple proof of the generalization of Israel’s Theorem. Gen. Relat. Gravit. 8, 695–698 (1977)

    Article  MATH  Google Scholar 

  17. Woolgar, E.: The rigid Horowitz-Myers conjecture. arXiv: 1602.06197v1

  18. Wang, X.: The mass of asymptotically hyperbolic manifolds. J. Differ. Geom. 57(2), 273–299 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, X.: On the uniqueness of the AdS spacetime. Acta Math. Sin. 21(4), 917–922 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first author is partially supported by Simons Foundation Collaboration Grant for Mathematicians #312820. The second author would like to thank Professor Mu-Tao Wang for his constant encouragement, Po-Ning Chen and Pei-Ken Hung for helpful discussions. We would also like to thank Professor Luen-Fai Tam for his comments on the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Michigan State University, East Lansing, MI, USA

    Xiaodong Wang & Ye-Kai Wang

Authors
  1. Xiaodong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ye-Kai Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ye-Kai Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, X., Wang, YK. Brendle’s Inequality on Static Manifolds. J Geom Anal 28, 152–169 (2018). https://doi.org/10.1007/s12220-017-9814-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-017-9814-3

Keywords

Mathematics Subject Classification

Search

Navigation