Advertisement

A remark on the rigidity of Poincaré–Einstein manifolds

  • Simon RaulotEmail author
Article
  • 24 Downloads

Abstract

In this paper, we give an optimal inequality relating the relative Yamabe invariant of a certain compactification of a conformally compact Poincaré–Einstein manifold with the Yamabe invariant of its boundary at infinity. As an application, we obtain an elementary proof (without any additional assumption) of the rigidity of the hyperbolic space as the only conformally compact Poincaré–Einstein manifold with the round sphere as its conformal infinity.

Keywords

Poincaré-Einstein manifold Yamabe invariants Uniqueness 

Mathematics Subject Classification

Differential Geometry 53C24 53C80 

References

  1. 1.
    Almaraz, S.M.: An existence theorem of conformal scalar-flat metrics on manifolds with boundary. Pac. J. Math 248(1), 1–22 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Anderson, M.T.: Boundary regularity, uniqueness and non-uniqueness for AH Einstein metrics on \(4\)-manifolds. Adv. Math. 179(2), 205–249 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Anderson, M.T.: Einstein matrics with prescribed conformal infinity on \(4\)-manifolds. Geom. Funct. Anal. 18, 305–366 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Anderson, M.T., Herzlich, M.: Unique continuation results for Ricci curvature and applications. J. Geom. Phys. 58(2), 179–207 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Andersson, L., Dahl, M.: Scalar curvature rigidity for asymptotically locally hyperbolic manifolds. Ann. Global. Anal. Geom. 16(1), 1–27 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bahri, A., Coron, J.M.: On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain. Commun. Pure Appl. Math. 41(3), 253–294 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Besse, A.: Einstein Manifolds. Springer, New York (1987)CrossRefzbMATHGoogle Scholar
  8. 8.
    Bicquard, O.: Désingularisation de métriques d’Einstein II. Invent. Math. 204(2), 473–504 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, S.Y.S.: Conformal deformation to scalar flat metrics with constant mean curvature on the boundary in higher dimensions, arXiv:0912.1302
  10. 10.
    Chen, X., Lai, M., Wang, F.: Escobar-Yamabe compactifications for Poincaré-Einstein manifolds and rigidity theorems, arXiv:1712.02540
  11. 11.
    Cherrier, P.: Problèmes de Neumann non linéaires sur les variétés riemanniennes. J. Funct. Anal. 57(2), 154–206 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chruściel, P., Delay, E.: Non-singular space-time with a negative cosmological constant II. Static solutions of the Einstein-Maxwell equations. Lett. Math. Phys 107(8), 1391–1407 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chruściel, P., Delay, E., Lee, J.M., Skinner, D.N.: Boundary regularity of conformally compact Einstein metrics. J. Differ. Geom. 69(1), 111–136 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chruściel, P.T., Herzlich, M.: The mass of asymptotically hyperbolic Riemannian manifolds. Pac. J. Math 212, 231–264 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dutta, S., Javaheri, M.: Rigidity of conformally compact manifolds with the round sphere as conformal infinity. Adv. Math. 224, 525–538 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Escobar, J.F.: Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary. Ann. Math. 136(1), 1–50 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Escobar, J.F., Addendum: Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary. Ann. Math. (2) 136(1), 1–50 (1992), Ann. Math. (2) 139(2), 749–750 (1994)Google Scholar
  18. 18.
    Escobar, J.F.: The Yamabe problem on manifolds with boundary. J. Differ. Geom. 35(1), 21–84 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Graham, C.R.: Volume renormalization for singular Yamabe metrics. Proc. Am. Math. Soc. 145(4), 1781–1792 (2017)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Graham, C.R., Lee, J.M.: Einstein metrics with prescribed conformal infinity on the ball. Adv. Math. 87, 186–225 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gursky, M.J., Han, Q.: Non-existence of Poincaré-Einstein manifolds with prescribed conformal infinity. Geom. Funct. Anal. 27(4), 863–879 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gursky, M.J., Han, Q., Stolz, S.: An invariant related to the existence of conformally compact Einstein fillings, arXiv:1801.04474
  23. 23.
    Hijazi, O., Montiel, S.: Supersymmetric rigidity of asymptotically locally hyperbolic manifolds. Int. J. Math. 25(3), 1450020 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lee, J.M., Parker, T.H.: The Yamabe problem. Bull. Am. Math. Soc. 17(1), 37–91 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Loewner, C., Nirenberg, L.: Partial Differential Equations Invariant Under Conformal and Projective Transformations, Contributions to Analysis (a collection of papers dedicated to Lipman Bers), pp. 245–272. Academic Press, New York (1994)Google Scholar
  26. 26.
    Li, G., Qing, J., Shi, Y.: Gap phenomena and curvature estimates for conformally compact Einstein manifolds. Trans. Am. Math. Soc. 369(6), 4385–4413 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Marques, F.C.: Existence results for the Yamabe problem on manifolds with boundary. Indiana Univ. Math. J. 54(6), 1599–1620 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Marques, F.C.: Conformal deformations to scalar-flat metrics with constant mean curvature on the boundary. Commun. Anal. Geom. 15(2), 381–405 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Mayer, M., Ndiaye, C.B.: Barycenter technique and the Riemann mapping problem of Cherrier-Escobar. J. Differ. Geom. 107(3), 519–560 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Obata, M.: The conjectures on conformal transformations of Riemannian manifolds. J. Differ. Geom 6, 247–258 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Qing, J.: On the rigidity for conformally compact Einstein manifolds. Int. Math. Res. Not. 21, 1141–1153 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Wang, X.: The mass of asymptotically hyperbolic manifolds. J. Differ. Geom. 57, 273–299 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys. 80(3), 381–402 (1981)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Wang, Q., Xia, C.: Sharp bounds for the first non-zero Stekloff eigenvalues. J. Funct. Anal. 257, 2635–2644 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques R. SalemUMR 6085 CNRS-Université de RouenSaint-Étienne-du-RouvrayFrance

Personalised recommendations