Inventiones mathematicae

, Volume 194, Issue 2, pp 381–513 | Cite as

Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov)



In this paper we develop a general, systematic, microlocal framework for the Fredholm analysis of non-elliptic problems, including high energy (or semiclassical) estimates, which is stable under perturbations. This framework, described in Sect. 2, resides on a compact manifold without boundary, hence in the standard setting of microlocal analysis.

Many natural applications arise in the setting of non-Riemannian b-metrics in the context of Melrose’s b-structures. These include asymptotically de Sitter-type metrics on a blow-up of the natural compactification, Kerr-de Sitter-type metrics, as well as asymptotically Minkowski metrics.

The simplest application is a new approach to analysis on Riemannian or Lorentzian (or indeed, possibly of other signature) conformally compact spaces (such as asymptotically hyperbolic or de Sitter spaces), including a new construction of the meromorphic extension of the resolvent of the Laplacian in the Riemannian case, as well as high energy estimates for the spectral parameter in strips of the complex plane. These results are also available in a follow-up paper which is more expository in nature (Vasy in Uhlmann, G. (ed.) Inverse Problems and Applications. Inside Out II, 2012).

The appendix written by Dyatlov relates his analysis of resonances on exact Kerr-de Sitter space (which then was used to analyze the wave equation in that setting) to the more general method described here.

Mathematics Subject Classification

35L05 35P25 58J47 83C57 


  1. 1.
    Baskin, D.: A parametrix for the fundamental solution of the Klein-Gordon equation on asymptotically de Sitter spaces. J. Funct. Anal. 259(7), 1673–1719 (2010) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bieri, L.: Part I: Solutions of the Einstein vacuum equations. In: Extensions of the Stability Theorem of the Minkowski Space in General Relativity. AMS/IP Studies in Advanced Mathematics, vol. 45, pp. 1–295. Am. Math. Soc., Providence (2009) Google Scholar
  3. 3.
    Bieri, L., Zipser, N.: Extensions of the Stability Theorem of the Minkowski Space in General Relativity. AMS/IP Studies in Advanced Mathematics, vol. 45. Am. Math. Soc., Providence (2009) MATHGoogle Scholar
  4. 4.
    Blue, P., Soffer, A.: Phase space analysis on some black hole manifolds. J. Funct. Anal. 256(1), 1–90 (2009) MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bony, J.-F., Häfner, D.: Decay and non-decay of the local energy for the wave equation on the de Sitter-Schwarzschild metric. Commun. Math. Phys. 282(3), 697–719 (2008) CrossRefMATHGoogle Scholar
  6. 6.
    Borthwick, D., Perry, P.: Scattering poles for asymptotically hyperbolic manifolds. Trans. Am. Math. Soc. 354(3), 1215–1231 (2002) (electronic) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cardoso, F., Vodev, G.: Uniform estimates of the resolvent of the Laplace-Beltrami operator on infinite volume Riemannian manifolds. II. Ann. Henri Poincaré 3(4), 673–691 (2002) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Carter, B.: Global structure of the Kerr family of gravitational fields. Phys. Rev. 174, 1559–1571 (1968) CrossRefMATHGoogle Scholar
  9. 9.
    Christodoulou, D., Klainerman, S.: The Global Nonlinear Stability of the Minkowski Space. Princeton Mathematical Series, vol. 41. Princeton University Press, Princeton (1993) MATHGoogle Scholar
  10. 10.
    Dafermos, M., Rodnianski, I.: A proof of Price’s law for the collapse of a self-gravitating scalar field. Invent. Math. 162(2), 381–457 (2005) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Dafermos, M., Rodnianski, I.: The wave equation on Schwarzschild-de Sitter space times. arXiv:0709.2766 (2007)
  12. 12.
    Dafermos, M., Rodnianski, I.: The red-shift effect and radiation decay on black hole spacetimes. Commun. Pure Appl. Math. 62, 859–919 (2009) MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Dafermos, M., Rodnianski, I.: The black hole stability problem for linear scalar perturbations. arXiv:1010.5137 (2010)
  14. 14.
    Dafermos, M., Rodnianski, I.: Decay of solutions of the wave equation on Kerr exterior space-times I–II: The cases of |a|≪m or axisymmetry. arXiv:1010.5132 (2010)
  15. 15.
    Datchev, K., Vasy, A.: Gluing semiclassical resolvent estimates via propagation of singularities. Int. Math. Res. Not. 2012, 5409–5443 (2012) MathSciNetMATHGoogle Scholar
  16. 16.
    Datchev, K., Vasy, A.: Propagation through trapped sets and semiclassical resolvent estimates. Ann. Inst. Fourier, to appear. arXiv:1010.2190
  17. 17.
    Dimassi, M., Sjöstrand, J.: Spectral Asymptotics in the Semi-classical Limit. London Mathematical Society Lecture Note Series, vol. 268. Cambridge University Press, Cambridge (1999) CrossRefMATHGoogle Scholar
  18. 18.
    Donninger, R., Schlag, W., Soffer, A.: A proof of Price’s law on Schwarzschild black hole manifolds for all angular momenta. Adv. Math. 226(1), 484–540 (2011) MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Dyatlov, S.: Exponential energy decay for Kerr–de Sitter black holes beyond event horizons. Math. Res. Lett. 18(5), 1023–1035 (2011) MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Dyatlov, S.: Quasi-normal modes and exponential energy decay for the Kerr-de Sitter black hole. Commun. Math. Phys. 306(1), 119–163 (2011) MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Dyatlov, S.: Asymptotic distribution of quasi-normal modes for Kerr-de Sitter black holes. Ann. Henri Poincaré 13, 1101–1166 (2012) MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Fefferman, C., Graham, C.R.: Conformal invariants. In: The Mathematical Heritage of Élie Cartan, Lyon, 1984. Astérisque Numero Hors Serie, pp. 95–116 (1985) Google Scholar
  23. 23.
    Finster, F., Kamran, N., Smoller, J., Yau, S.-T.: Decay of solutions of the wave equation in the Kerr geometry. Commun. Math. Phys. 264(2), 465–503 (2006) MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Finster, F., Kamran, N., Smoller, J., Yau, S.-T.: Linear waves in the Kerr geometry: a mathematical voyage to black hole physics. Bull., New Ser., Am. Math. Soc. 46(4), 635–659 (2009) MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Friedlander, F.G.: Radiation fields and hyperbolic scattering theory. Math. Proc. Camb. Philos. Soc. 88(3), 483–515 (1980) CrossRefMATHGoogle Scholar
  26. 26.
    Graham, C.R., Zworski, M.: Scattering matrix in conformal geometry. Invent. Math. 152(1), 89–118 (2003) MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Guillarmou, C., Hassell, A., Sikora, A.: Resolvent at low energy III: The spectral measure. arXiv:1009.3084 (2010)
  28. 28.
    Guillarmou, C.: Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds. Duke Math. J. 129(1), 1–37 (2005) MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Haber, N., Vasy, A.: Propagation of singularities around a Lagrangian submanifold of radial points. arXiv:1110.1419 (2011)
  30. 30.
    Hassell, A., Melrose, R.B., Vasy, A.: Spectral and scattering theory for symbolic potentials of order zero. Adv. Math. 181, 1–87 (2004) MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Hassell, A., Melrose, R.B., Vasy, A.: Microlocal propagation near radial points and scattering for symbolic potentials of order zero. Anal. Partial Differ. Equ. 1, 127–196 (2008) MathSciNetMATHGoogle Scholar
  32. 32.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators, vols. 1–4. Springer, Berlin (1983) CrossRefGoogle Scholar
  33. 33.
    Kay, B.S., Wald, R.M.: Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation 2-sphere. Class. Quantum Gravity 4(4), 893–898 (1987) MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Lindblad, H., Rodnianski, I.: Global existence for the Einstein vacuum equations in wave coordinates. Commun. Math. Phys. 256(1), 43–110 (2005) MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Lindblad, H., Rodnianski, I.: The global stability of Minkowski space-time in harmonic gauge. Ann. Math. 171(3), 1401–1477 (2010) MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Marzuola, J., Metcalfe, J., Tataru, D., Tohaneanu, M.: Strichartz estimates on Schwarzschild black hole backgrounds. Commun. Math. Phys. 293(1), 37–83 (2010) MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Mazzeo, R., Melrose, R.B.: Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. J. Funct. Anal. 75, 260–310 (1987) MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Mazzeo, R.: Elliptic theory of differential edge operators. I. Commun. Partial Differ. Equ. 16(10), 1615–1664 (1991) MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Melrose, R.B.: Spectral and Scattering Theory for the Laplacian on Asymptotically Euclidean Spaces. Dekker, New York (1994) Google Scholar
  40. 40.
    Melrose, R.B., Sá Barreto, A., Vasy, A.: Asymptotics of solutions of the wave equation on de Sitter-Schwarzschild space. arXiv:0811.2229 (2008)
  41. 41.
    Melrose, R.B., Sá Barreto, A., Vasy, A.: Analytic continuation and semiclassical resolvent estimates on asymptotically hyperbolic spaces. arXiv:1103.3507 (2011)
  42. 42.
    Melrose, R.B., Vasy, A., Wunsch, J.: Diffraction of singularities for the wave equation on manifolds with corners. Astérisque, to appear. arXiv:0903.3208 (2009)
  43. 43.
    Melrose, R.B.: The Atiyah-Patodi-Singer Index Theorem. Research Notes in Mathematics, vol. 4. AK Peters, Wellesley (1993) MATHGoogle Scholar
  44. 44.
    Nonnenmacher, S., Zworski, M.: Quantum decay rates in chaotic scattering. Acta Math. 203(2), 149–233 (2009) MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Polarski, D.: On the Hawking effect in de Sitter space. Class. Quantum Gravity 6(5), 717–722 (1989) MathSciNetCrossRefGoogle Scholar
  46. 46.
    Sá Barreto, A., Wunsch, J.: The radiation field is a Fourier integral operator. Ann. Inst. Fourier (Grenoble) 55(1), 213–227 (2005) MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Sá Barreto, A., Zworski, M.: Distribution of resonances for spherical black holes. Math. Res. Lett. 4(1), 103–121 (1997) MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Springer, Berlin (1987) CrossRefMATHGoogle Scholar
  49. 49.
    Tataru, D.: Local decay of waves on asymptotically flat stationary spacetimes. arXiv:0910.5290 (2009)
  50. 50.
    Tataru, D., Tohaneanu, M.: A local energy estimate on Kerr black hole backgrounds. Int. Math. Res. Not. 2011(2), 248–292 (2011) MathSciNetMATHGoogle Scholar
  51. 51.
    Taylor, M.E.: Partial Differential Equations. Basic Theory. Texts in Applied Mathematics, vol. 23. Springer, New York (1996) MATHGoogle Scholar
  52. 52.
    Vasy, A.: Propagation of singularities in three-body scattering. Astérisque 262 (2000) Google Scholar
  53. 53.
    Vasy, A.: The wave equation on asymptotically de Sitter-like spaces. Adv. Math. 223, 49–97 (2010) MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Vasy, A.: Microlocal analysis of asymptotically hyperbolic spaces and high energy resolvent estimates. In: Uhlmann, G. (ed.) Inverse Problems and Applications. Inside Out II. MSRI Publications, vol. 60. Cambridge University Press, Cambridge (2012) Google Scholar
  55. 55.
    Vasy, A.: Analytic continuation and high energy estimates for the resolvent of the Laplacian on forms on asymptotically hyperbolic spaces. arXiv:1206.5454 (2012)
  56. 56.
    Vasy, A.: The wave equation on asymptotically Anti-de Sitter spaces. Anal. Partial Differ. Equ. 5, 81–144 (2012) MathSciNetMATHGoogle Scholar
  57. 57.
    Vasy, A., Zworski, M.: Semiclassical estimates in asymptotically Euclidean scattering. Commun. Math. Phys. 212, 205–217 (2000) MathSciNetCrossRefMATHGoogle Scholar
  58. 58.
    Vodev, G.: Local energy decay of solutions to the wave equation for nontrapping metrics. Ark. Mat. 42(2), 379–397 (2004) MathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    Wald, R.M.: Note on the stability of the Schwarzschild metric. J. Math. Phys. 20(6), 1056–1058 (1979) MathSciNetCrossRefGoogle Scholar
  60. 60.
    Wang, F.: Radiation field for vacuum Einstein equation. PhD thesis, Massachusetts Institute of Technology (2010) Google Scholar
  61. 61.
    Wunsch, J., Zworski, M.: Resolvent estimates for normally hyperbolic trapped sets. Ann. Henri Poincaré 12(7), 1349–1385 (2011) MathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    Yagdjian, K., Galstian, A.: Fundamental solutions for the Klein-Gordon equation in de Sitter spacetime. Commun. Math. Phys. 285(1), 293–344 (2009) MathSciNetCrossRefMATHGoogle Scholar
  63. 63.
    Zworski, M.: Lectures on Semiclassical Analysis. Am. Math. Soc., Providence (2012) Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA

Personalised recommendations