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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)

  • András Vasy
Article

Abstract

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 

Notes

Acknowledgements

A.V. is very grateful to Maciej Zworski, Richard Melrose, Semyon Dyatlov, Mihalis Dafermos, Gunther Uhlmann, Jared Wunsch, Rafe Mazzeo, Kiril Datchev, Colin Guillarmou, Andrew Hassell, Dean Baskin and Peter Hintz for very helpful discussions, for their enthusiasm for this project and for carefully reading parts of this manuscript. Special thanks are due to Semyon Dyatlov in this regard who noticed an incomplete argument in an earlier version of this paper in holomorphy considerations, and to Mihalis Dafermos, who urged the author to supply details to the argument at the end of Sect. 6, which resulted in the addition of Sect. 3.3, as well as the part of Sect. 3.2 and Sect. 7 covering the complex absorption, to the main body of the argument, and that of Sect. 2.7 for stability considerations. A.V. is also very grateful to the three anonymous referees whose detailed reports improved the manuscript significantly.

A.V. gratefully acknowledges partial support from the National Science Foundation under grants number DMS-0801226 and DMS-1068742 and from a Chambers Fellowship at Stanford University, as well as the hospitality of Mathematical Sciences Research Institute in Berkeley. S.D. is grateful for partial support from the National Science Foundation under grant number DMS-0654436.

References

  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) MathSciNetCrossRefzbMATHGoogle 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) zbMATHGoogle Scholar
  4. 4.
    Blue, P., Soffer, A.: Phase space analysis on some black hole manifolds. J. Funct. Anal. 256(1), 1–90 (2009) MathSciNetCrossRefzbMATHGoogle 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) CrossRefzbMATHGoogle Scholar
  6. 6.
    Borthwick, D., Perry, P.: Scattering poles for asymptotically hyperbolic manifolds. Trans. Am. Math. Soc. 354(3), 1215–1231 (2002) (electronic) MathSciNetCrossRefzbMATHGoogle 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) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Carter, B.: Global structure of the Kerr family of gravitational fields. Phys. Rev. 174, 1559–1571 (1968) CrossRefzbMATHGoogle 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) zbMATHGoogle 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) MathSciNetCrossRefzbMATHGoogle 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) MathSciNetCrossRefzbMATHGoogle 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) MathSciNetzbMATHGoogle 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) CrossRefzbMATHGoogle 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) MathSciNetCrossRefzbMATHGoogle 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) MathSciNetCrossRefzbMATHGoogle 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) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Dyatlov, S.: Asymptotic distribution of quasi-normal modes for Kerr-de Sitter black holes. Ann. Henri Poincaré 13, 1101–1166 (2012) MathSciNetCrossRefzbMATHGoogle 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) MathSciNetCrossRefzbMATHGoogle 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) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Friedlander, F.G.: Radiation fields and hyperbolic scattering theory. Math. Proc. Camb. Philos. Soc. 88(3), 483–515 (1980) CrossRefzbMATHGoogle Scholar
  26. 26.
    Graham, C.R., Zworski, M.: Scattering matrix in conformal geometry. Invent. Math. 152(1), 89–118 (2003) MathSciNetCrossRefzbMATHGoogle 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) MathSciNetCrossRefzbMATHGoogle 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) MathSciNetCrossRefzbMATHGoogle 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) MathSciNetzbMATHGoogle 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) MathSciNetCrossRefzbMATHGoogle 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) MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Lindblad, H., Rodnianski, I.: The global stability of Minkowski space-time in harmonic gauge. Ann. Math. 171(3), 1401–1477 (2010) MathSciNetCrossRefzbMATHGoogle 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) MathSciNetCrossRefzbMATHGoogle 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) MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Mazzeo, R.: Elliptic theory of differential edge operators. I. Commun. Partial Differ. Equ. 16(10), 1615–1664 (1991) MathSciNetCrossRefzbMATHGoogle 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) zbMATHGoogle Scholar
  44. 44.
    Nonnenmacher, S., Zworski, M.: Quantum decay rates in chaotic scattering. Acta Math. 203(2), 149–233 (2009) MathSciNetCrossRefzbMATHGoogle 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) MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Sá Barreto, A., Zworski, M.: Distribution of resonances for spherical black holes. Math. Res. Lett. 4(1), 103–121 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Springer, Berlin (1987) CrossRefzbMATHGoogle 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) MathSciNetzbMATHGoogle Scholar
  51. 51.
    Taylor, M.E.: Partial Differential Equations. Basic Theory. Texts in Applied Mathematics, vol. 23. Springer, New York (1996) zbMATHGoogle 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) MathSciNetCrossRefzbMATHGoogle 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) MathSciNetzbMATHGoogle Scholar
  57. 57.
    Vasy, A., Zworski, M.: Semiclassical estimates in asymptotically Euclidean scattering. Commun. Math. Phys. 212, 205–217 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Vodev, G.: Local energy decay of solutions to the wave equation for nontrapping metrics. Ark. Mat. 42(2), 379–397 (2004) MathSciNetCrossRefzbMATHGoogle 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) MathSciNetCrossRefzbMATHGoogle 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) MathSciNetCrossRefzbMATHGoogle 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

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