Advertisement

The Journal of Geometric Analysis

, Volume 24, Issue 4, pp 2135–2185 | Cite as

Asymptotics of ACH-Einstein Metrics

  • Yoshihiko Matsumoto
Article

Abstract

We study the boundary asymptotics of ACH metrics which are formally Einstein. In terms of the partially integrable almost CR structure induced on the boundary at infinity, existence and uniqueness of such formal asymptotic expansions are studied. It is shown that there always exist formal solutions to the Einstein equation if we allow logarithmic terms, and that a local CR-invariant tensor arises as the obstruction to the existence of a log-free solution. Some properties of this new CR invariant, the CR obstruction tensor, are discussed.

Keywords

ACH metrics The Einstein equation Partially integrable almost CR manifolds The CR obstruction tensor 

Mathematics Subject Classification

32V05 53A55 

References

  1. 1.
    Barletta, E., Dragomir, S.: Differential equations on contact Riemannian manifolds. Ann. Scoula Norm. Super. Pisa, Cl. Sci. (4) 30, 63–95 (2001) MathSciNetzbMATHGoogle Scholar
  2. 2.
    Blair, D.E., Dragomir, S.: Pseudohermitian geometry on contact Riemannian manifolds. Rend. Mat. Appl. (7) 22, 275–341 (2002) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Biquard, O.: Métriques d’Einstein asymptotiquement symétriques. Astérisque 265 (2000). vi+109 pp. Google Scholar
  4. 4.
    Biquard, O., Herzlich, M.: A Burns-Epstein invariant for ACHE 4-manifolds. Duke Math. J. 126, 53–100 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Biquard, O., Herzlich, M.: Analyse sur un demi-espace hyperbolique et poly-homogénéité locale. Preprint. arXiv:1002.4106v1
  6. 6.
    Branson, T.P.: Sharp inequalities, the functional determinant, and the complementary series. Trans. Am. Math. Soc. 347, 3671–3742 (1995) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Čap, A., Schichl, H.: Parabolic geometries and canonical Cartan connections. Hokkaido Math. J. 29, 453–505 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Čap, A., Slovák, J.: Parabolic Geometries I: Background and General Theory. Mathematical Surveys and Monographs, vol. 154 (2009). x+628 pp. Google Scholar
  9. 9.
    Cheng, S.Y., Yau, S.T.: On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman’s equation. Commun. Pure Appl. Math. 33, 507–544 (1980) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Epstein, C.L., Melrose, R.B., Mendoza, G.A.: Resolvent of the Laplacian on strictly pseudoconvex domains. Acta Math. 167, 1–106 (1991) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Farris, F.: An intrinsic construction of Fefferman’s CR metric. Pac. J. Math. 123, 33–45 (1986) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fefferman, C.L.: Monge–Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains. Ann. Math. (2) 103, 395–416 (1976). Erratum 104, 393–394 (1976) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fefferman, C., Graham, C.R.: Conformal invariants. In: The Mathematical Heritage of Élie Cartan (Lyon, 1984), Astérisque, 1985, hors série, pp. 95–116 Google Scholar
  14. 14.
    Fefferman, C., Graham, C.R.: The Ambient Metric. Annals of Mathematics Studies, vol. 178, Princeton University Press, Princeton (2012). x+113 pp. zbMATHGoogle Scholar
  15. 15.
    Gover, A.R., Graham, C.R.: CR invariant powers of the sub-Laplacian. J. Reine Angew. Math. 583, 1–27 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Graham, C.R.: Higher asymptotics of the complex Monge–Ampère equation. Compos. Math. 64, 133–155 (1987) zbMATHGoogle Scholar
  17. 17.
    Graham, C.R., Hirachi, K.: The ambient obstruction tensor and Q-curvature. In: AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries. IRMA Lect. Math. Theo. Phys., vol. 8, pp. 59–71. Eur. Math. Soc., Zürich (2005) Google Scholar
  18. 18.
    Graham, C.R., Lee, J.M.: Einstein metrics with prescribed conformal infinity on the ball. Adv. Math. 87, 186–225 (1991) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Graham, C.R., Zworski, M.: Scattering matrix in conformal geometry. Invent. Math. 152, 89–118 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Guillarmou, C., Sá Barreto, A.: Scattering and inverse scattering on ACH manifolds. J. Reine Angew. Math. 622, 1–55 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hislop, P.D., Perry, P.A., Tang, S.-H.: CR-invariants and the scattering operator for complex manifolds with boundary. Anal. PDE 1, 197–227 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kuranishi, M.: PDEs associated to the CR embedding theorem. In: Analysis and Geometry in Several Complex Variables, Katata, 1997. Trends Math., pp. 129–157. Birkhäuser, Boston (1999) CrossRefGoogle Scholar
  23. 23.
    Lee, J.M.: Pseudo-Einstein structures on CR manifolds. Am. J. Math. 110, 157–178 (1988) CrossRefzbMATHGoogle Scholar
  24. 24.
    Lee, J., Melrose, R.: Boundary behaviour of the complex Monge–Ampère equation. Acta Math. 148, 159–192 (1982) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Mizner, R.I.: Almost CR structures, f-structures, almost product structures and associated connections. Rocky Mt. J. Math. 23, 1337–1359 (1993) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Seshadri, N.: Approximately Einstein ACH metrics. Bull. Soc. Math. Fr. 137, 63–91 (2009) MathSciNetzbMATHGoogle Scholar
  27. 27.
    Tanaka, N.: A Differential Geometric Study on Strongly Pseudo-Convex Manifolds. Lectures in Mathematics, vol. 9. Department of Mathematics, Kyoto Univ., Tokyo (1975) zbMATHGoogle Scholar
  28. 28.
    Tanno, S.: Variational problems on contact Riemannian manifolds. Trans. Am. Math. Soc. 314, 349–379 (1989) MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Webster, S.M.: Pseudo-Hermitian structures on a real hypersurface. J. Differ. Geom. 13, 25–41 (1978) MathSciNetzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2013

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

Personalised recommendations