Inhomogeneous Ambient Metrics

  • C. Robin Graham
  • Kengo Hirachi
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 144)


The ambient metric, introduced in [FG1], has proven to be an important object in conformal geometry. To a manifold M of dimension n with a conformai class of metrics [g] of signature (p, q) it associates a diffeomorphism class of formal expansions of metrics \( \tilde g \) of signature (p + 1, q + 1) on a space Open image in new window of dimension n + 2. This generalizes the realization of the conformal sphere Sn as the space of null lines for a quadratic form of signature (n + 1, 1), with associated Minkowski metric \( \tilde g \) on ℝn+2. The ambient space Open image in new window carries a family of dilations with respect to which \( \tilde g \) is homogeneous of degree 2. The other conditions determining \( \tilde g \) are that it be Ricci-flat and satisfy an initial condition specified by the conformal class [g].


Normal Form Bergman Kernel Conformal Class Smooth Part Complete Contraction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A1]
    S. Alexakis, On conformally invariant differential operators in odd dimensions, Proc. Natl. Acad. Sci. USA 100 (2003), 4409–4410.MATHCrossRefMathSciNetGoogle Scholar
  2. [A2]
    S. Alexakis, On conformally invariant differential operators, math.DG/0608771.Google Scholar
  3. [BEGo]
    T.N. Bailey, M.G. Eastwood, AND A.R. Gover, Thomas’s structure bundle for conformal, projective and related structures, Rocky Mountain J. Math. 24 (1994), 1191–1217.MATHCrossRefMathSciNetGoogle Scholar
  4. [BEGr]
    T.N. Bailey, M.G. Eastwood, AND C.R. Graham, Invariant theory for conformal and CR geometry, Ann. Math. 139 (1994), 491–552.MATHCrossRefMathSciNetGoogle Scholar
  5. [BG]
    T.N. Bailey AND A.R. Gover, Exceptional invariants in the parabolic invariant theory of conformal geometry, Proc. A.M.S. 123(1995), 2535–2543.MATHCrossRefMathSciNetGoogle Scholar
  6. [CG]
    A. Čap AND A.R. Gover, Standard tractors and the conformal ambient metric construction, Ann. Global Anal. Geom. 24 (2003), 231–259.MATHCrossRefMathSciNetGoogle Scholar
  7. [EG]
    M.G. Eastwood AND C.R. Graham, Invariants of conformal densities, Duke Math. J. 63 (1991), 633–671.MATHCrossRefMathSciNetGoogle Scholar
  8. [Fl]
    C. Fefferman, Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains, Ann. Math. 103 (1976), 395–416; Correction: Ann. Math. 104 (1976), 393–394.CrossRefMathSciNetGoogle Scholar
  9. [F2]
    C. Fefferman, Parabolic invariant theory in complex analysis, Adv. Math. 31 (1979), 131–262.MATHCrossRefMathSciNetGoogle Scholar
  10. [FG1]
    C. Fefferman AND C.R. Graham, Conformai invariants, in The mathematical heritage of Élie Cartan (Lyon, 1984). Astérisque, 1985, Numero Hors Serie, pp. 95–116.Google Scholar
  11. [FG2]
    C. Fefferman AND C.R. Graham, The ambient metric, in preparation.Google Scholar
  12. [Go]
    A.R. Gover, Invariant theory and calculus for conformai geometries, Adv. Math. 163 (2001), 206–257.MATHCrossRefMathSciNetGoogle Scholar
  13. [Grl]
    C.R. Graham, Scalar boundary invariants and the Bergman kernel, Complex Analysis II, Proceedings, Univ. of Maryland 1985–86, Springer Lecture Notes 1276, 108–135.Google Scholar
  14. [Gr2]
    C.R. Graham, Higher asymptotics of the complex Monge-Ampère equation, Comp. Math. 64 (1987), 133–155.MATHGoogle Scholar
  15. [H]
    K. Hirachi, Construction of boundary invariants and the logarithmic singularity of the Bergman kernel, Ann. Math. 151 (2000), 151–191.MATHCrossRefMathSciNetGoogle Scholar
  16. [K]
    S. Kichenassamy, On a conjecture of Fefferman and Graham, Adv. Math. 184 (2004), 268–288.MATHCrossRefMathSciNetGoogle Scholar
  17. [L]
    J. Lee, Higher asymptotics of the complex Monge-Ampère equation and geometry of CR-manifolds, MIT Ph.D. thesis, 1982.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • C. Robin Graham
    • 1
  • Kengo Hirachi
    • 2
  1. 1.Department of MathematicsUniversity of WashingtonSeattle
  2. 2.Graduate School of Mathematical SciencesUniversity of TokyoTokyoJapan

Personalised recommendations