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 


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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

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