Ambient Metric Construction of CR Invariant Differential Operators

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


These notes are based on my lectures at IMA, in which I tried to explain basic ideas of the ambient metric construction by studying the Szegö kernel of the sphere. The ambient metric was introduced in Fefferman [F] in his program of describing the boundary asymptotic expansion of the Bergman kernel of strictly pseudoconvex domain. This can be seen as an analogy of the description of the heat kernel asymptotic in terms of local Riemannian invariants. The counterpart of the Riemannian invariants for the Bergman kernel is invariants of the CR structure of the boundary. Thus the program consists of two parts:
  1. (1)

    Construct local invariants of CR structures;

  2. (2)

    Prove that (1) gives all invariants by using the invariant theory.

In the case of the Szegö kernel, (1) is replaced by the construction of local invariants of the Levi form that are invariant under scaling by CR pluriharmonic functions. We formulate the class of invariants in Sections 2 and 3. To simplify the presentation, we confine ourself to the case of the sphere in ℂn. It is the model case of the ambient metric construction and the basic tools already appears in this setting. We construct invariants (formulated as CR invariant differential operators) by using the ambient space in Section 4 and then explain, in Section 5, how to prove that we have got all.


Volume Form Invariant Theory Formal Power Series Ambient Space Real Hypersurface 
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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Kengo Hirachi
    • 1
  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

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