DIFFERENTIAL GEOMETRY/PDE SEMINAR

Thursday, November 20, 2003 Padelford C-36 1:30 pm

Volume of strictly pseudoconvex domains with respect to the Bergman volume form

Kengo Hirachi (University of Tokyo)

Recently there has been much progress made on the renormalized volume for conformally compact Einstein manifolds in the context of the AdS/CFT correspondence, where the volume asymptotic expansion of complete Einstein manifolds is related to conformal invariants of the boundary at infinity.

In this talk, I consider an analogous problem in complex/CR geometry. For a bounded strictly pseudoconvex domain D in a complex manifold, we consider the Bergman volume form BdV. Take a defining function r > 0 of D and consider the volume V (D_{t}) of the relatively compact subset D_{t} = {r > t} with respect to BdV. It then turns out that V (D_{t}) diverges as t → 0 and admits an asymptotic expansion as in the case of conformally compact Einstein manifolds. I show that the coefficient of log t in the expansion is a biholomorphic invariant of D and is given by the integral of a local pseudohermitian invariant on the boundary ∂D; I will also discuss the relation between this pseudohermitian invariant and Branson’s Q-curvature.

For more information about this seminar, visit the DG/PDE Seminar Web page (from the Math Department home page, www.math.washington.edu, follow the link Seminars, Colloquia, and Conferences).

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