The Log term in the Bergman and Szeg\H o kernels in strictly pseudoconvex domains in \mathbb C²

2016/06/19 by Peter Ebenfelt, Ebenfelt, Peter
Mathematics · #32T15 #32V15 #Algebraic and Geometric Analysis #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1606.05871

openalex publication_date 2016/06/19 · openalex created_date 2016/07/22 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider bounded strictly pseudoconvex domains D⊂ \mathbb C² with smooth boundary M=M³:=∂ D. If we consider the asymptotic expansion of the Bergman kernel on the diagonal K_B∼ \fracϕ_Bρⁿ⁺¹+ψ_Blogρ, where ρ>0 is a Fefferman defining equation for D, then it is well known that the trace of the log term bψ_B:=(ψ_B)|_M on M does not determine the CR geometry of M locally; e.g., the vanishing of bψ_B on an open subset of M does not imply that M is locally spherical there. Nevertheless, the main result in this paper is that if D⊂ \mathbb C² is assumed to have transverse symmetry, then the global vanishing of bψ_B on M implies that M is locally spherical. A similar result is proved for the Szeg\H o kernel.

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