The Log term in the Bergman and Szeg\H o kernels in strictly pseudoconvex domains in $\mathbb C^2$

TL;DR

This paper shows that for strictly pseudoconvex domains in a7^2 with transverse symmetry, the global vanishing of the log term in the Bergman and Szegf3 kernels implies the boundary is locally spherical, linking kernel behavior to CR geometry.

Contribution

It establishes a new global geometric implication from the vanishing of the log term in the Bergman and Szegf3 kernels under transverse symmetry in a7^2 domains.

Findings

Vanishing of the log term implies local sphericity under symmetry.

Results extend understanding of kernel asymptotics and CR geometry.

Connects kernel asymptotics with geometric properties of boundaries.

Abstract

In this paper, we consider bounded strictly pseudoconvex domains $D\subset \mathbb C^2$ with smooth boundary $M=M^3:=\partial D$. If we consider the asymptotic expansion of the Bergman kernel on the diagonal $$ K_B\sim \frac{\phi_B}{\rho^{n+1}}+\psi_B\log\rho, $$ where $\rho>0$ is a Fefferman defining equation for $D$, then it is well known that the trace of the log term $b\psi_B:=(\psi_B)|_M$ on $M$ does not determine the CR geometry of $M$ locally; e.g., the vanishing of $b\psi_B$ 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 $D\subset \mathbb C^2$ is assumed to have transverse symmetry, then the global vanishing of $b\psi_B$ on $M$ implies that $M$ is locally spherical. A similar result is proved for the Szeg\H o kernel.