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

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 $$K_B\sim \frac{{\varphi }_B}{{\rho }^{n+1}}+{\psi }_B\log \rho ,$$ 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.

Cited by in corpus (1)

• Bounded strictly pseudoconvex domains in ${\mathbb{C}}^2$ with obstruction flat boundary