The Log Term in the Bergman and Szegő 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 $M=M^3:=\partial D$, and the asymptotic expansion of the Bergman kernel on the diagonal

where $\rho >0$ is a Fefferman defining equation for $D$. The Ramadanov Conjecture states that if the log term ${\psi }_B$ vanishes to infinite order on $M$, then $M$ is locally spherical. In ${\mathbb{C}}^2$, the validity of this conjecture is known and follows from work of Boutet de Monvel, Burns, and Graham; indeed, it suffices that ${\psi }_B=O({\rho }^2)$ locally on $M$ to conclude that $M$ is locally spherical. On the other hand, it is also known that the boundary values alone 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. The main result in this paper, however, 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 Szegő kernel.