The Log Term in the Bergman and Szegő Kernels in Strictly Pseudoconvex Domains in $\mathbb C^2$

Ebenfelt, Peter

Documenta mathematica, Tome 23 (2018), pp. 1659-1676.

Résumé

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 $$K_B\sim \frac{\phi_{B}}{\rho^{n+1}}+\psi_B\log\rho,$$ 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.

Détail

Classification : 32T15, 32V15
Keywords: Bergman and Szegő kernels, log term on boundary, Cartan curvature