Analytic regularity for the Bergman kernel

Françis, Gabor; Hanges, Nicholas

Journées équations aux dérivées partielles (1998), article no. 5, 11 p.

Let $\Omega \subset {ℝ}^2$ be a bounded, convex and open set with real analytic boundary. Let $T_{\Omega }\subset {ℂ}^2$ be the tube with base $\Omega ,$ and let $ℬ$ be the Bergman kernel of $T_{\Omega }$. If $\Omega$ is strongly convex, then $ℬ$ is analytic away from the boundary diagonal. In the weakly convex case this is no longer true. In this situation, we relate the off diagonal points where analyticity fails to the Trèves curves. These curves are symplectic invariants which are determined by the CR structure of the boundary of $T_{\Omega }$. Note that Trèves curves exist only when $\Omega$ has at least one weakly convex boundary point.

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