Rigidity theorem by the minimal point of the Bergman kernel

Abstract

We use the Suita conjecture (now a theorem) to prove that for any domain $\Omega \subset \mathbb{C}$ its Bergman kernel $K(\cdot ,\cdot )$ satisfies $K(z_0,z_0)=Volume(\Omega {)}^{-1}$ for some $z_0\in \Omega$ if and only if $\Omega$ is either a disk minus a (possibly empty) closed polar set or $\mathbb{C}$ minus a (possibly empty) closed polar set. When $\Omega$ is bounded with $C^{\infty }$-boundary, we provide a simple proof of this using the zero set of the Szegö kernel. Finally, we show that this theorem fails to hold in ${\mathbb{C}}^n$ for $n>1$ by constructing a bounded complete Reinhardt domain (with algebraic boundary) which is strongly convex and not biholomorphic to the unit ball ${\mathbb{B}}^n\subset {\mathbb{C}}^n$.

9 pages, final version to appear in The Journal of Geometric Analysis