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 $K(z_0, z_0) = \hbox{Volume}(\Omega)^{-1}$ 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 Szeg\"o 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 $\mathbb{B}^n \subset \mathbb{C}^n$.