Rigidity theorem by the minimal point of the Bergman kernel

We use the Suita conjecture (now a theorem) to prove that for any domain $\mathrm{\Omega }\subset \mathbb{C}$ its Bergman kernel $K(\cdot ,\cdot )$ satisfies $K(z_0,z_0)=\text{Volume}(\mathrm{\Omega }{)}^{-1}$ for some $z_0\in \mathrm{\Omega }$ if and only if $\mathrm{\Omega }$ is either a disk minus a (possibly empty) closed polar set or $\mathbb{C}$ minus a (possibly empty) closed polar set. When $\mathrm{\Omega }$ is bounded with $C^{\mathrm{\infty }}$-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 ${\mathbb{B}}^n\subset {\mathbb{C}}^n$.