Let $D \subseteq \mathbb{C}^n$ be a bounded domain and let $t \in D$. We say that $D$ is a minimal domain with center $t$ if for each biholomorphism $F:D \to D' \subseteq \mathbb{C}^n$ such that $JF(t) = 1$ ($JF$ denotes the complex jacobian), we have $vol(D) \leq vol(D')$. I need to prove that a necessary and sufficient criterion for a domain $D$ to be minimal with center $t$ is $K(z,t) = \frac{1}{vol(D)}$, where $K$ is the Bergman kernel function of $D$. The sufficiency follows from the second extremal property of the Bergman kernel function mentioned here. How do I prove the converse?
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eom.springer.deis broken, but the article can now be found at encyclopediaofmath.org/wiki/Bergman_kernel_function. $\endgroup$