most recent 30 from http://mathoverflow.net 2013-05-19T22:40:53Z http://mathoverflow.net/feeds/question/65121 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65121/question-on-bergman-minimal-domains Jaikrishnan 2011-05-16T08:30:29Z 2011-05-16T08:30:29Z

<p>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 <a href="http://eom.springer.de/b/b015560.htm" rel="nofollow">here</a>. How do I prove the converse?</p>