An essay on the Bergman metric and balanced domains (Q2712253)

The main result is a quantitative version of a theorem of \textit{M. Jarnicki} and \textit{P. Pflug} [Ann. Pol. Math. 50, No.~2, 219-222 (1989; Zbl 0701.32002)] to the effect that the Bergman distance is complete for every bounded, balanced, pseudoconvex domain in \(\mathbb{C}^n\) having continuous Minkowski function. Namely, if the Minkowski function of such a domain \(\Omega\) is Hölder continuous, then there are positive constants \(A\) and \(B\) such that the Bergman distance \(\text{dist}_\Omega\) and the Euclidean distance \(\delta_\Omega\) to the boundary satisfy the inequality \(\text{dist}_\Omega(0,z) > A \log|\log(B\delta_\Omega(z))|-1\) for all \(z\) in~\(\Omega\). The proof is based on a quantitative estimate for the Bergman distance due to \textit{K. Diederich} and the author [Ann. Math. (2) 141, No. 1, 181-190 (1995; Zbl 0828.32002)] and the following second result of the paper: for every bounded, balanced, pseudoconvex domain in \(\mathbb{C}^n\) with Minkowski function~\(h\), there exist a constant~\(\epsilon\) between \(0\) and~\(1\) and a positive constant~\(C\) such that the function \(-(1-h(z)^2)^\epsilon e^{-C|z|^2}\) is plurisubharmonic.NEWLINENEWLINEFor the entire collection see [Zbl 0957.00034].