Holomorphic invariant strongly pseudoconvex complex Finsler metrics

TL;DR

This paper classifies invariant strongly pseudoconvex complex Finsler metrics on unit domains, showing uniqueness on the ball and constructing infinitely many on the polydisk with similar properties to the Bergman metric.

Contribution

It proves the uniqueness of invariant strongly pseudoconvex Finsler metrics on the unit ball and constructs a family of such metrics on the polydisk, expanding understanding of invariant complex Finsler geometry.

Findings

Unique invariant strongly pseudoconvex Finsler metric on the unit ball is the Poincaré-Bergman metric.

Infinitely many invariant strongly convex Finsler metrics exist on the polydisk.

Constructed metrics are strongly convex Kähler-Berwald and resemble the Bergman metric.

Abstract

Let $B_n$ and $P_n$ be the unit ball and the unit polydisk in $\mathbb{C}^n$ with $n\geq 2$ respectively. Denote $\mbox{Aut}(B_n)$ and $\mbox{Aut}(P_n)$ the holomorphic automorphism group of $B_n$ and $P_n$ respectively. In this paper, we prove that $B_n$ admits no $\mbox{Aut}(B_n)$-invariant strongly pseudoconvex complex Finsler metric other than a constant multiple of the Poincar$\acute{\mbox{e}}$-Bergman metric, while $P_n$ admits infinite many $\mbox{Aut}(P_n)$-invariant complete strongly convex complex Finsler metrics other than the Bergman metric. The $\mbox{Aut}(P_n)$-invariant complex Finsler metrics are explicitly constructed which depend on a real parameter $t\in [0,+\infty)$ and integer $k\geq 2$. These metrics are proved to be strongly convex K\"ahler-Berwald metrics, and they posses very similar properties as that of the Bergman metric on $P_n$. As applications, the existence of $\mbox{Aut}(M)$-invariant strongly convex complex Finsler metrics is also investigated on some Siegel domains of the first and the second kind which are biholomorphic equivalently to the unit polydisc in $\mathbb{C}^n$. We also give a characterization of strongly convex K\"ahler-Berwald spaces and give a de Rahm type decomposition theorem for strongly convex K\"ahler-Berwald spaces.