Holomorphic invariant strongly pseudoconvex complex Finsler metrics

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.