Normal families of holomorphic mappings between complex Finsler manifolds

TL;DR

This paper extends classical theorems on normal families of holomorphic mappings to the setting of complex Finsler manifolds, establishing new results on their properties and automorphism groups.

Contribution

It proves a Montel-type theorem for holomorphic mappings between complex Finsler manifolds, generalizing known results from Hermitian manifolds.

Findings

The integrated form of a complex Finsler metric is inner.

Completeness of the Finsler distance relates to compactness of bounded subsets.

Extension of classical theorems A-F to complex Finsler manifolds.

Abstract

In this paper, we find that the integrated form $d_F$ of a complex Finsler metric $F$ is inner. The distance $d_F$ is complete if and only if every closed bounded subset of a complex manifold $M$ is compact. We prove a version of theorem for normal families of holomorphic mappings between two complex Finsler manifolds, i.e, the theorem of Montel in complex Finsler manifolds. Our results extend the basic theorem of strongly negatively curved families for a Hermitian manifold [Wu, Acta Math. 119(1967), 193-233] or [Grauert, Reckziegel, Math. Z. 89(1965), 108-125]. As applications, we obtain a complex Finsler version of theorems $A$-$F$ in [Wu, Acta Math. 119(1967), 193-233], including the Cartan-Carath\'eodory-Kaup-Wu theorem, the theorem of the automorphism group on a complex Finsler manifold and some rigid results.