Normal families of holomorphic mappings between complex Finsler manifolds

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.