A Schwarz lemma for weakly K\"ahler-Finsler manifolds

TL;DR

This paper establishes a Schwarz lemma for weakly Kähler-Finsler manifolds, providing conditions under which holomorphic maps between such manifolds are necessarily constant, extending classical results in complex geometry.

Contribution

It introduces a Schwarz lemma for weakly Kähler-Finsler manifolds and proves holomorphic mappings are constant under curvature conditions, extending existing complex geometric theories.

Findings

Holomorphic mappings from certain Finsler manifolds are constant under curvature bounds.

Distance function estimates are established for convex complex Finsler manifolds.

A Schwarz lemma is derived for weakly Kähler-Finsler manifolds.

Abstract

In this paper, we first establish several theorems about the estimation of distance function on real and strongly convex complex Finsler manifolds and then obtain a Schwarz lemma from a strongly convex weakly K\"ahler-Finsler manifold into a strongly pseudoconvex complex Finsler manifold. As applications, we prove that a holomorphic mapping from a strongly convex weakly K\"ahler-Finsler manifold into a strongly pseudoconvex complex Finsler manifold is necessary constant under an extra condition. In particular, we prove that a holomorphic mapping from a complex Minkowski space into a strongly pseudoconvex complex Finsler manifold such that its holomorphic sectional curvature is bounded from above by a negative constant is necessary constant.