Schwarz lemma from a K\"ahler manifold into a complex Finsler manifold

TL;DR

This paper establishes a Schwarz lemma for holomorphic maps from a K"ahler manifold with curvature bounds into a complex Finsler manifold with negative curvature, leading to rigidity results.

Contribution

It introduces a Schwarz lemma in the setting of K"ahler to complex Finsler manifolds with curvature bounds, extending classical results.

Findings

Schwarz lemma for holomorphic maps between specified manifolds

Liouville type rigidity result for such maps

Rigidity theorem for bimeromorphic mappings

Abstract

Suppose that $M$ is a K\"ahler manifold with a pole such that its holomorphic sectional curvature is bounded from below by a constant and its radial sectional curvature is also bounded from below. Suppose that $N$ is a strongly pseudoconvex complex Finsler manifold such that its holomorphic sectional curvature is bounded from above by a negative constant. In this paper, we establish a Schwarz lemma for holomorphic mappings $f$ form $M$ into $N$. As applications, we obtain a Liouville type rigidity result for holomorphic mappings $f$ from $M$ into $N$, as well as a rigidity theorem for bimeromorphic mappings from a compact complex manifold into a compact complex Finsler manifold.