A uniformization theorem in complex Finsler geometry

TL;DR

This paper constructs examples of weakly K"ahler Finsler metrics that are not K"ahler and proves a uniformization theorem for unitary invariant complex Randers metrics with constant holomorphic curvature.

Contribution

It provides the first explicit examples of non-K"ahler weakly K"ahler Finsler metrics and establishes a uniformization theorem in this context.

Findings

Existence of non-K"ahler weakly K"ahler Finsler metrics.

Construction of examples within unitary invariant complex Randers metrics.

A uniformization theorem for these metrics with constant holomorphic curvature.

Abstract

In complex Finsler geometry, an open problem is: does there exist a weakly K\"ahler Finsler metric which is not K\"ahler? In this paper, we give an affirmative answer to this open problem. More precisely, we construct a family of the weakly K\"ahler Finsler metrics which are non-K\"ahler. The examples belong to the unitary invariant complex Randers metrics. Furthermore, a uniformization theorem of the unitary invariant complex Randers metrics with constant holomorphic curvature is proved under the weakly K\"ahler condition.