A uniformization theorem in complex Finsler geometry

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.