The Holomorphic Curvature On Unitary Invariant Complex Finsler Metrics

In this paper,for the unitary invariant complex Finsler metrics,we choose an appropriate transformation and get a simple expression under the weakly K(?)hler condition.Besides,we also get the simple holomorphic curvature expression under the same transformation.In complex Finsler geometry,an open problem is: does there exist a weakly K(?)hler Finsler metric which is not K(?)hler? In this paper,we construct a family of the weakly K(?)hler Finsler metrics which are non-K(?)hler.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(?)hler condition.