  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.
