On unitary invariant strongly pseudoconvex complex Finsler metrics

We consider a class of complex Finsler metrics of the form F=rϕ(t,s) with r=‖v‖2r=‖v‖2, t=‖z‖2t=‖z‖2 and s=|〈z,v〉|2r for z   in a domain D⊂CnD⊂Cn and v∈Tz1,0D. Complex Finsler metrics of this form are unitary invariant. We prove that F is a complex Berwald metric if and only if it comes from a Hermitian metric; F is a Kähler Finsler metric if and only if it comes from a Kähler metric. We obtain the necessary and sufficient condition for F to be weakly complex Berwald metrics and weakly Kähler Finsler metrics, respectively. Our results show that there are lots of weakly complex Berwald metrics which are unitary invariant. We also prove that, module a positive constant, a strongly convex complex Finsler metric F is locally projectively flat or dually flat if and only if F is the complex Euclidean metric.