On unitary invariant strongly pseudoconvex complex Finsler metrics

We consider a class of complex Finsler metrics of the form F = root r phi(t, s) with r = parallel to nu parallel to(2), t = parallel to z parallel to(2) and s = vertical bar < z,nu >vertical bar(2)/r for z in a domain D subset of C-n and nu is an element of T-z(1,0) D. 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 Kahler Finsler metric if and only if it comes from a Kahler metric. We obtain the necessary and sufficient condition for F to be weakly complex Berwald metrics and weakly Kaller 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. (C) 2015 Elsevier B.V. All rights reserved.