Geometry of holomorphic invariant strongly pseudoconvex complex Finsler metrics on the classical domains

In this paper, a class of holomorphic invariant metrics are introduced on the irreducible classical domains of type I-IV , which by their very definitions are strongly pseudoconvex complex Finsler metrics in the strict sense of M. Abate and G. Patrizio [2]. These metrics are of particular interesting since they are holomorphic invariant non-Hermitian quadratic complex Finsler metrics found so far in literature which enjoy good regularity as well as convexity and can be explicitly expressed so as to admit differential geometric studies. These metrics are explicitly constructed via deformations of the corresponding Bergman metrics on the irreducible classical domains and they are all proved to be complete K\"ahler-Berwald metrics. Moreover, these metrics enjoy very similar curvature properties as that of the Bergman metrics on the irreducible classical domains, namely they all have negative holomorphic sectional curvatures and non-positive holomorphic bisectional curvatures. From the view point of complex analysis, these metrics are analogue of Bergman metrics in complex Finsler geometry without Hermitian quadratic restrictions in the philosophy of S. S. Chern [8]; furthermore, they are all K\"ahler Finsler-Einstein metrics in the sense of T. Aikou [5]