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In this paper, a class of holomorphic invariant metrics is introduced on theirreducible classical domains of type I-IV, which are strongly pseudoconvexcomplex Finsler metrics in the strict sense of M. Abate and G. Patrizio[2].These metrics are of particular interest in several complex variables sincethey are holomorphic invariant complex Finsler metrics found so far inliterature which enjoy good regularity as well as strong pseudoconvexity andcan be explicitly expressed so as to admit differential geometric studies. Theyare, however, not necessarily Hermitian quadratic as that of the Bergmanmetrics. These metrics are explicitly constructed via deformation of thecorresponding Bergman metric on the irreducible classical domains of type I-IV,respectively, and they are all proved to be complete Kahler-Berwald metrics.They enjoy very similar curvature properties as that of the Bergman metric onthe irreducible classical domains, namely their holomorphic sectionalcurvatures are bounded between two negative constants, and their holomorphicbisectional curvatures are always non positive and bounded below by negativeconstants, respectively. From the viewpoint of complex analysis, these metricsare analogues of Bergman metrics in complex Finsler geometry which do notnecessarily have Hermitian quadratic restrictions in the viewpoint of S.-S.Chern[7].

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