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

Abstract

In this paper, a class of holomorphic invariant metrics is introduced on the irreducible classical domains of types I–IV, which are strongly pseudoconvex complex Finsler metrics in the strict sense of Abate and Patrizio (1994). These metrics are of particular interest in several complex variables since they are holomorphic invariant complex Finsler metrics found so far in literature which enjoy good regularity as well as strong pseudoconvexity and can be explicitly expressed so as to admit differential geometry studies. They are, however, not necessarily Hermitian quadratic as the Bergman metrics. These metrics are explicitly constructed via deformation of the corresponding Bergman metric on the irreducible classical domains of types I–IV, respectively, and they are all proved to be complete Kähler-Berwald metrics. They enjoy very similar curvature properties as those of the Bergman metric on the irreducible classical domains, i.e., their holomorphic sectional curvatures are bounded between two negative constants and their holomorphic bisectional curvatures are always nonpositive and bounded below by negative constants, respectively. From the viewpoint of complex analysis, these metrics are analogues of Bergman metrics in complex Finsler geometry which do not necessarily have Hermitian quadratic restrictions in the sense of Chern (1996).