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

TL;DR

This paper introduces a new class of holomorphic invariant strongly pseudoconvex complex Finsler metrics on classical domains, which are explicitly constructed, complete, and exhibit curvature properties similar to Bergman metrics, expanding the tools in complex geometry.

Contribution

The paper constructs explicit, complete, holomorphic invariant complex Finsler metrics on classical domains, extending the class of metrics beyond Hermitian quadratic forms like Bergman metrics.

Findings

Metrics are explicitly constructed via deformation of Bergman metrics.

All metrics are complete Kahler-Berwald and have bounded negative holomorphic sectional curvatures.

Holomorphic bisectional curvatures are non-positive and bounded below by negative constants.

Abstract

In this paper, a class of holomorphic invariant metrics is introduced on the irreducible classical domains of type I-IV, which are strongly pseudoconvex complex Finsler metrics in the strict sense of M. Abate and G. Patrizio[2]. 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 geometric studies. They are, however, not necessarily Hermitian quadratic as that of the Bergman metrics. These metrics are explicitly constructed via deformation of the corresponding 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 on the irreducible classical domains, namely their holomorphic sectional curvatures are bounded between two negative constants, and their holomorphic bisectional curvatures are always non positive 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 viewpoint of S.-S. Chern[7].