Complete complex Finsler metrics and uniform equivalence of the Kobayashi metric

Abstract

In this paper, first of all, according to Lu's and Zhang's works about the curvature of the Bergman metric on a bounded domain and the properties of the squeezing functions, we obtain that Bergman curvature of the Bergman metric on a bounded strictly pseudoconvex domain with $C^2$-boundary or bounded convex domain is bounded. Secondly, by the property of curvature symmetry on a K\"ahler manifold, we have the property: if holomorphic sectional curvature of a K\"ahler manifold is bounded, we can deduce that its sectional curvature is bounded. After that, applying to the Schwarz lemma from a complete K\"ahler manifold into a complex Finsler manifold, we get that a bounded strictly pseudoconvex domain with $C^2$-boundary or bounded convex domain admit complete strongly pseudoconvex complex Finsler metrics such that their holomorphic sectional curvature is bounded from above by a negative constant. Finally, by the Schwarz lemma from a complete K\"ahler manifold into a complex Finsler manifold, we prove the uniform equivalences of the Kobayashi metric and Carath\'eodory metric on a bounded strongly convex domain with smooth boundary.