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 observe that Bergman curvatures of the Bergman metric on a bounded strictly pseudoconvex domain with $C^2$-boundary or bounded convex domain are bounded. Applying to the Schwarz lemma from a complete Kähler manifold into a complex Finsler manifold, we get that a bounded strictly pseudoconvex domain with $C^2$-boundary or bounded convex domain admits 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 Kähler manifold into a complex Finsler manifold, we prove the uniform equivalences of the Kobayashi metric and Carathéodory metric on a bounded strongly convex domain with smooth boundary.