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.