Some aspects of the Kobayashi and Caratheodory metrics on pseudoconvex domains

Abstract

The purpose of this article is to consider two themes, both of which emanate from and involve the Kobayashi and the Caratheodory metric. First, we study the biholomorphic invariant introduced by B. Fridman on strongly pseudoconvex domains, on weakly pseudoconvex domains of finite type in C², and on convex finite type domains in Cⁿ using the scaling method. Applications include an alternate proof of the Wong–Rosay theorem, a characterization of analytic polyhedra with noncompact automorphism group when the orbit accumulates at a singular boundary point, and a description of the Kobayashi balls on weakly pseudoconvex domains of finite type in C² and convex finite type domains in Cⁿ in terms of Euclidean parameters. Second, a version of Vitushkin’s theorem about the uniform extendability of a compact subgroup of automorphisms of a real analytic strongly pseudoconvex domain is proved for C¹-isometries of the Kobayashi and Caratheodory metrics on a smoothly bounded strongly pseudoconvex domain.