Some aspects of the Kobayashi and Carath\'eodory metrics on pseudoconvex domains

TL;DR

This paper explores the properties of Kobayashi and Carathéodory metrics on pseudoconvex domains, providing new proofs, characterizations, and descriptions of automorphism groups and metric balls using advanced complex analysis techniques.

Contribution

It introduces a biholomorphic invariant for various pseudoconvex domains, offers an alternative proof of the Wong-Rosay theorem, and characterizes automorphism groups and metric balls in these domains.

Findings

New biholomorphic invariant for pseudoconvex domains

Alternate proof of Wong-Rosay theorem

Description of Kobayashi balls in finite type domains

Abstract

The purpose of this article is to consider two themes both of which emanate from and involve the Kobayashi and the Carath\'eodory metric. First we study the biholomorphic invariant introduced by B. Fridman on strongly pseudoconvex domains, on weakly pseudoconvex domains of finite type in $\mathbf C^2$ and on convex finite type domains in $\mathbf C^n$ 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 $ \mathbf{C}^2$ and convex finite type domains in $ \mathbf{C}^n $ 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 $C^1$-isometries of the Kobayashi and Carath\'eodory metrics on a smoothly bounded strongly pseudoconvex domain.