Localization of the Kobayashi metric and applications

Abstract

In this paper we introduce a new class of domains— log-type convex domains, which have no boundary regularity assumptions. Then we will localize the Kobayashi metric in log-type convex subdomains. As an application, we prove a local version of continuous extension of rough isometric maps between two bounded domains with log-type convex Dini-smooth boundary points. Moreover we prove that the Teichmuller space $${\mathcal {T}}_{g,n}$$ is not biholomorphic to any bounded pseudoconvex domain in $$\mathbb C^{3g-3+n}$$ which is locally log-type convex near some boundary point.