On isometries of the Kobayashi and Carath\'{e}odory metrics

TL;DR

This paper investigates the properties of isometries with respect to Kobayashi and Carathéodory metrics on complex domains, establishing their behavior akin to holomorphic maps and exploring their extension and non-existence in certain cases.

Contribution

It proves a metric version of Poincaré's theorem, provides examples suggesting non-isometric mappings between certain domains, and establishes boundary extension results for isometries.

Findings

No isometry exists between the unit ball and the polydisc.

Isometries extend continuously to boundary under certain conditions.

No isometry between strongly pseudoconvex and certain weakly pseudoconvex domains.

Abstract

This article considers isometries of the Kobayashi and Carath\'{e}od-ory metrics on domains in $ \mathbf{C}^n $ and the extent to which they behave like holomorphic mappings. First we prove a metric version of Poincar\'{e}'s theorem about biholomorphic inequivalence of $ \mathbf{B}^n $, the unit ball in $ \mathbf{C}^n $ and $ \Delta^n $, the unit polydisc in $ \mathbf{C}^n $ and then provide few examples which \textit{suggest} that $ \mathbf{B}^n $ cannot be mapped isometrically onto a product domain. In addition, we prove several results on continuous extension of isometries $ f : D_1 \rightarrow D_2 $ to the closures under purely local assumptions on the boundaries. As an application, we show that there is no isometry between a strongly pseudoconvex domain in $ \mathbf{C}^2 $ and certain classes of weakly pseudoconvex finite type domains in $ \mathbf{C}^2 $.