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

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 $.