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Abstract

This article considers $ C^1$-smooth isometries of the Kobayashi and Carathéodory metrics on domains in $ \mathbb{C}^n $ and the extent to which they behave like holomorphic mappings. First we provide an example which suggests that $ \mathbb{B}^n $ cannot be mapped isometrically onto a product domain. In addition, we prove several results on continuous extension of $ C^0$-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 $ C^0$-isometry between a strongly pseudoconvex domain in $ \mathbb{C}^2 $ and certain classes of weakly pseudoconvex finite type domains in $ \mathbb{C}^2 $.