IM PAN
INSTITUTE OF MATHEMATICS, POLISH ACADEMY OF SCIENCES
ANNALES
POLONICI MATHEMATICI
ISSN: 0066-2216(p) 1730-6272(e)
 

On isometries of the Kobayashi and Carathéodory metrics
Prachi Mahajan1
Ann. Polon. Math. 104 (2012), 121-151
doi:10.4064/ap104-2-2
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 $.

MSC (2010): Primary 32F45; Secondary 32Q45.
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  1. Department of Mathematics
    Indian Institute of Science
    Bangalore 560 012, India