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 $.
