On isometries of the Kobayashi and Carathéodory metrics

Prachi Mahajan¹

¹Department of Mathematics Indian Institute of Science Bangalore 560 012, India

Annales Polonici Mathematici, Tome 104 (2012) no. 2, pp. 121-151

Résumé

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 $.

Détail

DOI : 10.4064/ap104-2-2

Keywords: article considers smooth isometries kobayashi carath odory metrics domains mathbb extent which behave holomorphic mappings first provide example which suggests mathbb cannot mapped isometrically product domain addition prove several results continuous extension isometries rightarrow closures under purely local assumptions boundaries application there isometry between strongly pseudoconvex domain mathbb certain classes weakly pseudoconvex finite type domains mathbb