On isometries of the Kobayashi and Carathéodory metrics

2010/09/13 by Mahajan, Prachi
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1009.2414

Abstract

This article considers isometries of the Kobayashi and Carathéod-ory metrics on domains in Cⁿ and the extent to which they behave like holomorphic mappings. First we prove a metric version of Poincaré's theorem about biholomorphic inequivalence of Bⁿ , the unit ball in Cⁿ and Δⁿ , the unit polydisc in Cⁿ and then provide few examples which suggest that Bⁿ cannot be mapped isometrically onto a product domain. In addition, we prove several results on continuous extension of isometries f : D₁ → D₂ 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 C² and certain classes of weakly pseudoconvex finite type domains in C² .

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