On the holomorphicity of isometries of intrinsic metrics in complex analysis

Details On the holomorphicity of isometries of intrinsic metrics in complex analysis On the holomorphicity of isometries of intrinsic metrics in complex analysis

2005-05-13

arxiv.org/abs/math/0505284v2

Mathematics

Complex Variables

21 pages

Scientific paper

Let $\1$ and $\2$ be $\s$ domains in $\Cn$ and $f: \1 \rt \2$ an isometry for the Kobayashi or Carath\'eodory metrics. Suppose that $f$ extends as a $C^1$ map to $ \bar \om_1$. We then prove that $f|_{\partial \1}: \partial \1 \rt \partial \2$ is a CR or anti-CR diffeomorphism. It follows that $\1$ and $\2$ must be biholomorphic or anti-biholomorphic. The main tool is a metric version of the Pinchuk rescaling technique.

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