On the holomorphicity of isometries of intrinsic metrics in complex analysis

Seshadri, Harish; Verma, Kaushal

On the holomorphicity of isometries of intrinsic metrics in complex analysis

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Description

Let $\1$ and $\2$ be $\s$ domains in $\Cn$ and $f: \1 \rt \2$ an isometry for the Kobayashi or Carathéodory 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.
21 pages