On isometries of the carathéodory and Kobayashi metrics on strongly pseudoconvex domains

Let ${\Omega }_1$ and ${\Omega }_2$ be $\mathrm{strongly}\mathrm{pseudoconvex}$ domains in ${ℂ}^n$ and $f:{\Omega }_1\to {\Omega }_2$ an isometry for the Kobayashi or Carathéodory metrics. Suppose that $f$ extends as a $C^1$ map to ${\overline{\Omega }}_1$. We then prove that ${f|}_{\partial {\Omega }_1}:\partial {\Omega }_1\to \partial {\Omega }_2$ is a CR or anti-CR diffeomorphism. It follows that ${\Omega }_1$ and ${\Omega }_2$ must be biholomorphic or anti-biholomorphic.

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