On the holomorphicity of isometries of intrinsic metrics in complex analysis

Let $\text{1}$ and $\text{2}$ be $\text{s}$ domains in $\text{Cn}$ and $f:\text{1}\text{rt}\text{2}$ an isometry for the Kobayashi or Carath'eodory metrics. Suppose that $f$ extends as a $C^1$ map to ${\overline{\text{om}}}_1$. We then prove that $f|_{\partial \1}: \partial \1 \rt \partial \2$isaCRoranti-CRdiffeomorphism\mathrm{.}Itfollowsthat$\1$and$\2$ must be biholomorphic or anti-biholomorphic. The main tool is a metric version of the Pinchuk rescaling technique.