On the holomorphicity of isometries of intrinsic metrics in complex analysis

TL;DR

This paper proves that isometries for intrinsic metrics between certain complex domains extend smoothly to the boundary and are either holomorphic or anti-holomorphic, implying the domains are biholomorphically equivalent.

Contribution

It establishes boundary regularity and holomorphicity of isometries for Kobayashi and Carathéodory metrics using a metric rescaling approach.

Findings

Isometries extend as CR or anti-CR diffeomorphisms to the boundary.

Domains are biholomorphic or anti-biholomorphic if such isometries exist.

Introduces a metric version of the Pinchuk rescaling technique.

Abstract

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.