# Conformal equivalence of visual metrics in pseudoconvex domains

## TL;DR

This paper demonstrates that boundary extensions of isometries between smooth strongly pseudoconvex domains are conformal relative to the sub-Riemannian metric, providing new insights into the geometric structure of these domains and their biholomorphic mappings.

## Contribution

It refines existing estimates to show conformality of boundary extensions and offers an alternative proof of Fefferman's theorem using hyperbolic geometry techniques.

## Key findings

- Boundary extensions are conformal with respect to the sub-Riemannian metric.
- Provides an alternative proof of Fefferman's boundary regularity result.
- Utilizes hyperbolic geometric methods inspired by Mostow's rigidity theorem.

## Abstract

We refine estimates introduced by Balogh and Bonk, to show that the boundary extensions of isometries between smooth strongly pseudoconvex domains in $\C^n$ are conformal with respect to the sub-Riemannian metric induced by the Levi form. As a corollary we obtain an alternative proof of a result of Fefferman on smooth extensions of biholomorphic mappings between pseudoconvex domains. The proofs are inspired by Mostow's proof of his rigidity theorem and are based on the asymptotic hyperbolic character of the Kobayashi or Bergman metrics and on the Bonk-Schramm hyperbolic fillings.