An N-dimensional version of the Beurling-Ahlfors extension

TL;DR

This paper introduces an explicit integral operator to extend monotone quasiconformal mappings from n to n+1 dimensions, refining the classical Beurling-Ahlfors extension in one dimension.

Contribution

It provides a new explicit integral extension method for monotone quasiconformal maps in higher dimensions, generalizing and refining the classical Beurling-Ahlfors extension.

Findings

Extension preserves monotonicity and quasiconformality in higher dimensions

Explicit integral operator for extension is constructed

Refinement of the Beurling-Ahlfors extension in one dimension

Abstract

We extend monotone quasiconformal mappings from dimension n to n+1 while preserving both monotonicity and quasiconformality. The extension is given explicitly by an integral operator. In the case n=1 it yields a refinement of the Beurling-Ahlfors extension.