Schoenflies problem for area preserving biLipschitz mappings

TL;DR

This paper proves that boundary-preserving biLipschitz mappings of the disk can be extended to the entire plane while maintaining area-preserving properties, addressing a classical geometric problem.

Contribution

It establishes an extension theorem for area-preserving biLipschitz boundary maps to the whole plane, advancing the understanding of geometric mappings.

Findings

Extension of boundary maps to the entire plane while preserving area.

BiLipschitz mappings can be extended without losing area-preserving properties.

Addresses the classical Schoenflies problem in the context of area-preserving maps.

Abstract

We prove that any biLipschitz mapping of the boundary of the unit disk onto the boundary of the domain of the same area can be extended to a biLipschitz mapping of the whole plane which preserves the area of any measurable subset.