Sharp distortion growth for bilipschitz extension of planar maps

TL;DR

This paper proves that any bilipschitz embedding of a line into the plane can be extended to the entire plane with only a linear increase in distortion, addressing a key aspect of the bilipschitz extension problem.

Contribution

It establishes a linear bound on the distortion growth when extending bilipschitz maps from lines to the plane, advancing understanding of bilipschitz extension properties.

Findings

Extension with linear distortion bound for bilipschitz maps

Quantitative analysis of bilipschitz extension growth

Addresses a fundamental problem in geometric function theory

Abstract

This note addresses the quantitative aspect of the bilipschitz extension problem. The main result states that any bilipschitz embedding of $\mathbb R$ into $\mathbb R^2$ can be extended to a bilipschitz self-map of $\mathbb R^2$ with a linear bound on the distortion.