Planar Bilipschitz Extension from Separated Nets

TL;DR

This paper proves that bilipschitz maps from separated nets in the plane can be extended to the entire plane with controlled distortion, solving longstanding open problems in geometric analysis.

Contribution

It establishes polynomial bounds for extending bilipschitz maps from separated nets in b2 to b2, generalizing previous results and answering open questions.

Findings

Extension of bilipschitz maps from b2 to b2 with polynomial bounds

Extension results for separated nets in b2

Resolution of a decades-old open problem in geometric analysis

Abstract

We prove that every $L$-bilipschitz mapping $\mathbb{Z}^2\to\mathbb{R}^2$ can be extended to a $C(L)$-bilipschitz mapping $\mathbb{R}^2\to\mathbb{R}^2$ and provide a polynomial upper bound for $C(L)$. Moreover, we extend the result to every separated net in $\mathbb{R}^2$ instead of $\mathbb{Z}^2$, with the upper bound gaining a polynomial dependence on the separation and net constants associated to the given separated net. This answers an Oberwolfach question of Navas from 2015 and is also a positive solution of the two-dimensional form of a decades old open (in all dimensions at least two) problem due to Alestalo, Trotsenko and V\"ais\"al\"a.