BiLipschitz embeddings of spheres into jet space Carnot groups not admitting Lipschitz extensions

TL;DR

The paper constructs biLipschitz embeddings of spheres into jet space Carnot groups that cannot be extended Lipschitzly to the ball, revealing limitations in extension properties within these geometric structures.

Contribution

It introduces explicit biLipschitz embeddings of spheres into jet space Carnot groups that lack Lipschitz extensions, using novel constructions based on jet maps of smooth functions.

Findings

Constructed biLipschitz embeddings for all n,k ≥ 1

Proved non-existence of Lipschitz extensions for these embeddings

Applied factorization and modification of existing arguments

Abstract

For all $k,n\ge 1$, we construct a biLipschitz embedding of $\mathbb{S}^n$ into the jet space Carnot group $J^k(\mathbb{R}^n)$ that does not admit a Lipschitz extension to $\mathbb{B}^{n+1}$. Let $f:\mathbb{B}^n\to \mathbb{R}$ be a smooth, positive function with $k^{th}$-order derivatives that are approximately linear near $\partial \mathbb{B}^n$. The embedding is given by taking the jet of $f$ on the upper hemisphere and the jet of $-f$ on the lower hemisphere, where we view $\mathbb{S}^n$ as two copies of $\mathbb{B}^n$. To prove the lack of a Lipschitz extension, we apply a factorization result of Wenger and Young for $n=1$ and modify an argument of Rigot and Wenger for $n\ge 2$.