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

2017/12/15 by Jung, Derek
#53C17 #58A20 #FOS: Mathematics #Geometric Topology (math.GT) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1712.05508

Abstract

For all k,n≥ 1, we construct a biLipschitz embedding of \mathbbSⁿ into the jet space Carnot group J^k(ℝⁿ) that does not admit a Lipschitz extension to \mathbbBⁿ⁺¹. Let f:\mathbbBⁿ→ ℝ be a smooth, positive function with k^th-order derivatives that are approximately linear near ∂ \mathbbBⁿ. 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 \mathbbSⁿ as two copies of \mathbbBⁿ. 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≥ 2.

Related

• Lipschitz extensions into Jet space Carnot groups
• Lipschitz non-extension theorems into jet space Carnot groups
• BiLipschitz decomposition of Lipschitz maps between Carnot groups
• Dimension results for mappings of jet space Carnot groups
• Lipschitz and biLipschitz Maps on Carnot Groups