Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 261-280

BILIPSCHITZ EMBEDDINGS OF SPHERES INTO JET SPACE CARNOT GROUPS NOT ADMITTING LIPSCHITZ EXTENSIONS

Derek Jung

University of Illinois at Urbana-Champaign, Department of Mathematics
1409 West Green Street, Urbana, IL 61801, U.S.A.; djjung2 'at' illinois.edu

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

2010 Mathematics Subject Classification: Primary 53C17, 58A20; Secondary 30L05, 26A16, 22E25.

Key words: Sub-Riemannian geometry, biLipschitz embeddings, jet spaces, Carnot groups, Lipschitz extensions.

Reference to this article: D. Jung: BiLipschitz embeddings of spheres into jet space Carnot groups not admitting Lipschitz extensions. Ann. Acad. Sci. Fenn. Math. 44 (2019), 261-280.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4420