Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 45, 2020, 931–955

BI-LIPSCHITZ EMBEDDINGS OF HEISENBERG SUBMANIFOLDS INTO EUCLIDEAN SPACES

Vasileios Chousionis, Sean Li, Vyron Vellis and Scott Zimmerman

University of Connecticut, Department of Mathematics
Storrs, CT 06269-1009, U.S.A.; vasileios.chousionis 'at' uconn.edu

University of Connecticut, Department of Mathematics
Storrs, CT 06269-1009, U.S.A.; sean.li 'at' uconn.edu

University of Connecticut, Department of Mathematics
Storrs, CT 06269-1009, U.S.A.; vyron.vellis 'at' uconn.edu

University of Connecticut, Department of Mathematics
Storrs, CT 06269-1009, U.S.A.; scott.zimmerman 'at' uconn.edu

Abstract. The Heisenberg group H equipped with a sub-Riemannian metric is one of the most well known examples of a doubling metric space which does not admit a bi-Lipschitz embedding into any Euclidean space. In this paper we investigate which subsets of H bi-Lipschitz embed into Euclidean spaces. We show that there exists a universal constant L > 0 such that lines L-bi-Lipschitz embed into R³ and planes L-bi-Lipschitz embed into R⁴. Moreover, C^1,1 2-manifolds without characteristic points as well as all C^1,1 1-manifolds locally L-bi-Lipschitz embed into R⁴ where the constant L is again universal. We also consider several examples of compact surfaces with characteristic points and we prove, for example, that Korányi spheres bi-Lipschitz embed into R⁴ with a uniform constant. Finally, we show that there exists a compact, porous subset of H which does not admit a bi-Lipschitz embedding into any Euclidean space.

2010 Mathematics Subject Classification: Primary 30L05; Secondary 53C17.

Key words: Heisenberg group, bi-Lipschitz embedding, porous sets.

Reference to this article: V. Chousionis, S. Li, V. Vellis and S. Zimmerman: Bi-Lipschitz embeddings of Heisenberg submanifolds into Euclidean spaces. Ann. Acad. Sci. Fenn. Math. 45 (2020), 931–955.

Full document as PDF file

https://doi.org/10.5186/aasfm.2020.4551