Bi-Lipschitz embeddings of Heisenberg submanifolds into Euclidean spaces

The Heisenberg group $\mathbb{H}$ 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 \textit{subsets} of $\mathbb{H}$ bi-Lipschitz embed into Euclidean spaces. We show that there exists a universal constant $L>0$ such that lines $L$-bi-Lipschitz embed into $\mathbb{R}^3$ and planes $L$-bi-Lipschitz embed into $\mathbb{R}^4$. Moreover, $C^{1,1}$ $2$-manifolds without characteristic points as well as all $C^{1,1}$ $1$-manifolds locally $L$-bi-Lipschitz embed into $\mathbb{R}^4$ where the constant $L$ is again universal. We also consider several examples of compact surfaces with characteristic points and we prove, for example, that Kor\'{a}nyi spheres bi-Lipschitz embed into $\mathbb{R}^4$ with a uniform constant. Finally, we show that there exists a compact, porous subset of $\mathbb{H}$ which does not admit a bi-Lipschitz embedding into any Euclidean space.