A variant of Gromov's problem on H\"older equivalence of Carnot groups

Abstract

It is unknown if there exists a locally $\alpha$-H\"older homeomorphism $f:\mathbb{R}^3\to \mathbb{H}^1$ for any $\frac{1}{2}< \alpha\le \frac{2}{3}$, although the identity map $\mathbb{R}^3\to \mathbb{H}^1$ is locally $\frac{1}{2}$-H\"older. More generally, Gromov asked: Given $k$ and a Carnot group $G$, for which $\alpha$ does there exist a locally $\alpha$-H\"older homeomorphism $f:\mathbb{R}^k\to G$? Here, we equip a Carnot group $G$ with the Carnot-Carath\'eodory metric. In 2014, Balogh, Hajlasz, and Wildrick considered a variant of this problem. These authors proved that if $k>n$, there does not exist an injective, $(\frac{1}{2}+)$-H\"older mapping $f:\mathbb{R}^k\to \mathbb{H}^n$ that is also locally Lipschitz as a mapping into $\mathbb{R}^{2n+1}$. For their proof, they use the fact that $\mathbb{H}^n$ is purely $k$-unrectifiable for $k>n$. In this paper, we will extend their result from the Heisenberg group to model filiform groups and Carnot groups of step at most three. We will now require that the Carnot group is purely $k$-unrectifiable. The main key to our proof will be showing that $(\frac{1}{2}+)$-H\"older maps $f:\mathbb{R}^k\to G$ that are locally Lipschitz into Euclidean space, are weakly contact. Proving weak contactness in these two settings requires understanding the relationship between the algebraic and metric structures of the Carnot group. We will use coordinates of the first and second kind for Carnot groups.