Constructing Hölder maps to Carnot groups

Abstract:

In this paper, we construct Hölder maps to Carnot groups equipped with a Carnot metric, especially the first Heisenberg group $\mathbb {H}$. Pansu and Gromov [Carnot-Carathéodory spaces seen from within, Birkhäuser, Basel, 1996] observed that any surface embedded in $\mathbb {H}$ has Hausdorff dimension at least $3$, so there is no $\alpha$-Hölder embedding of a surface into $\mathbb {H}$ when $\alpha >\frac {2}{3}$. Züst [Anal. Geom. Metr. Spaces 3 (2015), pp. 73–92] improved this result to show that when $\alpha >\frac {2}{3}$, any $\alpha$-Hölder map from a simply-connected Riemannian manifold to $\mathbb {H}$ factors through a metric tree. In the present paper, we show that Züst’s result is sharp by constructing $(\frac {2}{3}-\epsilon )$-Hölder maps from $D^2$ and $D^3$ to $\mathbb {H}$ that do not factor through a tree. We use these to show that if $0<\alpha < \frac {2}{3}$, then the set of $\alpha$-Hölder maps from a compact metric space to $\mathbb {H}$ is dense in the set of continuous maps and to construct proper degree-1 maps from $\mathbb {R}^3$ to $\mathbb {H}$ with Hölder exponents arbitrarily close to $\frac {2}{3}$.