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