A solution of Gromov's H\"older equivalence problem for the Heisenberg group

We show that a map with H\"older exponent bigger than $1/2$ from a quasi-convex metric space with vanishing first Lipschitz homology into the Sub-Riemannian Heisenberg group factors through a tree. In particular, if the domain contains a disk, such a map can't be injective. This gives an answer to a question of Gromov for the simplest nontrivial case. The same tools allow to improve on a result of Borisov and it is shown that an isometric immersion of class $C^{1,\alpha}$ of a Riemannian surface with positive Gauss curvature into $\mathbb{R}^3$ has bounded extrinsic curvature if $\alpha > 1/2$.