Some properties of Hölder surfaces in the Heisenberg group

Abstract

It is a folk conjecture that for $\alpha>1/2$ there is no $\alpha$-Hölder surface in the subRiemannian Heisenberg group. Namely, it is expected that there is no embedding from an open subset of $\mathbb{R}^{2}$ into the Heisenberg group that is Hölder continuous of order strictly greater than $1/2$. The Heisenberg group here is equipped with its Carnot–Carathéodory distance. We show that, in the case that such a surface exists, it cannot be of essential bounded variation and it intersects some vertical line in at least a topological Cantor set.