Some properties of H\"older surfaces in the Heisenberg group
Metric Geometry
2012-05-02 v1 Differential Geometry
Geometric Topology
Abstract
It is a folk conjecture that for alpha > 1/2 there is no alpha-Hoelder surface in the subRiemannian Heisenberg group. Namely, it is expected that there is no embedding from an open subset of R^2 into the Heisenberg group that is Hoelder continuous of order strictly greater than 1/2. The Heisenberg group here is equipped with its Carnot-Caratheodory 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.
Keywords
Cite
@article{arxiv.1205.0228,
title = {Some properties of H\"older surfaces in the Heisenberg group},
author = {Enrico Le Donne and Roger Züst},
journal= {arXiv preprint arXiv:1205.0228},
year = {2012}
}
Comments
18 pages, 1 figure