Some properties of H\"older surfaces in the Heisenberg group

TL;DR

This paper investigates the properties of Hölder surfaces in the Heisenberg group, providing evidence that such surfaces, if they exist beyond a certain regularity, must have complex intersection properties and unbounded variation.

Contribution

It offers new insights into the geometric and measure-theoretic properties of hypothetical Hölder surfaces in the Heisenberg group, supporting the conjecture that high-regularity embeddings are unlikely.

Findings

Such surfaces cannot have essential bounded variation.

They intersect some vertical lines in at least a topological Cantor set.

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.