Some properties of H\"older surfaces in the Heisenberg group
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 R2 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.
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