  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.
