It is a folk conjecture that for α>1/2 there is no a- Hölder 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 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. © 2014 University of Illinois.
Some properties of hölder surfaces in the heisenberg group
Le Donne, Enrico;
2013-01-01
Abstract
It is a folk conjecture that for α>1/2 there is no a- Hölder 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 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. © 2014 University of Illinois.File in questo prodotto:
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