Illinois Journal of Mathematics

Some properties of Hölder surfaces in the Heisenberg group

Enrico Le Donne and Roger Züst

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It is a folk conjecture that for $\alpha>1/2$ there is no $\alpha$-Hölder surface in the subRiemannian Heisenberg group. Namely, it is expected that there is no embedding from an open subset of $\mathbb{R}^{2}$ 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.

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Illinois J. Math., Volume 57, Number 1 (2013), 229-249.

First available in Project Euclid: 23 June 2014

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Primary: 53C17: Sub-Riemannian geometry 49Q15: Geometric measure and integration theory, integral and normal currents [See also 28A75, 32C30, 58A25, 58C35] 28A75: Length, area, volume, other geometric measure theory [See also 26B15, 49Q15] 26A16: Lipschitz (Hölder) classes


Le Donne, Enrico; Züst, Roger. Some properties of Hölder surfaces in the Heisenberg group. Illinois J. Math. 57 (2013), no. 1, 229--249. doi:10.1215/ijm/1403534494.

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