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Some properties of Hölder surfaces in the Heisenberg group. (English) Zbl 1294.53033
Summary: It is a folk conjecture that for \(\alpha>1/2\) there is no \(\alpha\)-Hölder surface in the sub-Riemannian 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 when such a surface exists, it cannot be of essential bounded variation and it intersects some vertical line in at least a topological Cantor set.

53C17 Sub-Riemannian geometry
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
28A75 Length, area, volume, other geometric measure theory
26A16 Lipschitz (Hölder) classes
Full Text: Euclid arXiv
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