Roughness of level sets of differentiable maps on Heisenberg group

arXiv1110.3634MaRDI QIDQ6228410

Artem Kozhevnikov

Publication date: 17 October 2011

Abstract: We investigate metric properties of level sets of horizontally differentiable maps defined on the first Heisenberg group $(Bbb{H}^1,d_{cc})$ equipped with the standard sub-Riemannian structure. In particular, we present an exhaustive analysis in a new case of a map $Fin{C}_H^1(Bbb{H}^1,Bbb{R}^2)$ with surjective horizontal differential (an analogue of the classical implicit function theorem). Among other results, we show that a level set of such map is locally a simple curve of Hausdorff sub-Riemannian dimension 2, but, surprisingly, in general its two-dimensional Hausdorff measure can be zero or infinity. Therefore, those level sets (called extsf{vertical curves}) can be of rough nature and not belong to the class of intrinsic regular manifolds.

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