Roughness of level sets of differentiable maps on Heisenberg group

Abstract

We investigate metric properties of level sets of horizontally differentiable maps defined on the first Heisenberg group $({\mathbb{H}}^1,d_{cc})$ equipped with the standard sub-Riemannian structure. In particular, we present an exhaustive analysis in a new case of a map $F\in C_H^1({\mathbb{H}}^1,{\mathbb{R}}^2)$ 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 \textsf{vertical curves}) can be of rough nature and not belong to the class of intrinsic regular manifolds.

arXiv commentin french, 54 pages, 7 figures

Cited by in corpus (2)

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