Calculus of Variations and Geometric Measure Theory
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Z. Balogh

Size of charecteristic sets and functions with presribed gradient

created on 11 Apr 2003
modified on 28 Apr 2004


Published Paper

Inserted: 11 apr 2003
Last Updated: 28 apr 2004

Journal: Crelle's Journal
Volume: 564
Pages: 63-83
Year: 2003

to appear


The characteristic set $C(S)$ of a codimension 1 submanifold $S$ in the Heisenberg group $\hn$ consists of those points where the tangent space of $S$ coincides with the space spanned by the left invariant horizontal vector fields of $\hn$. We prove that if $S$ is $C^1$ smooth then $C(S)$ has vanishing $2n+1$-dimensional Hausdorff measure with respect to the Heisenberg metric. If $S$ is $C^2$ smooth then $C(S)$ has Hausdorff dimension less or equal than $n$, both with respect to the Euclidean and Heisenberg metrics. On the other hand, $C(S)$ can have a positive $2n$-dimensional Hausdorff measure with respect to the Euclidean metric even for hypersurfaces of class $\cap_{0< \alpha < 1} C^{1,\alpha}$. This is shown by constructing a function of class $\cap_{0< \alpha < 1} C^{1,\alpha}$ with a prescribed gradient on a large measure set.

Keywords: Hausdorff measure, Heisenberg group, Characteristic set


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