Preprint

Inserted: 10 nov 2026
Last Updated: 10 nov 2026

Year: 2026

Abstract:

We investigate the local integrability of the horizontal mean curvature near characteristic points of surfaces in the first Heisenberg group. We prove that, for every embedded surface of class $C^{2,\alpha}$, with $\alpha>0$, the horizontal mean curvature is locally integrable with respect to (any) Riemannian surface measure, without any assumption on the structure of the characteristic set. We further establish that the horizontal unit normal belongs to $BV_{\mathrm{loc}}$ and characterize its $W^{1,1}_{\mathrm{loc}}$ regularity in terms of the vanishing one-dimensional Hausdorff measure of the characteristic set. Finally, we answer an open question by Danielli, Garofalo and Nhieu by constructing a $C^2$ surface with a unique characteristic point whose horizontal mean curvature fails to be locally integrable, thus showing that the additional H\"older regularity cannot, in general, be replaced by $C^2$ regularity.

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