Integrability of the sub-Riemannian mean curvature at degenerate characteristic points in the Heisenberg group

TL;DR

This paper investigates the conditions under which the sub-Riemannian mean curvature is integrable near characteristic points in the Heisenberg group, introducing a new concept of mildly degenerate points and confirming integrability for real-analytic surfaces with discrete characteristic sets.

Contribution

It introduces the concept of mildly degenerate characteristic points and proves local integrability of mean curvature for real-analytic surfaces with discrete characteristic sets in the Heisenberg group.

Findings

Mean curvature is integrable near mildly degenerate characteristic points.

Real-analytic surfaces with discrete characteristic sets have locally integrable mean curvature.

Partial answer to a question by Danielli-Garofalo-Nhieu on mean curvature integrability.

Abstract

We address the problem of integrability of the sub-Riemannian mean curvature of an embedded hypersurface around isolated characteristic points. The main contribution of this note is the introduction of a concept of mildly degenerate characteristic point for a smooth surface of the Heisenberg group, in a neighborhood of which the sub-Riemannian mean curvature is integrable (with respect to the perimeter measure induced by the Euclidean structure). As a consequence we partially answer to a question posed by Danielli-Garofalo-Nhieu in [Danielli D., Garofalo N., Nhieu D.M., Proc. Amer. Math. Soc., 2012], proving that the mean curvature of a real-analytic surface with discrete characteristic set is locally integrable.