在sub-Riemannian geometry中，horizontal mean curvature在曲面上的可積性扮演著重要的角色，並且在許多研究中常被假設是對的。但是，在Heisenberg group中，存在著一個C2 曲面，使得這個曲面上的horizontal mean curvature不是局部可積的。
本篇論文探討的問題是：在Heisenberg group 的曲面上，horizontal mean curvature在isolated characteristic point 附近的局部可積性。我們主要探討在Heisenberg group裡面其中一類的曲面，並建立一些在這些曲面上，horizontal mean curvature
可積性的充份條件。

The integrability of the horizontal mean curvature H plays a crucial role in
sub-Riemannian geometry and is often assumed in many works. However, in the
first Heisenberg group, which serves as a fundamental example of sub-Riemannian
manifolds, there exists a C2 surface where H for which fails to be locally integrable
near the characteristic set.
In this paper, we investigate the local integrability of H for surfaces near isolated
characteristic points in the first Heisenberg group, as conjectured in [DGN12]. We
study a specific class of surfaces and establish sufficient conditions that support the
validity of the conjecture.

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