Sub-Riemannian constant mean curvature surfaces in the Heisenberg group as limits (Q2505615)

It is well-known that the \(3\)-dimensional Heisenberg group \(H\) is a non-commutative nilpotent Lie group, with a left invariant non-elliptic Heisenberg (sub-) Laplacian \(\bigtriangleup\) and associated sub-Riemannian metric \(g_0\), denoted by \((H, g_0)\). Let \((H, g_u)\) be a family of Riemannian manifolds, \(u > 0\). In this paper, for each \(u > 0\), by considering some invariant surfaces with constant mean curvature in \((H, g_u)\), the author shows that \((H, g_0)\) can be recovered as a limit of \((H, g_u)\), and \(\bigtriangleup\) can be seen as a limit of (elliptic) Laplace-Beltrami operators. Based on the above result, the author gives a definition of the mean curvature of a hypersurface in \((H, g_0)\) as the limit of the mean curvature in \((H, g_u)\).