The construction of rotation surfaces of constant mean curvature and the corresponding Lagrangians (Q436065)

The second order differential equation characterizing constant mean curvature surfaces in \(S^3\), with a metric \(g_{\alpha , \beta }\) depending on two positive scalars, is put in correspondence with the variational equation of a Lagrangian \(L\) of the type \(L=\) kinetic energy + potential \(G(\theta )\). The potential associated to the \(S^1\)-equivariant case is found: \(G(\theta )=\alpha \sqrt{\alpha +\beta }\pi H\cos 2\theta \) and as application it is shown that the torus \(S^1(\frac{\sqrt{2\alpha +\beta }}{2})\times S^1(\frac{\sqrt{2\alpha +\beta }}{2})\) is minimally embedded in \((S^3, g_{\alpha , \beta })\).