Constant mean curvature surfaces of Delaunay type along a closed geodesic (Q2046683)

The author proves existence of constant mean curvature (CMC) Delaunay surfaces of unduloïd type embedded, along a closed simple geodesic, in a given \(3\)-dimensional Riemannian manifold \(M\) by \[X(s, \theta) = (\phi(s)\cos\theta, \phi(s)\sin\theta, \psi(s)), (s,\theta)\in\mathbb R\times S^1,\] where \[\phi' + (\phi^2 + \psi^2)^2 = \phi^2, \psi' = \phi^2 + \tau,\] and \(\tau\) is a constant involved in the initial conditions \(\phi(0) = \frac12(1 \sqrt{1 - 4\tau})\), \(\psi(0) = 0\).