Uniqueness of the Catenoid and the Delaunay Surface via a One-Parameter Family of Touching Spheres

Abstract

We provide a new characterization of the catenoid and the Delaunay surface among constant mean curvature surfaces touching a one-parameter family of spheres along arcs in a 3-dimensional Euclidean space. Furthermore we prove a higher-dimensional analogue for constant mean curvature hypersurfaces. We also give a generalization of the classical theorem due to Joachimsthal. © 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.