A constant mean curvature annulus tangent to two identical spheres is Delauney

Abstract

We show that a compact embedded annulus of constant mean curvature in ℝ 3 tangent to two spheres of the same radius along its boundary curves and having nonvanishing Gaussian curvature is part of a Delaunay surface. In particular, if the annulus is minimal, it is part of a catenoid. We also show that a compact embedded annulus of constant mean curvature with negative meeting a sphere tangentially and a plane at a constant contact angle ≥ π/2 (in the case of positive Gaussian curvature) or ≤ π/2 (in the negative case) is part of a Delaunay surface. Thus, if the contact angle is ≥ π/2 and the annulus is minimal, it is part of a catenoid.