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

Sung-ho Park

Vol. 251 (2011), No. 1, 197–206

DOI: 10.2140/pjm.2011.251.197

Abstract

We show that a compact embedded annulus of constant mean curvature in ℝ³ 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.

Keywords

minimal annulus, contact angle, sphere, parallel surface

Mathematical Subject Classification 2000

Primary: 53A10

Milestones

Received: 7 February 2010

Accepted: 12 April 2010

Published: 22 April 2011