Symmetric surfaces of constant mean curvature in 𝕊³

Ryan Hynd, Sung-ho Park and John McCuan

Vol. 241 (2009), No. 1, 63–115

DOI: 10.2140/pjm.2009.241.63

Abstract

We introduce two notions of symmetry for surfaces in 𝕊³. The first, special spherical symmetry, generalizes the notion of rotational symmetry, and we classify all complete surfaces of constant mean curvature having this symmetry. These surfaces turn out to also be rotationally symmetric, so our characterization answers a question first posed by Hsiang in 1982 and also considered by several authors since. From this point of view, these are the Delaunay surfaces of 𝕊³.

Our second notion of symmetry, spherical symmetry, is a substantial, and we believe important, technical generalization of special spherical symmetry. We classify all compact surfaces of constant mean curvature having this symmetry. We show in particular that the only compact embedded minimal surfaces possessing this kind of symmetry are the great spheres and the Clifford torus.

We derive from our classification theorem a special case of Lawson’s conjecture that the only embedded minimal torus in 𝕊³ is the Clifford torus.

Keywords

Delaunay surfaces, constant mean curvature, Willmore surfaces, Clifford torus, minimal surfaces, submanifolds in space forms, Lawson’s conjecture

Mathematical Subject Classification 2000

Primary: 53A10

Milestones

Received: 12 March 2008

Revised: 19 January 2009

Accepted: 27 January 2009

Published: 1 May 2009