Symmetric surfaces of constant mean curvature in 𝕊3

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.