Rotationally invariant constant mean curvature surfaces in homogeneous 3-manifolds

TL;DR

This paper classifies rotationally invariant constant mean curvature surfaces in certain homogeneous 3-manifolds, providing explicit descriptions, new examples, and insights into their geometric properties and isoperimetric relations.

Contribution

It offers a comprehensive classification of invariant CMC surfaces in Berger spheres and SL(2,R), including explicit descriptions and new Delaunay-type examples.

Findings

All CMC spheres in these spaces are explicitly described.

Some CMC spheres are not embedded.

Certain CMC spheres do not solve the isoperimetric problem.

Abstract

We classify constant mean curvature surfaces invariant by a 1-parameter group of isometries in the Berger spheres and in the special linear group Sl(2, R). In particular, all constant mean curvature spheres in those spaces are described explicitly, proving that they are not always embedded. Besides new examples of Delaunay-type surfaces are obtained. Finally the relation between the area and volume of these spheres in the Berger spheres is studied, showing that, in some cases, they are not solution to the isoperimetric problem.