Screw motion surfaces of constant mean curvature in homogeneous 3-manifolds

TL;DR

This paper classifies constant mean curvature surfaces invariant under screw motions in homogeneous 3-manifolds, including new results for Berger spheres and introducing a novel family called tubes.

Contribution

It provides the first classification of such surfaces in Berger spheres and introduces a new family called tubes, extending previous classifications.

Findings

Complete classification of screw motion CMC surfaces in $ ext{E}( ext{kappa}, au)$.

First classification results for Berger spheres.

Introduction of a new family of surfaces called tubes.

Abstract

We study the geometry of non-minimal surfaces of supercritical constant mean curvature invariant under screw motions in the homogeneous 3-manifolds $\mathbb{E}(\kappa,\tau)$ including the space-forms of non-negative curvature. We give a complete classification, thereby unifying and extending various previous results. We give the first classification for the Berger sphere case, and we exhibit a new family of screw motion CMC surfaces, called tubes.