Constant mean curvature spheres in homogeneous three-spheres

TL;DR

This paper classifies all immersed constant mean curvature spheres in homogeneous three-spheres, establishing existence and uniqueness for each mean curvature value, regardless of the specific homogeneous metric.

Contribution

It provides a complete classification and uniqueness result for constant mean curvature spheres in arbitrary homogeneous three-spheres.

Findings

Existence of a unique constant mean curvature H-sphere for each H in real numbers.

Classification applies to all homogeneous metrics on three-spheres.

Results extend known classifications to more general homogeneous spaces.

Abstract

We give a complete classification of the immersed constant mean curvature spheres in a three-sphere with an arbitrary homogenous metric, by proving that for each $H\in\mathbb{R}$, there exists a constant mean curvature $H$-sphere in the space that is unique up to an ambient isometry.