Invariant constant mean curvature tubes in homogeneous spaces

TL;DR

This paper investigates the global geometry of constant mean curvature tubes in homogeneous spaces, focusing on their embeddedness, foliation properties, and numerical analysis of isoperimetric profiles.

Contribution

It provides new insights into the geometry of invariant CMC tubes, including embeddedness criteria, foliation results, and numerical studies of isoperimetric profiles in compact cases.

Findings

Established conditions for embeddedness of CMC tubes.

Proved foliation results for these tubes.

Numerically analyzed the isoperimetric profile in compact homogeneous spaces.

Abstract

We study the global geometry of families of tubes of constant mean curvature invariant under screw-motions in homogeneous $\mathbb{E}(\kappa,\tau)$-spaces. In particular, we study embeddedness and prove a foliation result. Moreover, we study numerically the isoperimetric profile in the compact case.