Invariant constant mean curvature tubes in homogeneous spaces

We study the global geometry of families of tubes of constant mean curvature invariant under screw-motions in homogeneous E ( κ , τ ) -spaces. In particular, we study embeddedness and prove a foliation result. Moreover, we numerically analyze the isoperimetric profile in the compact case. • We prove existence of a continuous family of CMC tubes around any screw motion geodesic in E ( κ , τ ) , that converges to this geodesic. • For κ > 0 , we prove that a subfamily always foliates an open set of ambient space. In some cases the foliation is global. • For κ ≤ 0 , we prove the embeddedness of some tubes. • Unlike all other tubes previously described in the literature, some of these tubes do not admit a dihedral symmetry of order 4. • For Heisenberg space Nil 3 we proof a partial uniqueness result.