Cylinders as Left Invariant CMC Surfaces in ${\mathrm{Sol}}_3$ and $E(\kappa ,\tau )$-Spaces Diffeomorphic to ${\mathbb{R}}^3$

In the present paper we give a geometric proof for the existence of cylinders with constant mean curvature $H>H(X)$ in certain simply connected homogeneous three-manifolds $X$ diffeomorphic to ${\mathbb{R}}^3$, which always admit a Lie group structure. Here, $H(X)$ denotes the critical value for which constant mean curvature spheres in $X$ exist. Our cylinders are generated by a simple closed curve under a one-parameter group of isometries, induced by left translations along certain geodesics. In the spaces ${\mathrm{Sol}}_3$ and ${\stackrel{}{\mathrm{PSL}}}_2(\mathbb{R})$ we establish existence of new properly embedded constant mean curvature annuli. We include computed examples of cylinders in ${\mathrm{Sol}}_3$ generated by non-embedded simple closed curves.