Cylinders as left invariant CMC surfaces in Sol3 and E(κ,τ)-spaces diffeomorphic to R3

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 R3, 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 Sol3 and PSL˜2(R) we establish existence of new properly embedded constant mean curvature annuli. We include computed examples of cylinders in Sol3 generated by non-embedded simple closed curves.