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

Abstract

ABSTRACT. 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 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 Sol 3 and PSL 2 (R) we establish existence of new properly embedded constant mean curvature annuli. We include computed examples of cylinders in Sol 3 generated by non-embedded simple closed curves.