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Abstract. In this article, we study complete surfaces Σ, isometrically immersed in the product space H × R or S × R having positive extrinsic curvature K e . Let K i denote the intrinsic curvature of Σ. Assume that the equation aK i + bK e = c holds for some real constants a = , b > and c. e main result of this article state that when such a surface is a topological sphere it is rotational.

Abstract. We study the existence of multi-graphs which are immersed in E 3 (κ, τ ), having constant mean curvature H, where E 3 (κ, τ ) is a homogeneous, simply connected 3-manifold whose isometry group has dimension 4.

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A screw motion surfaces in P SL 2 (R, τ) is either a minimal or a constant mean curvature surface which is invariant by helicoidal isometries. In this paper, we study the geometric behavior of such screw motion surfaces.