On complete constant mean curvature vertical multigraphs in E(κ,τ)

We prove that any complete surface with constant mean curvature in a homogeneous space E(κ,τ) which is transversal to the vertical Killing vector field is, in fact, a vertical graph. As a consequence we get that any orientable, parabolic, complete, immersed surface with constant mean curvature H in E(κ,τ) (different from a horizontal slice in S^2xR) is either a vertical cylinder or a vertical graph (in both cases, it must be 4H^2+κ\leq0).

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