In this paper we study constant mean curvature surfaces in
a product space, M2 × R, where M2 is a complete Riemannian
manifold. We assume the angle function ν = hN, @
@t i does not
change sign on . We classify these surfaces according to the
infimum c() of the Gaussian curvature of the projection of .
When H 6= 0 and c() ≥ 0, then is a cylinder over a complete
curve with curvature 2H. If H = 0 and c() ≥ 0, then must be
a vertical plane or is a slice M2 ×{t}, or M2 ≡ R2
with the flat
metric and is a tilted plane (after possibly passing to a covering
space).
When c() < 0 and H >
p
−c()/2, then is a vertical cylinder
over a complete curve of M2 of constant geodesic curvature
2H. This result is optimal.
We also prove a non-existence result concerning complete multigraphs
in M2 × R, when c(M2) < 0.