A general halfspace theorem for constant mean curvature surfaces

In this paper, we prove a general halfspace theorem for constant mean curvature surfaces. Under certain hypotheses, we prove that, in an ambient space M, any constant mean curvature H0 surface on one side of a constant mean curvature H0 surface Σ0 is an equidistant surface to Σ0. The main hypotheses of the theorem are that Σ0 is parabolic and the mean curvature of the equidistant surfaces to Σ0 evolves in a certain way.