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 $M^3$, any constant mean curvature $H_0$ surface on one side of a constant mean curvature $H_0$ surface $\Sigma_0$ is an equidistant surface to $\Sigma_0$. The main hypotheses of the theorem are that $\Sigma_0$ is parabolic and the mean curvature of the equidistant surfaces to $\Sigma_0$ evolves in a certain way.