Constant mean curvature surfaces in 𝑀²×𝐑

The subject of this paper is properly embedded H − H- surfaces in Riemannian three manifolds of the form M 2 × R M^2\times \mathbf {R} , where M 2 M^2 is a complete Riemannian surface. When M 2 = R 2 M^2={\mathbf R}^2 , we are in the classical domain of H − H- surfaces in R 3 {\mathbf R}^3 . In general, we will make some assumptions about M 2 M^2 in order to prove stronger results, or to show the effects of curvature bounds in M 2 M^2 on the behavior of H − H- surfaces in M 2 × R M^2\times \mathbf {R} .