On the angle of complete CMC hypersurfaces in Riemannian product spaces

We deal with complete two-sided hypersurfaces immersed with constant mean curvature in a Riemannian product space R×MnR×Mn. First, when the fiber MnMn is compact with positive sectional curvature, we apply Bochner's technique jointly with Omori–Yau's generalized maximum principle to establish sufficient conditions which assure that such a hypersurface ΣnΣn is a slice {t}×Mn{t}×Mn, provided that its (not necessarily constant) angle has some suitable boundedness. Afterwards, when MnMn is complete with nonnegative sectional curvature, we proceed with our technique in order to guarantee that a complete constant mean curvature hypersurface immersed in R×MnR×Mn is minimal. Moreover, we characterize the uniqueness of entire vertical graphs Σn(u)Σn(u) with constant mean curvature into R×MnR×Mn, in terms of the norm of the gradient of the function u   which determines such a graph. A nontrivial example of entire vertical graph with constant angle in R×H2R×H2 is also given, showing that our results do not hold when the fiber of the ambient product space has negative sectional curvature.