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The main point of this paper is that, under suitable conditions on the meancurvature and the Ricci curvature of the ambient space, we can extendChoi-Schoen's Compactness Theorem to compact embedded minimal surfaces tosimple immersed compact H-surfaces in a Riemannian manifold with positive Riccicurvature (the mean curvature small depending on the Ricci curvature). Also, weprove that the space of convex embedded (fixed) constant mean curvaturehypersurfaces in a simply connected 1/4-pinched manifold is compact.

The space of Constant Mean Curvature surfaces in compact Riemannian Manifolds