Radius estimates for nearly stable H-hypersurfaces of dimension 2, 3, and 4

In this paper, we study the geometry of complete constant mean curvature (CMC) hypersurfaces immersed in an (n + 1)-dimensional Riemannian manifold N (n = 2, 3 and 4) with sectional curvatures uniformly bounded from below. We generalise radius estimates given by Rosenberg [Constant mean curvature surfaces in homogeneously regular 3-manifolds, Bull. Aust. Math. Soc. 74 (2006) 227-238] (n = 2) and by Elbert et al. [Stable constant mean curvature hypersurfaces, Proc. Amer. Math. Soc. 135 (2007) 3359-3366] and Cheng [On constant mean curvature hypersurfaces with finite index, Arch. Math. (Basel) 86 (2006) 365-374] (n = 3, 4) to nearly stable CMC hypersurfaces immersed in N. We also prove that certain CMC hypersurfaces effectively embedded in N must be proper.