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 [Formula: see text]-dimensional Riemannian manifold [Formula: see text] ([Formula: see text] and [Formula: see text]) 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] ([Formula: see text]) 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] ([Formula: see text]) to nearly stable CMC hypersurfaces immersed in [Formula: see text]. We also prove that certain CMC hypersurfaces effectively embedded in [Formula: see text] must be proper.