This paper is devoted to the study of constant-mean-curvature-hypersurfaces \(M^n\) (called here \(H\)-hypersurfaces) in a Riemannian manifold \(\mathcal N^{n+1} \; (n = 3, 4)\,\) which has sectional curvature uniformly bounded from below. If \(\text{sec} (\mathcal N)\) denotes the infimum of the sectional curvatures of \(\mathcal N\) and \(H\) the mean curvature vector of the immersion \(M^n \to \mathcal N^{n+1}, \) the authors prove the following:
Theorem: If \(M^n\) is a (strongly) stable and complete hypersurface, then for \(n = 3, 4 \) there exists a constant \(c = c (n, H, \text{sec} (\mathcal N))\) such that for every point \(p \in M \) one has \( \text{dist}_M (p, \partial M) \leq c,\) whenever \(| H| > 2 \sqrt{|\min\{0,\text{sec} (\mathcal N) \}| }.\)
The notion of strong stability they use is with regard to the stability operator \(L = \Delta + | A| ^2 + \text{Ric} (N), \) with \(A\) being the shape operator and \(\text{Ric} (N)\) the Ricci curvature of \(\mathcal N\) in the direction of the unit normal \(N\) to the hypersurface, whereby it is required that \(- \int_M u\, Lu \geq 0,\) for any smooth function \(u\) with compact support.
As an immediate corollary of this diameter estimate it follows that a complete stable \(H\)-hypersurface of \(\mathcal N^{n+1}\; (n = 3, 4)\,\) with \(| H| \) satisfying the inequality condition of above has a non-empty boundary, \(\partial M \neq 0\). In other words, there are no complete (strongly) stable \(H\)-hypersurfaces without boundary, provided that \(| H| \) is sufficiently large, and in particular no complete, strongly stable non-minimal \(H\)-hypersurfaces in \(\mathbb R^{n+1}\) without boundary. The above diameter estimate is also independently established by X. Cheng [Arch. Math. 86, No. 4, 365–374 (2006; Zbl 1095.53043)].