In this paper, several rigidity results for hypersurfaces with constant mean curvature in space forms are given, focusing in the case of the sphere \(\mathbb{S}^{n+1}(1)\). Techniques are based on H. Alencar and M. P. do Carmo [Proc. Am. Math. Soc. 120, No. 4, 1223–1229 (1994; Zbl 0802.53017)]. A key role is played by the umbilicity operator \(\phi:=HI-A\), where \(H\) is the constant mean curvature and \(A\) is the shape operator.
Given a hypersurface in \(\mathbb{S}^{n+1}(1)\) with constant mean curvature \(H\), assuming the upper bound \(|\phi|^2\leq B_{H,k}\), where \(B_{H,k}\) is the square of the positive root of the polynomial \[ p_{H,k}(x)=x^2+{{n(n-2k)}\over{\sqrt{nk(n-k)}}}\,Hx-n(H^2+1), \] and \(\text{tr}(\phi^3)\leq C_{n,k}|\phi|^3\), where \(C_{n,k}=(n-2k)/\sqrt{nk(n-k)}\), a first rigidity result is proven in Theorem 1.2: Either the hypersurface is totally umbilical or a generalized Clifford torus with constant mean curvature, i.e., a product of spheres.
Assuming an upper bound on \(|A|^2\) and a lower bound on \(\text{tr}(A^3)\), a second rigidity result in \(\mathbb{S}^{n+1}(1)\) is proven in Theorem 1.3. Theorem 1.2 is generalized to space forms in Theorem 1.4.
The main tools in the proofs are a Simons-type formula for the Laplacian of \(|\phi|^2\) (Lemma 2.1) and Lemma 2.2 related to Okamura’s inequality.