Rigidity theorems for hypersurfaces with constant mean curvature (Q487037)

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 \textit{H. Alencar} and \textit{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.