Rigidity theorems for hypersurfaces with constant mean curvature

Let M ⁿ be a compact oriented hypersurface of a unit sphere (mathbb{S}^{n + 1} ) (1) with constant mean curvature H. Given an integer k between 2 and n ? 1, we introduce a tensor ? related to H and to the second fundamental form A of M, and show that if |?|² ≤ B _H,k and tr(? ³) ≤ C _n,k |?|³, where B _H,k and C _n,k are numbers depending only on H, n and k, then either |?|² ≡ 0 or |?|² ≡ B _H,k . We characterize all M ⁿ with |?|² ≡ B _H,k . We also prove that if (left| A right|^2 leqslant 2sqrt {k(n - k)}) and tr(? ³) ≤ C _n,k |?|³ then |A|² is constant and characterize all M ⁿ with |A|² in the interval (left[ {0,2sqrt {kleft( {n - k} right)} } right] ) . We also study the behavior of |?|², with the condition additional tr(? ³) ≤ C _n,k |?|³, for complete hypersurfaces with constant mean curvature immersed in space forms and show that if sup _M |?|² = B _H,k and this supremum is attained in M ⁿ then M ⁿ is an isoparametric hypersurface with two distinct principal curvatures of multiplicities k y n ? k. Finally, we use rotation hypersurfaces to show that the condition on the trace of ? ³ is necessary in our results; more precisely, for each integer k with 2 ≤ k ≤ n ? 1 and (H geqslant 1/sqrt {2n - 1} ) there is a complete hypersurface M ⁿ in (mathbb{S}^{n + 1} ) (1) with constant mean curvature H such that sup _M |?|² = B _H,k , and this supremum is attained in M ⁿ , and which is not a product of spheres.