Rigidity theorems for hypersurfaces with constant mean curvature

Abstract

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 $\left| A \right|^2 \leqslant 2\sqrt {k(n - k)}$ and tr(⌽ ³) ≤ C _n,k |⌽|³ then |A|² is constant and characterize all M ⁿ with |A|² in the interval $\left[ {0,2\sqrt {k\left( {n - k} \right)} } \right]$.

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.