Abstract

Let $M$ be an $n$-dimensional closed hypersurface with constant mean curvature and constant scalar curvature in a unit sphere. Denote by $H$ and $S$ the mean curvature and the squared length of the second fundamental form. We prove that if $\alpha (n,H)\le S\le \alpha (n,H)+C_n{H}^2$, where $n\ge 4$ and $H\ne 0$, then $S=\alpha (n,H)$ and $M$ is a Clifford torus ${\mathbb{𝕊}}^{n-1}(r_1)\times {\mathbb{𝕊}}^1(r_2)$. Here

and $C_n$ is a positive constant explicitly depending on $n$. The emphasis is that our gap theorem imposes no restriction on the range of mean curvature. Moreover, we obtain gap theorems for complete hypersurfaces with constant mean curvature and constant scalar curvature in space forms.

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