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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
${\mathbb{\mathbb{S}}}^{n1}({r}_{1})\times {\mathbb{\mathbb{S}}}^{1}({r}_{2})$.
Here
$$\alpha (n,H)=n+\frac{{n}^{3}}{2(n1)}{H}^{2}\frac{n(n2)}{2(n1)}\sqrt{{n}^{2}{H}^{4}+4(n1){H}^{2}},$$ 
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.

Dedicated to Professor Hesheng Hu on
the occasion of her 95th birthday

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Keywords
complete hypersurface, gap theorem, mean curvature, scalar
curvature

Mathematical Subject Classification
Primary: 53C24, 53C40

Milestones
Received: 6 July 2021
Revised: 23 March 2022
Accepted: 23 April 2022
Published: 1 August 2022

