Let
M
n
M^n
be a closed hypersurface with constant mean curvature and constant scalar curvature in the unit sphere
S
n
+
1
\mathbb {S}^{n+1}
. Denote by
S
S
and
H
H
the squared length of the second fundamental form and the mean curvature of
M
M
, respectively. For a fixed integer
k
k
, where
1
≤
k
≤
n
−
2
1\leq k\leq n-2
, we prove that there exists a positive constant
γ
(
n
)
\gamma (n)
depending only on
n
n
, such that if
|
H
|
>
γ
(
n
)
|H|>\gamma (n)
and
α
k
(
n
,
H
)
≤
S
≤
α
k
+
1
(
n
,
H
)
\alpha _k(n,H)\leq S \leq \alpha _{k+1}(n,H)
, then either
S
=
α
k
(
n
,
H
)
S=\alpha _k(n,H)
and
M
M
must be the Clifford torus
S
n
−
k
(
1
1
+
λ
k
2
)
×
S
k
(
λ
k
1
+
λ
k
2
)
\mathbb {S}^{n-k}\big (\frac {1}{\sqrt {1+\lambda _k^2}}\big )\times \mathbb {S}^{k}\big (\frac {\lambda _k}{\sqrt {1+\lambda _k^2}}\big )
, or
S
=
α
k
+
1
(
n
,
H
)
S=\alpha _{k+1}(n,H)
and
M
M
is the Clifford torus
S
n
−
k
−
1
(
1
1
+
λ
k
+
1
2
)
×
S
k
+
1
(
λ
k
+
1
1
+
λ
k
+
1
2
)
\mathbb {S}^{n-k-1}\big (\frac {1}{\sqrt {1+\lambda _{k+1}^2}}\big )\times \mathbb {S}^{k+1}\big (\frac {\lambda _{k+1}}{\sqrt {1+\lambda _{k+1}^2}}\big )
. Here
α
k
(
n
,
H
)
=
n
+
n
3
2
k
(
n
−
k
)
H
2
−
n
(
n
−
2
k
)
2
k
(
n
−
k
)
n
2
H
4
+
4
k
(
n
−
k
)
H
2
\alpha _k(n,H)=n+\frac {n^3}{2k(n-k)}H^2-\frac {n(n-2k)}{2k(n-k)}\sqrt {n^2H^4+4k(n-k)H^2}
and
λ
k
=
n
|
H
|
+
n
2
H
2
+
4
k
(
n
−
k
)
2
(
n
−
k
)
\lambda _k=\frac {n|H|+\sqrt {n^2H^2+4k(n-k)}}{2(n-k)}
.