An optimal gap theorem for scalar curvature of CMC hypersurfaces in a sphere
Authors: Lei Li , Xu Hongwei , Xu Zhiyuan 
Published: May 2022

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper S Superscript n minus k Baseline left-parenthesis StartFraction 1 Over StartRoot 1 plus lamda Subscript k Superscript 2 Baseline EndRoot EndFraction right-parenthesis times double-struck upper S Superscript k Baseline left-parenthesis StartFraction lamda Subscript k Baseline Over StartRoot 1 plus lamda Subscript k Superscript 2 Baseline EndRoot EndFraction right-parenthesis">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper S Superscript n minus k minus 1 Baseline left-parenthesis StartFraction 1 Over StartRoot 1 plus lamda Subscript k plus 1 Superscript 2 Baseline EndRoot EndFraction right-parenthesis times double-struck upper S Superscript k plus 1 Baseline left-parenthesis StartFraction lamda Subscript k plus 1 Baseline Over StartRoot 1 plus lamda Subscript k plus 1 Superscript 2 Baseline EndRoot EndFraction right-parenthesis">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha Subscript k Baseline left-parenthesis n comma upper H right-parenthesis equals n plus StartFraction n cubed Over 2 k left-parenthesis n minus k right-parenthesis EndFraction upper H squared minus StartFraction n left-parenthesis n minus 2 k right-parenthesis Over 2 k left-parenthesis n minus k right-parenthesis EndFraction StartRoot n squared upper H Superscript 4 Baseline plus 4 k left-parenthesis n minus k right-parenthesis upper H squared EndRoot">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda Subscript k Baseline equals StartFraction n StartAbsoluteValue upper H EndAbsoluteValue plus StartRoot n squared upper H squared plus 4 k left-parenthesis n minus k right-parenthesis EndRoot Over 2 left-parenthesis n minus k right-parenthesis EndFraction">

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)}

.