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 APM  Vol.11 No.5 , May 2021
Lp p-Harmonic 1-Forms on δ-Stable Hypersurface in Space Form with Nonnegative Bi-Ricci Curvature
Abstract: In this paper, we investigate the space of Lp p-harmonic 1-forms on a complete noncompact orientable δ-stable hypersurface Mm that is immersed in space form with nonnegative BiRic curvature. We prove the nonexistence of Lp p-harmonic 1-forms on Mm. Moreover, we obtain some vanishing properties for this class of harmonic 1-forms.

1. Introduction

Let x : M m c m + 1 , be a complete noncompact orientable stable hypersurface M m immersed in space form c m + 1 with nonnegative BiRic curvature bounded from below. Fix a point x M and let { e 1 , , e m + n } be local orthogonal frame of c m + 1 such that { e 1 , , e m } are tangent fields of M m . Now we will use the following convention on the ranges of induces: 1 i , j , k , m and m + 1 α m + n . Let A denote the second fundamental form of x, is define by

A ( X , Y ) = α ¯ X Y , e α e α , X , Y T x M , (1)

where ¯ is the Levi-Civita connection on the ambient manifold c m + 1 . Here, we denote h i j α = ¯ e i e j , e α , then | A | 2 = α i , j ( h i j α ) 2 denote the square length of the norm of A and the mean curvature vector field H is define by

H = α H α e α = 1 m α i h i i α e α . (2)

The traceless second fundamental form Φ is defined by

Φ ( X , Y ) = A ( X , Y ) X , Y H , X , Y T x M , (3)

where , is the metric of Mm. A simple computational shows that

| Φ | 2 = | A | 2 m | H | 2 . (4)

In particular, if Φ 0 , then Mm is totally umbilical see ( [1] [2] [3] [4] ).

Definition 1.1. [5], Let Mm be an m-dimensional Riemannian manifold, μ , ν be orthonormal tangent vectors at a point p M m and D be the 2-plane generated by μ and ν . The bi-Ricci curvature of the plane D is defined by

B i R i c ( D ) = B i R i c ( μ , ν ) : = R i c ( μ , μ ) + δ R i c ( ν , ν ) R ( μ , ν , μ , ν ) , (5)

where δ > 0 , R ( μ , ν , μ , ν ) denotes the sectional curvature and B i R i c ( μ , ν ) , denotes the BiRic curvature in the direction μ , ν . Observe that when m = 3 , we have that

2 B i R i c ( μ , ν ) = R ( μ , ν , μ , ν ) . (6)

In general, BiRic is the sum of the sectional curvatures overall mutually orthogonal 2-planes containing at least one of these tangent vectors (see [6] ).

The vanishing theorems for Lp p-harmonic 1-forms on complete noncompact submanifolds have been studied extensively by many mathematicians from various points of views. There are some relations between the geometry and topology of a manifold and the space of Lp p-harmonic 1-forms. According to the decomposition theorem by Hodge-Rham [7], Lp p-harmonic 1-forms completely represent the Lp cohomology of the underlying manifold. The nonexistence of nontrivial Lp p-harmonic 1-forms on Mm implies that any codimension one cycle on Mm must disconnect Mm, also the uniqueness of the non-parabolic ends of the underlying manifold. In [8], Li considers hypersurface M m ( 2 m 5 ) with constant means curvature and then drives the same vanishing properties. In [9], Dung studied immersed hypersurface in a weighted Riemannian manifold with weighted BiRici curvature and proved that if such hypersurfaces are weighted stable then the space of L2 weighted harmonic 1-forms is trivial. In [10], Tanno studied a complete noncompact oriented stable minimal hypersurface immersed in a Riemannian manifold with nonnegative BiRic curvature and proved that there are no nontrivial L2 harmonic 1-forms on Mm. In [11], Cheng generalized

Li’s results by assuming that B i R i c m 5 4 H 2 , where H is the mean curvature

of Mm, and is normalized to be equal to the second fundamental form. In [5], the Author proves that there are no nontrivial L2 harmonic 1-forms on a strongly stable hypersurface Mm of a general Riemannian manifold when the bi-Ricci curvature of is no less than certain lower bound, which gives a topological obstruction for the stability of Mm. In [12], Palmer considered L2 harmonic forms on a complete oriented stable minimal hypersurface Mm in m + 1 , and proved that there exist no nontrivial L2 harmonic 1-forms on Mm. In this direction, many Authors give us various results for L2 harmonic 1-forms on stable minimal hypersurfaces (see [13] [14] ). In [15], the Author proved that the nonexistence of L2 harmonic 1-forms on a complete super stable minimal submanifold Mm in hyperbolic space.

The aim of this work is to investigate some vanishing theorems for Lp p-harmonic 1-forms on a complete noncompact orientable stable hypersurface that is immersed in space form with nonnegative BiRic curvature bounded from below.

2. Preliminaries

Let Mm be an m-dimensional Riemannian manifold and the Riemannian structure under a local coordinate system given by

d s 2 = g i j d x i d x j , (7)

where g is the Riemannian metric. We shall make use of the following conventions about indices:

1 = i , j , k , = m , (8)

and shall agree that repeated indices are summed over their ranges. Denote x i by i . The Riemannian curvature tensor R i j k l , the Ricci curvature tensor R i c i j and scalar curvature R ¯ are defined by (see [16] [17] )

R ( X , Y ) Z = ¯ X ¯ Y Z ¯ Y ¯ X Z ¯ [ X , Y ] Z , (9)

where ¯ denotes the Levi-Civita connection on M m and

R i j k l = R ( i , j ) l , k , R i c i j = k g p q R i p j q , R ¯ = 1 i , j n g i j R i c i j . (10)

The Weyl conformal curvature tensor W i j k l and Einstein tensor A i j are defined respectively by

W i j k l = R i j k l 1 m 2 ( R i c j k g i l R i c j l g i k R i c i l g j k ) + 1 ( m 1 ) ( m 2 ) R ¯ ( g i k g j l g i l g j k ) , (11)

and

A i j = R i c i j 1 m g i j R ¯ (12)

By direct computations, we obtain

| A | 2 = | R i c | 2 1 m R ¯ 2 , (13)

| W | 2 = | R | 2 4 m 2 | R i c | 2 + 2 ( m 1 ) ( m 2 ) R ¯ 2 . (14)

Now we define a new tensor B i j k l of type (0,4) as follows:

B i j k l = ( m 3 ) R i j k l ( m 2 ) W i j k l + 1 m 1 R ¯ ( g i k g j l g i l g j k ) . (15)

It is clear that B i j k l has all the symmetries of the curvature tensor R i j k l and the Weyl curvature W i j k l .

B i j k l = B j i k l = B i j l k = B j i l k = B k l i j . (16)

B i j k l = B i k l j + B i l j k = 0. (17)

By direct computations, the BiRici curvature of the plane generated by i , j

1 | i j | 2 B i j i j = 1 g i i g j j g i j 2 ( R i i g j j + g i i R j j 2 R i j g i j R i j i j ) . (18)

So BiRic behaves like a “sectional curvature” of the tensor B i j k l .

B i j k l = R i k g j l + R j l g i k R i l g j k R j k g i l R i j k l . (19)

From (19), we obtain

| B | 2 = | R | 2 + 4 ( m 1 ) | R i c | 2 + R ¯ 2 . (20)

And

| B i j k l ( 2 m 3 ) R ¯ m ( m 1 ) ( g i k g j l g i l g j k ) | 2 = | B | 2 2 ( 2 m 3 ) 2 m ( m 1 ) R ¯ 2 . (21)

Combining (13), (14) and (20), we obtain

| B | 2 = | W | 2 + 4 ( m 3 ) 2 m 2 | A | 2 + 2 ( 2 m 3 ) 2 m ( m 1 ) R ¯ 2 . (22)

From (21) and (22), we obtain

| B i j k l ( 2 m 3 ) R ¯ m ( m 1 ) ( g i k g j l g i l g j k ) | 2 = | W | 2 + 4 ( m 3 ) 2 m 2 | A | 2 . (23)

When the BiRic curvatures of all 2 planes are the same at a point, by the argument of polarization, we have

B i j k l = c ( g i k g j l g i l g j k ) . (24)

We get c = ( 2 m 3 ) R ¯ m ( m 1 ) . Therefore, W = A = 0 by (24) and the Riemannian curvature is constant.

3. The Estimation of the BiRic Curvature

Let M m c m + 1 be a complete noncompact orientable stable hypersurface M m immersed in space form c m + 1 . We shall make use of the following conventions about indices:

1 i , j , k , m , m + 1 α , β m + n .

Denote by ¯ , R ¯ , R i c and B i R i c the Levi-Civita connection, sectional curvature, Ric curvature and BiRic curvature of c m + 1 respectively.

The Gauss equation is

R i j k l = R ¯ i j k l + α ( h i k α h j l α h i l α h j k α ) . (25)

we have

B k l k l = R i c k k + R i c l l R k l k l = i ( R ¯ i k i k + R ¯ i l i l ) R ¯ k l k l . (26)

By the Gauss Equation (25), we have

R i c ( X , X ) = i R ¯ ( X , e i , X , e i ) + h ( X , X ) H i h ( e i , X ) 2 . (27)

Lemma 3.2. [9] Let ( h i j ) i , j = 1 m be a symmetric matrix m × m , m 3 .

And let H = i = 1 m h i i and S = | A | 2 = i , j = 1 m ( h i j ) 2 then

h ( X , X ) H i h ( X , e i ) 2 | X | 2 n 2 { 2 ( m 1 ) H 2 ( m 2 ) H ( m 1 ) ( m S H 2 ) m ( m 1 ) S } . (28)

Assume that X 0 . By the definition of the BiRic in Equation (5), we obtain

R i c ( X , X ) i R ¯ ( X , e i , X , e i ) ( δ S + φ ( H , S ) ) | X | 2 . (29)

Let us first assume that X 0 everywhere. By the definition, we have

i R ¯ ( X , e i , X , e i ) = ( B i R i c ( X | X | , N ) δ R i c ( N , N ) ) | X | 2 . (30)

Combining (29) with (30), we obtain

R i c ( X , X ) { B i R i c ( X | X | , N ) φ ( H , S ) δ ( R i c ( N , N ) + S ) } | X | 2 , (31)

where

φ ( H , S ) = ( m 1 m δ ) S 1 m 2 { 2 ( m 1 ) H 2 ( m 2 ) H ( m 1 ) ( m S H 2 ) } . (32)

From the Bochner formula [18], we have

Δ | ω | 2 = 2 ( | ω | 2 + R i c ( ω , ω ) ) . (33)

Since

Δ | ω | 2 = 2 ( | ω | Δ | ω | + | | ω | | 2 ) . (34)

Combining (33) with (34), we get

| ω | Δ | ω | R i c ( ω , ω ) = | ω | 2 | | ω | | 2 1 m 1 | | ω | | 2 . (35)

Inparticular, we know

R i c ( ω , ω ) ( B i R i c ( X | X | , N ) φ ( H , S ) δ ( R i c ( N , N ) + S ) ) | ω | 2 . (36)

We set q = R i c ( N , N ) + S , thus

R i c ( ω , ω ) ( B i R i c ( X | X | , N ) ( δ q + φ ( H , S ) ) ) | ω | 2 . (37)

4. The Structure of δ-Stable Hypersurfaces in c m + 1

In this section, we assume that c m + 1 is a complete noncompact oriented space form and Mm is a complete noncompact oriented stable hypersurface of c m + 1 . Adapt the same notations as in the previous section and the second fundamental form can be written as h = i , j h i j ω i ω j . We assume that the mean curvature vector is in the same direction as in e m + 1 . We have

H = 1 m i h i i 0. (38)

Definition 4.1. [19], Let x : M m m + 1 , m 3 , be a complete noncompact hypersurface immersed in a Riemannian manifold m + 1 . Then the first eigenvalue of the Laplacian of M is defined by

λ 1 ( M ) M φ 2 M | φ | 2 , (39)

for all smooth function φ C 0 ( M ) .

Definition 4.2. [11], Let Mm be a complete noncompact manifold and let H 0 , Mm is said to be strongly stable if

I ( φ ) = M ( | φ | 2 ( R i c ( N , N ) + S ) φ 2 ) d v 0, φ C 0 ( M ) , (40)

where C 0 is the smooth functions and d v is the volume form.

Definition 4.3. [11], For some number 0 < δ 1 , Mm is δ-stable if

I ( φ ) = M ( | φ | 2 δ ( R i c ( N , N ) + S ) φ 2 ) d v 0, φ C 0 ( M ) , (41)

where S is the square norm of the second fundamental form of Mm. Obviously, given δ 1 > δ 2 , δ1-stable implies δ2-stable. So, that Mm is stable implies that Mm is δ-stable.

Mm is said to be δ-stable or weakly δ-stable if I ( φ ) 0 , φ C 0 satisfying

M φ = 0. (42)

Remark. When H = 0 , i.e. Mm is minimal, then the immersion is called stable if it is in the strong sense, which is different from the stability of the hypersurfaces with constant mean curvature as said above.

5. The Vanishing Theorems

In this section, we presented some vanishing theorems as follows.

Theorem 5.1. Let x : M m c m + 1 , m 3 , be a complete noncompact orientable δ-stable minimal hypersurface M m immersed in space form c m + 1 with nonnegative BiRic curvature bounded from below. If

B i R i c ( Y , N ) ( m 1 m δ ) S .

Then there is no nontrivial Lp p-harmonic 1-form on Mm.

Proof: Using (35) and (37), we obtain

| ω | Δ | ω | 1 m 1 | | ω | | 2 + ( B i R i c ( X | X | , N ) ( δ q + φ ( H , S ) ) ) | ω | 2 . (43)

Since

| ω | p Δ | ω | p = p 1 p | | ω | p | 2 + p | ω | 2 p 2 | ω | Δ | ω | (44)

for any p > 0 . Combining (43) with (44), we get

| ω | p Δ | ω | p p 1 p | | ω | p | 2 + p m 1 | ω | 2 p 2 | | ω | | 2 + p ( B i R i c ( X | X | , N ) ( δ q + φ ( H , S ) ) ) | ω | 2 p (45)

Let η C 0 ( M ) be a smooth function with compact supported. Multiplying both sides of (45) by η 2 and integrating over M, we obtain

M η 2 | ω | p Δ f | ω | p ( 1 m 2 p ( m 1 ) ) M η 2 | | ω | p | 2 + p M ( B i R i c ( X | X | , N ) ( δ q + φ ( H , S ) ) ) η 2 | ω | 2 p (46)

Applying the divergence theorem, we obtain

M η 2 | ω | p Δ f | ω | p = M d i v ( η 2 | ω | p | ω | p ) M η 2 | | ω | p | 2 2 M η | ω | p η , | ω | p = M η 2 | | ω | p | 2 2 M η | ω | p η , | ω | p . (47)

Combining (46) with (47), we get

( 2 p ( m 1 ) ( m 2 ) p ( m 1 ) ) M η 2 | | ω | p | 2 p M ( B i R i c ( X | X | , N ) ( δ q + φ ( H , S ) ) ) η 2 | ω | 2 p 2 M η | ω | p η , | ω | p . (48)

( 2 p ( m 1 ) ( m 2 ) p ( m 1 ) ) M η 2 | | ω | p | 2 p M ( B i R i c ( X | X | , N ) φ ( H , S ) ) η 2 | ω | 2 p 2 M η | ω | p η , | ω | p + p δ M q η 2 | ω | 2 p . (49)

From definition (4.2), we obtain

M | φ | 2 M q φ 2 d v . (50)

Replacing φ by η | ω | p , we obtain

M | ( η | ω | p ) | 2 M q η 2 | ω | 2 p d v . (51)

Combining (49) with (51), we obtain

( 2 p ( m 1 ) ( m 2 ) p ( m 1 ) ) M η 2 | | ω | p | 2 p M ( B i R i c ( X | X | , N ) φ ( H , S ) ) η 2 | ω | 2 p 2 M η | ω | p η , | ω | p + p δ M | ( η | ω | p ) | 2 . (52)

( 2 p ( m 1 ) ( m 2 ) p ( m 1 ) + p δ ) M η 2 | | ω | p | 2 p M ( B i R i c ( X | X | , N ) φ ( H , S ) ) η 2 | ω | 2 p 2 ( p δ + 1 ) M η | ω | p η , | ω | p + p δ M | η | 2 | ω | 2 p . (53)

Note that

2 M η | ω | p η , | ω | p ε M η 2 | | ω | p | 2 + 1 ε M | η | 2 | ω | 2 p , (54)

for some constant ε > 0 .

( 2 p ( m 1 ) ( m 2 ) p ( m 1 ) + p δ | p δ + 1 | ε ) M η 2 | | ω | p | 2 p M ( B i R i c ( X | X | , N ) φ ( H , S ) ) η 2 | ω | 2 p + ( p δ + | p δ + 1 | ε ) M | η | 2 | ω | 2 p . (55)

Thus

A M η 2 | | ω | p | 2 + B M η 2 | ω | 2 p C M | η | 2 | ω | 2 p (56)

Set

A = 2 p ( m 1 ) ( m 2 ) p ( m 1 ) + p δ | p δ + 1 | ε ,

B = p ( B i R i c ( X | X | , N ) φ ( H , S ) )

C = p δ + | p δ + 1 | ε . (57)

Let B r be a geodesic ball of radius r > 0 on Mm centered at the point p. Choose a cut-off function η satisfying

{ η = 0 in M \ B 2 r , η = 1 in B r , | η | 2 r in B 2 r \ B r . (58)

Let 0 η 1 . Using (56) with (58), we obtain

A B r | | ω | p | 2 C ( 4 r 2 ) B 2 r \ B r | ω | 2 p . (59)

Taking r , we get | ω | = 0 , and | ω | = | X | is constant. Hence,

| ω | 2 = m m 1 | | ω | | 2 = 0 , B i R i c ( X | X | , N ) = φ ( H , S ) . (60)

By (60) we obtain

R i c ( ω , ω ) + δ ( R i c ( N , N ) + S ) = 0. (61)

Moreover, since | ω | = 0 , and | ω | = | X | is constant, the Bochner formula implies

R i c ( X , X ) = 0. (62)

Thus, by (62) we can deduce

R i c ( N , N ) + S = 0. (63)

Therefore, for any unite tangent vector Y, it follows from (31) and (63) that

R i c ( Y , Y ) B i R i c ( Y , N ) δ ( R i c ( N , N ) + S ) φ ( H , S ) = B i R i c ( Y , N ) φ ( H , S ) 0. (64)

Thus, using (32) with (64) we get

B i R i c ( Y , N ) ( m 1 m δ ) S 1 m 2 { 2 ( m 1 ) H 2 ( m 2 ) H ( m 1 ) ( m S H 2 ) } . (65)

Assume that Mm is a minimal stable hypersurface immersed in space form c m + 1 . Hence H = 0 , and this implies

B i R i c ( Y , N ) ( m 1 m δ ) S . (66)

Then there is no nontrivial Lp p-harmonic 1-forms on Mm. Hence we get the prove as assumption in theorem.

Corollary 5.2. Let x : M m c m + 1 , m 3 , be a complete noncompact orientable δ-stable minimal hypersurface Mm immersed in space form c m + 1 with nonnegative BiRic curvature bounded from below. If B i R i c φ ( H , S ) 0 for any positive number δ satisfy

δ m 1 m .

Then there is no nontrivial Lp p-harmonic 1-form on Mm.

Corollary 5.3. Let x : M m c m + 1 , m 3 , be a complete noncompact orientable δ-stable hypersurface Mm immersed in space form c m + 1 . If B i R i c = φ ( H , S ) = 0 , then one of the following conditions holds

1) M is minimal and S is totally geodesic.

2) M is minimal and δ = m 1 m .

Then there is no nontrivial Lp p-harmonic 1-form on Mm.

Theorem 5.4. Let x : M m c m + 1 , m 3 , be a complete noncompact orientable δ-stable minimal hypersurface Mm immersed in space form c m + 1 with nonnegative BiRic curvature bounded from below. If Mm satisfy

λ 1 ( M ) > B i R i c ( X | X | , N ) φ ( H , S ) δ .

Then there is no nontrivial Lp p-harmonic 1-form on Mm.

Proof: From the definition (4.1) and replacing φ by η | ω | p we get

λ 1 ( M ) M η 2 | ω | 2 p M | ( η | ω | p ) | 2 . (67)

Thus,

λ 1 M η 2 | ω | 2 p M η 2 | | ω | p | 2 + M | η | 2 | ω | 2 p + 2 M η | ω | p η , | ω | p . (68)

Using Cauchy-Schwartz inequality

2 | M η | ω | p η , | ω | p | s M η 2 | | ω | p | 2 + 1 s M | η | 2 | ω | 2 p , (69)

where s > 0 , using (68) with (69), and multiplying both said by B we get

B M η 2 | ω | 2 p B ( 1 + s ) λ 1 M η 2 | | ω | p | 2 + B ( 1 + 1 s ) λ 1 M | η | 2 | ω | 2 p . (70)

Compining (56) with (70), we get

D M η 2 | | ω | p | 2 E M | η | 2 | ω | 2 p . (71)

Set

D = A + B ( 1 + s ) λ 1 , E = C B ( 1 + 1 s ) λ 1 , (72)

for some constant E > 0

E = C B ( 1 + 1 s ) λ 1 > 0. (73)

Thus,

p δ + | p δ + 1 | ε > p ( B i R i c ( X | X | , N ) φ ( H , S ) ) ( 1 + 1 s ) λ 1 (74)

Choosing ε and s small enough, we get

λ 1 ( M ) > B i R i c ( X | X | , N ) φ ( H , S ) δ . (75)

Now we observe that

φ ( H , S ) = S m 2 ( m 1 ) H 2 m 2 + ( m 2 ) H ( m 1 ) ( m S H 2 ) m 2 S m . (76)

This implies

B i R i c ( X | X | , N ) = S m 0. (77)

Using (58) with (71), we obtain

D B r | | ω | p | 2 E ( 4 r 2 ) B 2 r \ B r | ω | 2 p . (78)

Taking r , we get ω = 0 . Then there are no nontrivial Lp p-harmonic 1-forms on Mm. Hence we get the conclusion.

On the other hand, Dung and Seo [3] proved that

m 1 m S 2 ( m 1 ) H 2 m 2 + ( m 2 ) H ( m 1 ) ( m S H 2 ) m 2 m 1 2 S .

In fact, in [3], Dung showed that

m 1 m S 2 ( m 1 ) H 2 m 2 + ( m 2 ) H ( m 1 ) ( m S H 2 ) m 2 = m 1 2 S m 1 2 m 2 ( ( m 2 ) m S H 2 m 1 + 1 ( m 1 + 1 ) H 2 ) 2 m 1 2 S . (79)

This implies that φ ( H , S ) ( m 1 2 δ ) S . Therefore, Theorem 5.4 implies the following conclusion.

Corollary 5.5. Let x : M m c m + 1 , m 3 , be a complete noncompact δ-stable minimal hypersurface immersed in space form c m + 1 with nonnegative BiRic curvature bounded from below. Suppose that one of the following conditions holds. Then there is no nontrivial Lp p-harmonic 1-form on Mm.

1) If B i R i c ( X | X | , N ) = S m = 0 , then S is totally geodesic.

2) If B i R i c = ( m 1 2 δ ) S = 0 , then either δ = m 1 2 or S is totally geodesic.

6. Conclusion

We investigated the space of Lp p-harmonic 1-forms on a complete noncompact orientable δ-stable hypersurfaces that are immersed in space form with nonnegative BiRic curvature. We proved the nonexistence of Lp p-harmonic 1-forms on Mm. Moreover, we obtained some vanishing properties for this class of harmonic 1-forms.

Acknowledgements

The authors would like to deeply thank all the reviewers for their insightful and constructive comments.

Cite this paper: Musa, B. and Liu, J. (2021) Lp p-Harmonic 1-Forms on δ-Stable Hypersurface in Space Form with Nonnegative Bi-Ricci Curvature. Advances in Pure Mathematics, 11, 427-439. doi: 10.4236/apm.2021.115029.
References

[1]   Marcos, P.C., Heudson, M. and Feliciano, V. (2014) L2-Harmonic 1-Forms on Submanifolds with Finite Total Curvature. Journal of Geometric Analysis, 24, 205-222.
https://doi.org/10.1007/s12220-012-9334-0

[2]   Seo, K. (2014) Lp Harmonic 1-Forms and the First Eigenvalue of a Stable Minimal Hypersurface. Pacific Journal of Mathematics, 268, 205-229.
https://doi.org/10.2140/pjm.2014.268.205

[3]   Nguyen, T.D. and Keomkyo, S. (2015) Vanishing Theorem for L2 Harmonic 1-Forms on Complete Submanifolds in a Riemannian Manifold. Journal of Mathematical Analysis and Applications, 423, 1594-1609.
https://doi.org/10.1016/j.jmaa.2014.10.076

[4]   Wenzhen, G. and Peng, Z. (2014) L2 Harmonic 1-Forms on Minimal Submanifolds in Sphere. Results in Mathematics, 65, 483-490.
https://doi.org/10.1007/s00025-013-0360-4

[5]   Lao, W.L. and Xiao, R.Z. (2008) On the Bi-Ricci Curvature and Some Applications. Houston Journal of Mathematics (University of Houston), 34, No. 2.

[6]   Shen, Y. and Ye, R.G. (1997) On the Geometry and Topology of Manifolds of Positive Bi-Ricci Curvature.

[7]   Bueler, P. and Igor, P. (2002) Hodge Theory and Cohomology with Compact Supports. Soochow Journal of Mathematics, 28, 33-55.

[8]   Li, H.Z. (1998) L2 Harmonic 1-Forms on a Completes Stable Hypersurface with Constant Mean Curvature. Kodai Mathematical Journal, 21, 1051-1061.
https://doi.org/10.2996/kmj/1138043830

[9]   Nguyen, T.D. and Van, D. (2019) Harmonic One Forms on Immersed Hypersurfaces in a Riemannian Manifold with Weighted Bi-Ricci Curvature Bounded from Below. Journal of Mathematical Analysis and Applications, 484, Article ID: 123693.

[10]   Shukichi, T. (1996) L2 Harmonic Forms and Stability of Minimal Hypersurfaces. Journal of the Mathematical Society of Japan, 48, 761-768.
https://doi.org/10.2969/jmsj/04840761

[11]   Cheng, X. (2000) L2 Harmonic Forms and Stability of Hypersurfaces with Constant Mean Curvature. Boletim da Sociedade Brasileria de Matema—Bulletin, 31, 225-239.
https://doi.org/10.1007/BF01244246

[12]   Bérnard, P. (1991) Stability of Minimal Hypersurfaces. Commentarii Mathematici Helvetici, 66, 185-188.
https://doi.org/10.1007/BF02566644

[13]   Cao, H.D., et al. (1997) The Structure of Stable Minimal Hypersurfaces in Rm+1. Mathematical Research Letters, 4, 637-644.
https://doi.org/10.4310/MRL.1997.v4.n5.a2

[14]   Shen, Y. and Ye, R.G. (1996) On Stable Minimal Surfaces in Manifolds of Positive Bi-Ricci Curvature. Duke Mathematical Journal, 85, 109-116.
https://doi.org/10.1215/S0012-7094-96-08505-1

[15]   Seo, K. (2008) Minimal Submanifolds with Small Total Scalar Curvature in Euclidean Space. Kodai Mathematical Journal, 31, 113-119.
https://doi.org/10.2996/kmj/1206454555

[16]   Peng, F.L. (1992) An Estimate on the Ricci Curvature of a Submanifold and Some Application. Proceedings of the American Mathematical Society, 114, 1051-1063.
https://doi.org/10.1090/S0002-9939-1992-1093601-7

[17]   Limoncu, M. (2012) The Bakry-émery Ricci Tensor and It’s Applications to Some Compactness Theorem. Mathematische Zeitschrift, 271, 715-722.
https://doi.org/10.1007/s00209-011-0886-7

[18]   Zhu, P. and Fang, S.W. (2014) A Gap Theorem on Submanifolds with Finite Total Curvature in Spheres. Journal of Mathematical Analysis and Applications, 413, 195-201.
https://doi.org/10.1016/j.jmaa.2013.11.064

[19]   Han, Y.B. (2018) p-Harmonic Forms on Complete Noncompact Submanifolds in Hyperbolic Spaces. Bulletin Mathématique de la Société des Sciences Mathématiques de la République, 61, 305-319.

 
 
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