J. Math. Anal. Appl. 423 (2015) 1594–1609

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Vanishing theorems for L2 harmonic 1-forms on complete submanifolds in a Riemannian manifold Nguyen Thac Dung a , Keomkyo Seo b,∗ a

Department of Mathematics, Mechanics, and Informatics (MIM), Hanoi University of Sciences (HUS–VNU), Vietnam National University, 334 Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam b Department of Mathematics, Sookmyung Women’s University, Hyochangwongil 52, Yongsan-ku, Seoul, 140-742, Republic of Korea

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 30 June 2014 Available online 31 October 2014 Submitted by H.R. Parks

Let M be an n-dimensional complete orientable noncompact hypersurface in a complete Riemannian manifold of nonnegative sectional curvature. For 2 ≤ n ≤ 6, we prove that if M satisfies the δ-stability inequality (0 < δ ≤ 1), then there is no nontrivial L2β harmonic 1-form on M for some constant β. We also provide sufficient conditions for complete hypersurfaces to satisfy the δ-stability inequality. Moreover, we prove a vanishing theorem for L2 harmonic 1-forms on M when M is an n-dimensional complete noncompact submanifold in a complete simply-connected Riemannian manifold N with sectional curvature KN satisfying that −k2 ≤ KN ≤ 0 for some constant k. © 2014 Elsevier Inc. All rights reserved.

Keywords: δ-Stability inequality L2 harmonic 1-form Traceless second fundamental form First eigenvalue

1. Introduction Let M n be an n-dimensional orientable minimal hypersurface in a Riemannian manifold N of nonnegative sectional curvature. We recall that a minimal hypersurface in a Riemannian manifold is called stable provided the second variation of the volume is nonnegative for any normal variation on a compact subset. More precisely, a minimal hypersurface M in a Riemannian manifold N is said to be called stable if for any f ∈ C0∞ (M ) 

  |∇f |2 − |A|2 + Ric(ν, ν) f 2 dv ≥ 0,

(1.1)

M

where A is the second fundamental form, Ric is the Ricci curvature of N , ν is the unit normal vector of M , and dv is the volume form on M . * Corresponding author. E-mail addresses: [email protected] (N.T. Dung), [email protected] (K. Seo). URL: http://sookmyung.ac.kr/~kseo (K. Seo). http://dx.doi.org/10.1016/j.jmaa.2014.10.076 0022-247X/© 2014 Elsevier Inc. All rights reserved.

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On the other hand, for a number 0 < δ ≤ 1, it is called δ-stable if any function f ∈ C0∞ (M ) satisfies 

  |∇f |2 − δ |A|2 + Ric(ν, ν) f 2 dv ≥ 0.

(1.2)

M

It is obvious that δ1 -stability implies δ2 -stability for 0 < δ2 < δ1 ≤ 1. In particular, if M is stable, then M is δ-stable for 0 < δ ≤ 1. There have been plenty of works on δ-stable complete minimal hypersurfaces in a Riemannian manifold. (See [3,8,10,19,27] and references therein for more details.) It is well-known that the only complete orientable stable minimal surface in R3 is a plane [4,7]. For 1/8 < δ, Kawai [15] showed that a δ-stable complete minimal surface in R3 should be a plane. Furthermore, do Carmo and Peng [5] proved that if a stable  complete minimal hypersurface M in the Euclidean space with M |A|2 dv < ∞, then M is a hyperplane. Later, Shen and Zhu [26] proved that an n-dimensional stable complete minimal hypersurface M in the  Euclidean space with M |A|n dv < ∞ is a hyperplane. Recently, Tam and Zhou proved that a complete n−2 n -stable minimal hypersurface whose second fundamental form satisfies some decay conditions in the Euclidean space is either a hyperplane or a catenoid. In case of complete orientable stable minimal hypersurfaces, several results on the nonexistence of L2 harmonic forms are well-known. Palmer [22] proved that if there exists a codimension one cycle on a complete minimal hypersurface M in the Euclidean space which does not separate M , then M is unstable by using the nonexistence of L2 harmonic 1-form. Thereafter, using Bochner’s vanishing technique, Miyaoka [20] showed that a complete orientable noncompact stable minimal hypersurface in a nonnegatively curved manifold has no nontrivial L2 harmonic 1-forms. In [32], Yun proved that if M ⊂ Rn+1 is a complete minimal hypersurface  with sufficiently small total scalar curvature M |A|n , then there is no nontrivial L2 harmonic 1-form on M . Yun’s result has been generalized into various ambient spaces [2,6,23–25]. For an n-dimensional complete orientable noncompact (not necessarily minimal) hypersurface M in a complete manifold N of nonnegative sectional curvature with 2 ≤ n ≤ 4, Kim and Yun [16] recently proved that if M satisfies the stability inequality (1.1), then there is no nontrivial L2 harmonic 1-form on M , which is an extension of a well-known fact in the case when M is a complete stable minimal hypersurface in N . In Section 2, motivated by this, we prove that if M is an n-dimensional complete noncompact (not necessarily minimal) hypersurface in a complete manifold N of nonnegative sectional curvature and M satisfies the δ-stability inequality (1.2) for a number 0 < δ ≤ 1, then there is no nontrivial L2β harmonic 1-form on M for some constant β. (See Theorem 2.6 for more details.) As a consequence, we extend Kim and Yun’s result into the case when n = 5, 6. In Section 3, we also provide sufficient condition for complete hypersurfaces to satisfy the δ-stability inequality in a Riemannian manifold. In Section 4, we deal with complete noncompact submanifold cases. For an n-dimensional complete noncompact submanifold M in a complete simply-connected Riemannian manifold N with sectional curvature KN satisfying that −k2 ≤ KN ≤ 0 for some constant k, it turns out that if the L2 norm φn of the traceless second fundamental form φ is sufficiently small and the first eigenvalue λ1 (M ) of the Laplacian is bigger than some constant depending only on k, n, and φn , then there is no nontrivial L2 harmonic 1-form on M . 2. Harmonic 1-forms on complete hypersurfaces of lower dimensions A complete manifold M is called non-parabolic if it has a positive Green function. Otherwise, M is called parabolic. We note that M is non-parabolic provided it has a non-constant positive superharmonic function on M . The following sufficient condition for parabolicity is well-known. Theorem. (See [11,12,14,28].) Let M be a complete manifold. If, for any point p ∈ M and a geodesic ball Bp (r) ⊂ M ,

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∞

r dr = ∞, Vol(Bp (r))

1

then M is parabolic. Using the above theorem, we see that if M is non-parabolic, then ∞

r dr < ∞, Vol(Bp (r))

(2.1)

1

and hence M has an infinite volume. Definition 2.1. Let M n be an n-dimensional orientable hypersurface in a Riemannian manifold N . We say the δ-stability inequality holds on M for 0 < δ ≤ 1 if any f ∈ C0∞ (M ) satisfies 

  |∇f |2 − δ |A|2 + Ric(ν, ν) f 2 dv ≥ 0.

M

It turns out that a complete orientable noncompact hypersurface in a complete manifold with nonnegative sectional curvature has an infinite volume if the δ-stability inequality holds for 0 < δ ≤ 1. Lemma 2.2. Let M n be a complete orientable noncompact hypersurface in a complete manifold N with nonnegative sectional curvature. If the δ-stability inequality holds on M for 0 < δ ≤ 1, then the volume of M is infinite. Proof. If M is non-parabolic, then M has an infinite volume by (2.1). We now assume that M is parabolic. Given 0 < δ ≤ 1, since the δ-stability inequality holds on M , we have 

 |∇ϕ| ≥ δ 2

M

  Ric(ν, ν) + |A|2 ϕ2

M

for any f ∈ C0∞ (M ). Let q := δ(|A|2 + Ric(ν, ν)) and let D ⊂ M be any bounded domain with smooth boundary. Denote by λq1 (D) the first eigenvalue of the Schrödinger operator Δ + q acting on functions vanishing on ∂D. The assumption that the δ-stability holds on M is equivalent to that λq1 (D) ≥ 0 for any bounded domain D ⊂ M . From the result in [7], it follows that there is a positive function u such that the equation Δu + qu = 0 on M . Since the sectional curvature of N is nonnegative, u is a positive superharmonic function on M . The parabolicity of M implies that u is constant. Hence |A| ≡ 0, which shows that M is totally geodesic in N . Thus M has nonnegative Ricci curvature, which gives the conclusion that M has an infinite volume [31]. 2 Let M n be an n-dimensional orientable submanifold in an (n + p)-dimensional Riemannian manifold N . Fix a point x ∈ M and choose any local orthonormal frame {e1 , · · · , en+p } such that {e1 , · · · , en } is an orthonormal basis of the tangent space Tx M and {en+1 , · · · , en+p } is an orthonormal basis of the normal space Nx M . For each α ∈ {n + 1, · · · , n + p}, define a linear map Aα : Tx M → Tx M by n+p

¯ X Y, eα , Aα X, Y = ∇

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¯ denotes the Levi-Civita connection on N . Then the (unwhere X, Y are tangent vector fields and ∇ normalized) mean curvature vector H is defined by n+p 

H=

(trace Aα )eα .

α=n+1

Define a linear map φα : Tx M → Tx M by φα X, Y = Aα X, Y − X, Y H, eα and a traceless bilinear map φ : Tx M × Tx M → Nx M by φ(X, Y ) =

n+p 

φα X, Y eα .

α=n+1

This map φ is called the traceless second fundamental form of M . Denote by A the second fundamental form. Then |φ|2 = |A|2 −

H2 . n

Note that H2 ≤ n. |A|2 In particular, if p = 1, then φ=A−

H g, n

where g is the induced metric on M . Lemma 2.3. Let b :=

√ (n−2)2 n−1 √ . 2n( n−1+1)2

Then we have

2(n − 1)|H|2 − (n − 2)

√

n − 1|H|

 n|A|2 − |H|2 ≥ −bn2 |A|2 .

(2.2)

Proof. If |A| = 0, then H = 0. Thus the inequality (2.2) is trivial. Now we assume that |A| > 0. The inequality (2.2) is equivalent to  √ H2 2(n − 1) H 2 (n − 2) n − 1 |H| n − − ≤ b. n2 |A| |A|2 n2 |A|2 √ We define fn (t) on [0, n ] by √ (n − 2) n − 1  2(n − 1) 2 fn (t) = t n − t2 − t . n2 n2 Suppose that there is a positive constant B such that B ≥ max[0,√n ] fn (t). Then  √ (n − 2) n − 1t n − t2 ≤ 2(n − 1)t2 + Bn2 ,

∀t ∈ [0,

√

n]

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or equivalently, (n − 2)2 (n − 1)x(n − x) ≤ 4(n − 1)2 x2 + 4B(n − 1)n2 x + B 2 n4 ,

(2.3)

where x := t2 for all x ∈ [0, n]. A simple computation shows that the inequality (2.3) holds true if √ (n − 2)2 n − 1 √ B≥ = b, 2n( n − 1 + 1)2 which gives the conclusion. 2 In the following, we need the Ricci curvature estimate for submanifolds in a Riemannian manifold which was done by Leung [17]. Lemma 2.4. (See [17].) Let M be an n-dimensional submanifold in a Riemannian manifold N with sectional curvature KN satisfying that K ≤ KN where K is a constant. Then the Ricci curvature RicM of M satisfies RicM ≥ (n − 1)K +

 √

n−1 2 1 2 |A| . 2(n − 1)|H| − (n − 2) n − 1|H| n|A|2 − |H|2 − n2 n

Using Lemma 2.3 and the above Ricci curvature estimate, one can obtain the following. Lemma 2.5. Let M n be a complete orientable noncompact hypersurface in N of nonnegative sectional curvature. Then √ RicM ≥ −

n−1 2 |A| . 2

(2.4)

Proof. By Lemma 2.3 and Lemma 2.4, we see RicM

√  (n − 2)2 n − 1 n−1 √ |A|2 + ≥− n 2n( n − 1 + 1)2 √ n−1 2 |A| , =− 2

which completes the proof. 2 Theorem 2.6. Let M n (2 ≤ n ≤ 6) be a complete orientable noncompact hypersurface in a complete manifold n−2 N with nonnegative sectional curvature. If the δ-stability inequality holds on M for some 2 √ ≤ δ ≤ 1, n−1 2β then there is no nontrivial L harmonic 1-form on M for any constant β satisfying     2δ 2δ n−2 n−2 √ 1− 1− √ <β< √ 1+ 1− √ . 2δ n − 1 2δ n − 1 n−1 n−1 Proof. We will prove Theorem 2.6 by using Fu–Yang’s arguments [10]. Let ω be an L2β harmonic 1-form on M . Then  |ω|2β dv < ∞.

Δω = 0 and M

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We shall use the same notation ω for a harmonic 1-form and its dual harmonic vector field in an abuse of notation. Bochner’s formula implies   Δ|ω|2 = 2 |∇ω|2 + RicM (ω, ω) . On the other hand, it follows that

2 

 Δ|ω|2 = 2 |ω|Δ|ω| + ∇|ω| . Thus we obtain

2

|ω|Δ|ω| − RicM (ω, ω) = |∇ω|2 − ∇|ω| . Since the Kato-type inequality

2 |∇ω|2 − ∇|ω| ≥

2 1

∇|ω| n−1

holds for any harmonic 1-form ω on an n-dimensional Riemannian manifold [29], |ω|Δ|ω| ≥

2 1

∇|ω| + RicM (ω, ω). n−1

(2.5)

Applying Lemma 2.5, we get

2 1

∇|ω| − |ω|Δ|ω| ≥ n−1

√

n−1 2 2 |A| |ω| . 2

Given any α > 0, we have

2

  |ω|α Δ|ω|α = |ω|α α(α − 1)|ω|α−2 ∇|ω| + α|ω|α−1 Δ|ω|

2

α − 1

= ∇|ω|α + α|ω|2α−2 ω|Δ|ω α √ 

α − 1

1

n−1 2 2 α 2 2α−2 2 ≥ ∇|ω| ∇|ω| − |A| |ω| + α|ω| α n−1 2 √ 

n−2 n − 1 2 2α α 2

∇|ω| |A| |ω| . −α ≥ 1− (n − 1)α 2

(2.6)

Choose any nonnegative number q and a smooth function φ with compactly support in M . Multiplying both sides of the inequality (2.6) by |ω|2qα φ2 and integrating over M , we obtain 

n−2 1− (n − 1)α



 ≤

|ω|

M

(2q+1)α 2

M

=α

√

−2 M

√ α

φ Δ|ω| + α

n−1 2 

2 |ω|2qα φ2 ∇|ω|α n−1 2

 |A|2 φ2 |ω|2(q+1)α M





|A|2 φ2 |ω|2(q+1)α − (2q + 1) M

  φ|ω|(2q+1)α ∇φ, ∇|ω|α .

M

2 |ω|2qα ∇|ω|α φ2

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Hence  2(q + 1) − √ ≤α

n−2 (n − 1)α

n−1 2



2 |ω|2qα ∇|ω|α φ2

M





  φ|ω|(2q+1)α ∇φ, ∇|ω|α .

|ω|2(q+1)α |A|2 φ2 − 2 M

(2.7)

M

On the other hand, since M satisfies the δ-stability inequality and N has nonnegative sectional curvature, we have 

 |∇φ| ≥ δ 2

M







Ric(ν, ν) + |A| φ ≥ δ 2

|A|2 φ2 .

2

M

M

Replacing φ by |ω|(1+q)α φ in the above inequality gives 



2 |ω|2qα ∇|ω|α φ2 +

|ω|2(q+1)α |A|2 φ2 ≤ (q + 1)2

δ M

M



+ 2(1 + q)

 |ω|2(q+1)α |∇φ|2 M



 |ω|(2q+1)α φ ∇φ, ∇|ω|α .

(2.8)

M

Combining (2.7) and (2.8), we obtain  2(q + 1) −

≤

α δ

√





2α δ

√

2 |ω|2qα ∇|ω|α φ2

M



n−1 (q + 1)2 2

 +

n−2 (n − 1)α

M



n−1 (q + 1) − 2 2





2 |ω|2qα ∇|ω|α φ2 +

|ω|2(q+1)α |∇φ|2 M

  |ω|(2q+1)α φ ∇φ, ∇|ω|α .

(2.9)

M

Given ε > 0, the Schwarz inequality implies 

2α δ

√



  |ω|(2q+1)α φ ∇φ, ∇|ω|α

n−1 (q + 1) − 2 2

M

√





α n−1

(q + 1)

2|ω|(2q+1)α φ|∇φ| ∇|ω|α ≤ 1 − δ 2 M

  

2 1 ≤ |D| ε |ω|2qα ∇|ω|α φ2 + |ω|2(q+1)α |∇φ|2 , ε M

M

where α D := 1 − δ

√

n−1 (q + 1). 2

(2.10)

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From the inequalities (2.9) and (2.10), it follows that  2(q + 1) − √ ≤

n−2 − (n − 1)α

n − 1 α(q + 1)2 2 δ



2 |ω|2qα ∇|ω|α φ2

M



n−1α 2 δ 

√

|ω|2(q+1)α |∇φ|2 M

+ |D|ε

2qα

∇|ω|

|ω|



|D| φ + ε

α 2 2

M

|ω|2(q+1)α |∇φ|2 , M

or equivalently,  2(q + 1) − √ ≤

n−2 − (n − 1)α

n − 1 α |D| + 2 δ ε

√



n − 1 α(q + 1)2 − |D|ε 2 δ

2 |ω|2qα ∇|ω|α φ2

M

 |ω|2(q+1)α |∇φ|2 .

(2.11)

M

Now let β := (1 + q)α and choose the numbers α and q such that n−2 − 2(q + 1) − (n − 1)α

√

n − 1 α(q + 1)2 > 0. 2 δ

Therefore, for a sufficiently small ε > 0, the inequality (2.11) implies that there is a constant C > 0 which depends on ε, δ, q, α such that 

2 |ω|2qα ∇|ω|α φ2 ≤ C

M

 |ω|2β |∇φ|2 ,

(2.12)

M

provided that n−2 − 2(q + 1) − (n − 1)α

√

n − 1 α(q + 1)2 > 0. 2 δ

(2.13)

Note that the inequality (2.13) is equivalent to n−2 − 2β − n−1

√ n − 1 β2 > 0, 2 δ

which is satisfied by the assumption     2δ n−2 n−2 2δ √ 1− 1− √ <β< √ 1+ 1− √ . n−1 n−1 2δ n − 1 2δ n − 1 Given R > 0 and a fixed point p ∈ M , we take a test function φ(r) defined on [0, ∞) such that φ ≥ 0, φ = 1 on [0, R] and φ = 0 in [2R, ∞) with |φ | ≤ R2 , where r(x) denotes the distance from p to x on M . Then the inequality (2.12) becomes  M

2 4C |ω|2qα ∇|ω|α ≤ 2 R

 |ω|2β . M

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Letting R → ∞, we conclude that |ω| is constant since ω is an L2β harmonic 1-form. However, since the volume of M is infinite by Lemma 2.2, we obtain that ω ≡ 0, which completes the proof. 2 As a direct consequence of Theorem 2.6, if δ = 1, that is, M satisfies the stability inequality (1.1), we have the following. Corollary 2.7. For 2 ≤ n ≤ 6, let M n be a complete orientable noncompact hypersurface in a complete manifold N with nonnegative sectional curvature. Let α1 and b be the same constants as in Theorem 2.6. If the stability inequality (1.1) holds on M , then there is no nontrivial L2β harmonic 1-form on M for any constant β satisfying     2 2 n−2 n−2 √ 1− 1− √ <β< √ 1+ 1− √ . n−1 2 n−1 n−1 2 n−1 √

Moreover, if we take (n−1)2n n−1 < δ ≤ 1 in Theorem 2.6, we have the following vanishing theorem for L2 harmonic 1-forms on such hypersurfaces M . Corollary 2.8. For 2 ≤ n ≤ 6, let M n be a complete orientable noncompact hypersurface in a complete √ manifold N with nonnegative sectional curvature. If M satisfies δ-stability inequality for some (n−1)2n n−1 < δ ≤ 1, then there is no nontrivial L2 harmonic 1-form on M . Proof. Taking β = 1 in Theorem 2.6 and using our assumption that

√ (n−1) n−1 2n

< δ ≤ 1, we see

    2δ n−2 n−2 2δ √ 1− 1− √ <1< √ 1+ 1− √ . n−1 2δ n − 1 n−1 2δ n − 1 Applying Theorem 2.6 gives the conclusion. 2 We remark that Corollary 2.7 and Corollary 2.8 can be regarded as extensions of the main result in [16]. For higher codimensional cases, we define the δ-super stability inequality as follows. Definition 2.9. Let M n be an n-dimensional orientable submanifold in the (n + p)-dimensional Euclidean space Rn+p . We say the δ-super stability inequality holds on M for 0 < δ ≤ 1 if any f ∈ C0∞ (M ) satisfies  |∇f |2 − δ|A|2 f 2 dv ≥ 0. M

The definition of super stability in the Euclidean space was introduced by Q. Wang [30]. In particular, when p = 1 and δ = 1, the concept of super stability is the same as the usual definition of stability. Using the same arguments as in the proof of Theorem 2.6, we obtain the following. n+p Corollary 2.10. Let M n (2 ≤ n ≤ 6) be a complete . If the δ-super √ orientable noncompact submanifold in R (n−1) n−1 2 stability inequality holds on M for some < δ ≤ 1, then there is no nontrivial L harmonic 1-form 2n on M .

3. Sufficient conditions for complete hypersurfaces to satisfy the δ-stability inequality Given a complete noncompact Riemannian manifold M , we consider a compact domain Ω ⊂ M . Let λ1 (Ω) > 0 denote the first eigenvalue of the Dirichlet boundary value problem

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Δf + λf = 0 in Ω f =0 on ∂Ω

where Δ denotes the Laplace operator on M . By the domain monotonicity principle, the first eigenvalue λ1 (M ) of a complete noncompact manifold M is defined by λ1 (M ) = inf λ1 (Ω), Ω

where the infimum is taken over all compact domains in M . In this section, we provide sufficient conditions for an n-dimensional complete hypersurface to satisfy the δ-stability inequality. More precisely, we prove that if the Ln or L∞ norm of the second fundamental form is sufficiently small in terms for an n-dimensional complete hypersurface M in a Riemannian manifold with nonpositive sectional curvature, then the δ-stability inequality holds on M . Theorem 3.1. Let M be an n-dimensional complete hypersurface in a Riemannian manifold N with sectional curvature KN satisfying that KN ≤ K ≤ 0 where K ≤ 0 is a constant. Assume that for 0 < δ ≤ 1 |A|2 ≤

λ1 (M ) − nK. δ

Then the δ-stability inequality holds on M . Proof. The assumption on the sectional curvature on N implies that Ric(ν, ν) ≤ nK. Since for any f ∈ C0∞ (M )  λ1 (M ) ≤

|∇f |2 , f2 M

M 

it follows from the assumption on |A|2 that 

  |∇f |2 − δ |A|2 + Ric(ν, ν) f 2 ≥

M





  λ1 (M ) − δ|A|2 + nK f 2 ≥ 0.

M

Therefore the δ-stability inequality on M holds for any 0 < δ ≤ 1. 2 It is well known that the following Sobolev inequality holds on a complete submanifold in a complete simply-connected manifold with nonpositive sectional curvature. Lemma 3.2. (See [13].) Let M n (n ≥ 3) be an n-dimensional complete submanifold in a complete simplyconnected manifold with nonpositive sectional curvature. Then for any φ ∈ W01,2 (M ) we have  |φ| M

2n n−2

n−2 n dv

 ≤ CS

 2 |∇φ|2 + φ|H| dv,

(3.1)

M

where CS is the Sobolev constant which depends only on n ≥ 3. Using the above Sobolev inequality (3.1), one obtains another sufficient condition for a complete hypersurface in a complete simply connected Riemannian manifold to have the δ-stability inequality.

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Theorem 3.3. Let M n (n ≥ 3) be an n-dimensional complete hypersurface in a complete simply connected Riemannian manifold N with nonpositive sectional curvature. Assume that for 0 < δ ≤ 1  |A|

n

n2

≤

1 , (n + δ)CS

M

where CS is a Sobolev constant in (3.1). Then the δ-stability inequality holds on M . Proof. From the Sobolev inequality (3.1) and Hölder inequality, it follows that for any f ∈ C0∞ (M ),  |f |

2n n−2

n−2 n

 ≤ CS

M

 2 |∇f |2 + f |H|

M



 ≤ CS

|∇f | + CS

|H|

2

M

n

n2  

M

|f |

2n n−2

n−2 n .

M

Thus  |f |

2n n−2

n−2 n

M

CS ≤ 1 − CS H2n

 |∇f |2 , M

since H2n ≤ nA2n ≤

n 1 < (n + δ)CS CS

by our assumption. Therefore 





1 − CS H2n |∇f | − δ |A| + Ric(ν, ν) f ≥ CS 2

2

 |f |

2

M

2n n−2

n−2 n

 −δ

M

 ≥

|A|2 f 2 M

1 − CS H2n − δA2n CS

  |f |

2n n−2

n−2 n

M

 ≥

1 − nCS A2n − δA2n CS

 

2n

|f | n−2

n−2 n

M

 ≥

1 − (n + δ)A2n CS

 

2n

|f | n−2

n−2 n

M

≥ 0, which gives the conclusion. 2 4. Vanishing theorem for L2 harmonic 1-forms on complete noncompact submanifolds As mentioned in the introduction, there are several vanishing theorems for L2 harmonic forms on complete noncompact stable minimal hypersurfaces. Recall that ω is an L2 harmonic 1-form on M if it satisfies  Δω = −(dδ + δd)ω = 0 and |ω|2 < ∞. M

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Even without assuming stability of minimal hypersurfaces, if M ⊂ Rn+1 is a complete minimal hypersurface  with sufficiently small quantity of M |A|n , then we still have a vanishing theorem for L2 harmonic 1-forms on M . (See [21,25,32] for details.) The analogue of this result is also true for a complete minimal hypersurface in hyperbolic space [24]. Later, it turned out that these gap theorems hold for more general submanifolds. Given an n-dimensional complete noncompact submanifold in Euclidean space, Carron [1] proved that there exists a constant c(n) such that if |A| ≤ c(n), then all spaces of L2 harmonic forms are trivial. Fu and Li [9] showed that for a complete noncompact submanifold M n ⊂ RN there also exists a constant d(n) such that if the Ln norm of the traceless second fundamental form φ is less than d(n) then there is no nontrivial L2 harmonic 1-form on M . More generally, let M be an n-dimensional complete noncompact submanifold in a complete simplyconnected Riemannian manifold N with sectional curvature KN satisfying −k2 ≤ KN ≤ 0 for some constant k. In [2], Cavalcante, Mirandola and Vitório recently proved that if the Ln norm of the traceless second fundamental form φ is sufficiently small and if the first eigenvalue λ1 (M ) of the Laplacian satisfies λ1 (M ) >

 (n − 1)2  2 k − inf |H|2 , n

then there is no nontrivial L2 harmonic 1-form on M . Note that the lower bound of λ1 (M ) depends on inf |H| in their result. In this section, we prove a similar vanishing theorem for L2 harmonic 1-forms on complete noncompact submanifolds under the same assumptions as in [2] except that the lower bound of λ1 (M ) depends on φn . More precisely, we prove Theorem 4.1. Let M n (n ≥ 3) be an n-dimensional complete noncompact submanifold in a complete simplyconnected Riemannian manifold N with sectional curvature KN satisfying that −k2 ≤ KN ≤ 0 where k is a constant. Assume that the traceless second fundamental form φ satisfies  φn <

1 . n(n − 1)CS

In the case k = 0, assume further that the first eigenvalue λ1 (M ) of M satisfies λ1 (M ) >

2n2 (n − 1)2 k2  , n3 − (n − 2)(n − 1) n(n − 1)CS φn − 2n(n − 1)CS φ2n

where CS is a Sobolev constant in (3.1). Then there is no nontrivial L2 harmonic 1-form on M . Proof. As in the proof of Theorem 2.6, it follows from the inequality (2.5) that for any harmonic 1-form ω |ω|Δ|ω| ≥

2 1

∇|ω| + RicM (ω, ω). n−1

Moreover, using Lemma 2.4 and the fact that |φ|2 = |A|2 −

|H|2 n ,

we see

 n − 2 n−1 2 |H|2 2 |φ| . Ric ≥ (n − 1) −k − n(n − 1)|φ||H| − 2 2 n n n

N.T. Dung, K. Seo / J. Math. Anal. Appl. 423 (2015) 1594–1609

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Combining these two inequalities, we have 

2 1

|H|2 2

|ω|Δ|ω| ≥ − k |ω|2 ∇|ω| + (n − 1) n−1 n2 n − 2 n−1 2 2 − n(n − 1)|φ||H||ω|2 − |φ| |ω| . n2 n

(4.1)

Consider a geodesic ball B(R) of radius R centered at x ∈ M . Choose a test function f satisfying 0 ≤ f ≤ 1, f ≡ 1 on B(R), f ≡ 0 on M \ B(2R), and |∇f | ≤ R1 . Multiplying both sides by a compactly supported function f 2 in BR ⊂ M and integrating over BR , we have 

 BR



2 1

∇|ω| f 2 + n−1

f 2 |ω|Δ|ω| ≥ BR

 |H|2 2 (n − 1) − k |ω|2 f 2 n2

BR



n − 2 n(n − 1)|φ||H||ω|2 f 2 − n2

− BR



n−1 2 2 2 |φ| |ω| f . n

BR

Applying the divergence theorem, we have 

  f |ω| ∇f, ∇|ω| −

0 ≤ −2 BR

+

n − 2 n(n − 1) n2

n n−1







∇|ω| 2 f 2 + n − 1 n

BR

 

|φ||H||ω|2 f 2 + (n − 1)

BR

 |φ|2 |ω|2 f 2 BR

k2 −

|H|2 |ω|2 f 2 . n2

BR

For any positive numbers α, β > 0, it follows from the Schwarz inequality



 

2 1

  2

2 f |ω| ∇f, ∇|ω| ≤ α f ∇|ω| + |∇f |2 |ω|2 α BR

BR

(4.2)

BR

and 

 |φ||H||ω| f ≤ β 2 2

2 BR

1 |H| |ω| f + β 2

 |φ|2 |ω|2 f 2 .

2 2

BR

BR

Combining these inequalities, we obtain  0≤

α−  +  +

n n−1



2 1 f 2 ∇|ω| + α

BR

 |∇f |2 |ω|2 BR

β(n − 2)  n−1 n(n − 1) − 2 2n n2 n − 2 n−1 n(n − 1) + 2βn2 n

BR



|H|2 |ω|2 f 2 BR

|φ|2 |ω|2 f 2 BR

 + k2 (n − 1)



|ω|2 f 2 .

(4.3)

N.T. Dung, K. Seo / J. Math. Anal. Appl. 423 (2015) 1594–1609

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Similarly, we have the following estimate: 

 |φ| |ω| f ≤ 2

|φ|

2 2

n

BR

n2  

BR



 2n |ω||f | n−2

n−2 n

BR

 ≤ CS

|φ|n

n2  

BR

 

∇ |ω|f 2 +

BR

 |φ|n

≤ CS



 |H|2 |ω|2 f 2

BR

n2 

 (1 + α)

BR

  

2

1 f 2 ∇|ω| + 1 + |∇f |2 |ω|2 α

BR

 |φ|n

+ CS

n2 

BR

BR

|H|2 |ω|2 f 2 .

(4.4)

BR

In the case k = 0, we need an additional estimate. Using the monotonicity of the first eigenvalue λ1 (BR ) of a ball BR , we observe that  |∇f |2 λ1 (M ) ≤ λ1 (BR ) ≤ BR (4.5) f2 BR for any f ∈ C0∞ (M ). Putting |ω|f for f in the inequality (4.5) and using the Schwarz inequality (4.2) gives 

 

2 1

f ∇|ω| + 1 + |∇f |2 |ω|2 . α



2

|ω| f ≤ (1 + α) 2 2

λ1 (M ) BR

BR

(4.6)

BR

Combining the inequalities (4.3), (4.4) and (4.6), we have  A

2 f 2 ∇|ω| + B

BR



 |H|2 |ω|2 f 2 ≤ C

BR

|∇f |2 |ω|2 ,

(4.7)

BR

where the constants A, B, C are defined by  2  n2   k (n − 1) n − 2 n n−1 n −α− + CS (1 + α) A= |φ| n(n − 1) + n−1 λ1 (M ) 2βn2 n BR

B=

n − 1 n − 2 − n(n − 1)β − CS n2 2n2

 |φ|n

n2 

n−1 n − 2 n(n − 1) + 2βn2 n



BR



1 k2 (n − 1) + CS C= + α λ1 (M )

 |φ|

n

n2 

n−1 n − 2 n(n − 1) + 2βn2 n



1 1+ . α

BR

Using our assumption on φn and the following arithmetic–geometric mean inequality β + CS φ2n

 1 ≥ 2 CS φn , β

(4.8)

√ we see that B > 0 for any β > 0. Take β = CS φn which makes equality in the inequality (4.8). By the assumptions on λ1 (M ) and φn , we can choose the number α > 0 small enough such that A > 0. Furthermore, it automatically follows that C > 0.

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Now letting R → ∞ in the inequality (4.7), we obtain |∇|ω|| ≡ 0 and |H||ω| ≡ 0. Since |∇|ω|| ≡ 0, we get |ω| ≡ constant. Suppose that |ω| is a nonzero constant. From the fact that |H||ω| ≡ 0, it follows that M is a minimal submanifold. However, since the volume of a complete minimal submanifold in a Riemannian  manifold of nonpositive sectional curvature is infinite, we have M |ω|2 = ∞. This is a contradiction to the assumption that ω is an L2 harmonic 1-form. Therefore ω ≡ 0, which completes the proof. 2 As a consequence, if the ambient space N is the Euclidean space, we obtain the following result. Corollary 4.2. Let M n (n ≥ 3) be an n-dimensional complete noncompact submanifold in the Euclidean space RN . If the traceless second fundamental form φ satisfies  1 , n(n − 1)CS

φn <

then there is no nontrivial L2 harmonic 1-form on M . It immediately follows from the above result that such M must have only one end. We remark that the upper bound of φn is less than the upper bound in Corollary 1.1 of [2], which is nonetheless a generalization of [9] and [18]. Moreover, when the ambient space N has a pinched nonpositive sectional curvature, we immediately obtain the following. Corollary 4.3. Let M n (n ≥ 3) be an n-dimensional complete noncompact submanifold in a complete simplyconnected Riemannian manifold N with sectional curvature KN satisfying that −k2 ≤ KN ≤ 0 for some constant k = 0. If  φn <

1 n(n − 1)CS

and λ1 (M ) >

2n2 (n − 1)2 k2 , n3 − n2 + 3n − 4

then there is no nontrivial L2 harmonic 1-form on M . We remark that the upper bound of φn and the lower bound of λ1 (M ) depend only on the dimension of M and the curvature of the ambient space, which is different from [2]. Acknowledgments The authors would like to thank the referee for the helpful comments and suggestions. The first author was supported in part by NAFOSTED under grant number 101.02-2014.49. A part of this paper was written during a stay of the first author at Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to express his sincerely thanks to staffs there for excellent working conditions and financial support. The second author was supported in part by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2013R1A1A1A05006277).

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