Gap theorems for minimal submanifolds of a hyperbolic space

J. Math. Anal. Appl. 436 (2016) 983–989

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

Gap theorems for minimal submanifolds of a hyperbolic space ✩ Changyu Xia a,b,∗ , Qiaoling Wang b a b

School of Mathematics and Statistics, Hubei University, Wuhan, 430062, China Departamento de Matemática, Universidade de Brasilia, 70910-900 Brasilia, DF, Brazil

a r t i c l e

i n f o

Article history: Received 31 October 2014 Available online 29 December 2015 Submitted by J. Xiao Keywords: Minimal submanifolds Hyperbolic space Gap theorems

a b s t r a c t This paper provides some gap theorems for complete immersed minimal submanifolds of dimension no less than five in a hyperbolic space. Namely, we show that an n (≥ 5)-dimensional complete immersed minimal submanifold M in a hyperbolic space is totally geodesic if the L2 norm of |A| on geodesic balls centered at some point p ∈ M has less than quadratic growth and if either supx∈M |A|2 (x) is not too large or the Ln norm of |A| on M is small, here, A is the second fundamental form of M . © 2016 Elsevier Inc. All rights reserved.

1. Introduction In a seminal paper, Simons [24] calculated the Laplacian of the squared length of the second fundamental form of a minimal submanifold in a space form. As an application of Simons’ formula, it follows that if M is an n-dimensional closed minimal submanifold in an (n + m)-dimensional unit sphere with squared norm of the second fundamental form less than n/(2 −1/m), then M is totally geodesic. Simons’ work has stimulated great developments of Riemannian submanifolds. Many interesting gap theorems for submanifolds have been proven during the past years (cf. [1–6,8,10–13,15,16,18–31], etc). It is natural to ask whether a Simons’s type pinching theorem holds for minimal submanifolds in a hyperbolic space. By definition, the hyperbolic space Hm is a (unique) simply connected complete m-dimensional Riemannian manifold with a constant negative sectional curvature −1. In this paper, we show that this is true if the L2 -norm on geodesic balls of the length of the second fundamental form of the minimal submanifold has less than quadratic growth and if the dimension of the submanifolds is not less than 5. Theorem 1.1. Let M be an n (≥ 5)-dimensional complete immersed minimal submanifold in Hn+m with second fundamental form A satisfying ✩

This work was partially supported by CNPq, Brazil, grants 307089/2014-2 and 306146/2014-2.

* Corresponding author at: Departamento de Matemática, Universidade de Brasilia, 70910-900 Brasilia, DF, Brazil. E-mail addresses: [email protected] (C. Xia), [email protected] (Q. Wang). http://dx.doi.org/10.1016/j.jmaa.2015.12.050 0022-247X/© 2016 Elsevier Inc. All rights reserved.

C. Xia, Q. Wang / J. Math. Anal. Appl. 436 (2016) 983–989

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 lim sup R→∞

Bp (R) R2

|A|2

= 0,

(1.1)

where Bp (R) denotes the geodesic ball of radius R centered at p ∈ M . Suppose that ⎧ 2 ⎨ (n+2)(n−1) − n, 4n 2  2  sup |A| (x) < D(n, m) ≡ 2 ⎩ 2 1+ mn (n−1) − n , x∈M 3 4

if m = 1, if m ≥ 2,

(1.2)

then M is totally geodesic. The pinching theorem has an Ln version. Indeed, it has been shown in [4] that a complete (resp. closed) n-dimensional minimal submanifold in a Euclidean space (resp. a sphere) is totally geodesic if the Ln norm of the second fundamental form of M is small enough. The proof of these results are nice applications of the Simons’s formula and the Sobolev inequality for minimal submanifolds in a Euclidean space or in a sphere. In this paper, we obtain a similar result for complete minimal submanifolds of a hyperbolic space. Theorem 1.2. Let M be an n (≥ 5)-dimensional complete immersed minimal submanifold in Hn+m and let A be the second fundamental form of M . Assume that (1.1) holds. There exists a positive constant C which depends only on n and m such that if

|A|n < C,

(1.3)

M

then M is totally geodesic. Remark 1.1. We believe that the restriction on the dimension of M and the condition (1.1) in Theorems 1.1 and 1.2 are not necessary. Remark 1.2. It is interesting to know the best possible pinching constant in the above results, especially in Theorem 1.1. 2. Proof of the results Before proving our results, let us recall some known facts we need. n+m Let M be a complete non-compact submanifold immersed in a simply connected space form M (κ) of constant curvature c. We adopt the usual convention on the range of the indices 1 ≤ A, B, C, . . . ≤ n + m,

1 ≤ i, j, k, . . . ≤ n,

n + 1 ≤ α, β, γ, . . . ≤ n + m.

n+m

Choose a local orthonormal frame {eA } in M (κ) such that, when restricted to M n , the vectors ei , B i = 1, · · · , n, are tangent to M n . Let {ωA } and {ωA } be the dual basis to {eA } and the connection forms n+m on M (κ), respectively. Restricting these forms to M n and using the Einstein’s summation convention we have j ωiα = hα ij ω ,

R`ijkl

α hα ij = hji ,  α α α = c (δik δjl − δil δjk ) + hα ik hjl − hil hjk , α

C. Xia, Q. Wang / J. Math. Anal. Appl. 436 (2016) 983–989

i j A = hα ij ω ⊗ ω ⊗ eα ,

985

− → 1 α H = h eα , n i ii

i j k ∇A = hα ijk ω ⊗ ω ⊗ ω ⊗ eα ,

α hα ijk = hikj ,

→ − where A, Rijkl , H and hα ijk are the second fundamental form, the components of the Riemannian curvature tensor, the mean curvature vector and the components of the covariant derivative of hα ij , respectively. Let

|A|2 =



hα ij

2

 2

  1  α H= hii , n α i

,

i,j,α

be the squared length of the second fundamental form and the mean curvature of M , respectively. When M is minimal, that is, H ≡ 0, we have the well-known Simons’ formula (cf. [8,24]):   2    α β  1 |A|2 = |∇A|2 + nc|A|2 + tr Aα Aβ − Aβ Aα − tr A A ) , 2 α,β

where |∇A|2 =

(2.1)

α,β

    2 and Aα = hα hα ijk ij n×n . i,j,k,α

The last terms in the above expression can be estimated as (cf. [8,16,24])      α β 2 tr Aα Aβ ) − tr A A − Aβ Aα ≤ b(m)|A|4 , α,β

(2.2)

α,β

with b(1) = 1 and b(m) = 32 if m ≥ 2. Xin and Yang proved the following estimate: Lemma 2.1. (See [26].) Let M be n-dimensional immersed submanifold with parallel mean curvature in n+m M (κ), then |∇A|2 − |∇|A||2 ≥

2 |∇|A||2 . nm

(2.3)

Recalling that |A|2 = 2|A| |A| + 2|∇|A||2 , using (2.2), (2.3) and taking κ = −1, we get the following Kato-type inequality for n-dimensional minimal submanifold of Hn+m : |A| |A| + b(m)|A|4 + n|A|2 ≥

2 |∇|A||2 . nm

(2.4)

Let φ be a function in C0∞ (M ). Multiplying (2.4) by φ2 and integrating on M , we deduce from the divergence theorem that 2 mn



|∇|A|| φ ≤ −

M

|∇|A|| φ − 2

2 2

M

φ|A| ∇φ, ∇|A| +

2 2

M

(b(m)|A|2 + n)|A|2 φ2 , M

that is 

2 1+ mn





|∇|A|| φ ≤ −2

M

φ|A| ∇φ, ∇|A| +

2 2

M

(b(m)|A|2 + n)|A|2 φ2 . M

The following eigenvalue estimate is a main tool in proving our results.

(2.5)

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Lemma 2.2. (See [9].) Let M be an n-dimensional complete minimal submanifold in Hn+m and let λ1 (M ) be the first eigenvalue of M which is defined by  λ1 (M ) =

◦

|∇f |2 . f2 M

M 

(2.6)

(n − 1)2 . 4

(2.7)

inf

f ∈H12 (M ), f =0

Then, we have λ1 (M ) ≥

We need the following well-known Sobolev inequality to prove Theorem 1.2: Lemma 2.3. (See [14].) Let M be an n-dimensional complete minimal submanifold in a Hadamard manifold N . There exists a positive constant a which depends only on n such that ⎛ ⎝



⎞(n−1)/n |ψ| n−1 ⎠ n

≤a

M

|∇ψ|,

(2.8)

M

◦

for any ψ ∈H12 (M ).  2 Proof of Theorem 1.1. From (1.2), we can find an  ∈ 0, (n−1) such that 8mn b(m)|A| (x) + n ≤ 2

 1+

2 mn



(n − 1)2

4

− , ∀x ∈ M.

(2.9)

It follows from (2.7) that

4 |A| φ ≤ (n − 1)2

|∇(φ|A|)|2

2 2

M

(2.10)

M

Substituting (2.9) and (2.10) into (2.5), we get  1+

2 mn





|∇|A||2 φ2 ≤ −2

M

M



 4 2 − φ|A| ∇φ, ∇|A| + 1 + |∇(|A|φ)|2 , mn (n − 1)2

(2.11)

M

that is, 4 (n − 1)2



|∇|A||2 φ2 ≤ 2δ M

φ|A| ∇φ, ∇|A| + (1 + δ)

M

|A|2 |∇φ|2 ,

(2.12)

M

where δ=

4 2 − > 0. mn (n − 1)2

(2.13)

Substituting

φ|A| ∇φ, ∇|A| ≤

2δ M

 (n − 1)2

|∇|A||2 φ2 + M

(n − 1)2 δ 2 

|A|2 |∇φ|2 M

C. Xia, Q. Wang / J. Math. Anal. Appl. 436 (2016) 983–989

987

into (2.12), we get

|∇|A||2 φ2 ≤

(n − 1)2 3

 1+δ+

(n − 1)2 δ 2 



M

|A|2 |∇φ|2 .

(2.14)

M

Fix a p ∈ M and choose φ to be a cut-off function with the properties 

1 0 ≤ φ ≤ 1, |∇φ| ≤ , R

1 on Bp (R),

φ=

(2.15)

0 on M \ Bp (2R).

Substituting the above φ into (2.14) we get



|∇|A|| φ ≤

|∇|A||2 φ2

2 2

M

Bp (R)

(n − 1)2 ≤ 3



(n − 1)2 δ 2 1+δ+ 



Bp (2R) R2

|A|2

.

(2.16)

Taking R → ∞ and using (1.1), we conclude that ∇|A| = 0, that is, |A| = d = const. If d = 0, we know from (1.1) that lim sup R→∞

Vol[Bp (R)] = 0. R2

(2.17)

It then follows from [7] (cf. [17]) that λ1 (M ) = 0 which contradicts with (2.7). Hence |A| = 0. 2 Proof of Theorem 1.2. Replacing ψ by ψ ⎛ ⎝

2(n−1) n−2

in (2.8), we get ⎞(n−2)/n

|ψ|

2n n−2

⎠

≤ a1

M

|∇ψ|2

(2.18)

M

for some positive constant a1 depending only on n. Taking ψ = |A|φ, φ ∈ C0∞ (M ), we get ⎛ ⎝

⎞(n−2)/n

(|A|φ)

2n n−2

⎠

≤ a1

M



Setting γ =

M

|A|n

 n2

|∇(|A|φ)|2 .

(2.19)

M

, we then get from the Hölder’s inequality that

⎛ |A|4 φ2 ≤ ⎝

M

M

⎞ n2 ⎛ |A|n ⎠ ⎝

≤ a1 γ

⎞ n−2 n

(|A|φ)

2n n−2

⎠

M

|∇(|A|φ)|2 .

(2.20)

M

Combining (2.5), (2.7) and (2.20), we have  1+

2 mn





|∇|A||2 φ2 ≤ −2

M

M

 φ|A| ∇φ, ∇|A| + ba1 γ +

4n (n − 1)2



|∇(|A|φ)|2 , M

(2.21)

C. Xia, Q. Wang / J. Math. Anal. Appl. 436 (2016) 983–989

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that is,  1+

2 −l mn







|∇|A||2 φ2 ≤ 2(l − 1) M

φ|A| ∇φ, ∇|A| + l M

|A|2 |∇φ|2 ,

(2.22)

M

where l = ba1 γ +

4n . (n − 1)2

(2.23)

We take the constant C in Theorem 1.2 as ⎛ C=⎝

1+

2 mn

−

4n (n−1)2

ba1

⎞ n2 ⎠ .

(2.24)

It is easy to see that if (1.3) holds then 1+

2 − l > 0. mn

(2.25)

1+

2 − l ≥ τ. mn

(2.26)

Hence, we can find a τ > 0 such that

Consequently, we have



|∇|A||2 φ2 ≤ 2(l − 1)

τ M

φ|A| ∇φ, ∇|A| + l

M

|A|2 |∇φ|2 .

(2.27)

M

For any ν > 0, it holds

2(l − 1)

φ|A| ∇φ, ∇|A| ≤ |l − 1|ν

M

|∇|A||2 φ2 + M

|l − 1| ν

|A|2 |∇φ|2 .

(2.28)

M

Thus, when |l − 1|ν ≤ τ2 , we can deduce from (2.27) and (2.28) that there exists a constant ρ > 0 such that



|∇|A||2 φ2 ≤ ρ M

|A|2 |∇φ|2 , ∀φ ∈ C0∞ (M ).

(2.29)

M

One can now use the same arguments as in the proof of the final part of Theorem 1.1 to show that M is totally geodesic. 2 Acknowledgments The authors are very grateful to Professor Steven Krantz and the referees for the encouragements and the valuable suggestions.

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