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Gap theorems on hypersurfaces in spheres
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Gap theorems on hypersurfaces in spheres ✩ Peng Zhu School of Mathematics and Physics, Jiangsu University of Technology, Changzhou, Jiangsu, 213001, PR China
a r t i c l e
i n f o
Article history: Received 28 January 2015 Available online xxxx Submitted by Y. Du Keywords: Gap theorem Scale-invariant total tracefree curvature The second fundamental form
a b s t r a c t We study a complete noncompact hypersurface M n isometrically immersed in an (n + 1)-dimensional sphere Sn+1 (n ≥ 3). We prove that there is no non-trivial L2 -harmonic 2-form on M , if the length of the second fundamental form is less than a fixed constant. We also showed that the same conclusion holds if the scale-invariant total tracefree curvature is bounded above by a small constant depending only on n. These results are generalized versions of the result of Cheng and Zhou on bounded harmonic functions with finite Dirichlet integral and the one of Fang and the author on L2 harmonic 1-forms. © 2015 Elsevier Inc. All rights reserved.
1. Introduction It is well known that harmonic forms give canonical representation to de Rham cohomology on compact manifolds. Suppose that (M n , g) is a complete Riemannian manifold of dimension n (n ≥ 3). Let {e1 , . . . , en } be locally defined orthogonal frame fields of tangent bundle T M . Denote the dual coframe fields by {e1 , . . . , en }. Suppose ω = ai1 ...ip eip ∧ · · · ∧ ei1 = aI ωI , θ = bi1 ...ip eip ∧ · · · ∧ ei1 = bI ωI , where the summation is being performed over the multi-index I = (i1 , . . . , ip ), ai1 ...ik ...il ...ip = −ai1 ...il ...ik ...ip and bi1 ...ik ...il ...ip = −bi1 ...il ...ik ...ip . Set |ω|2 =
a2I ,
I
✩
This work was partially supported by NSFC Grant 11471145 and Qing Lan Project. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.jmaa.2015.05.018 0022-247X/© 2015 Elsevier Inc. All rights reserved.
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|∇ω|2 =
n
|∇ei ω|2
i=1
and ω, θ =
aI bI .
I
The space L2 (∧p T ∗ M ) is a Hilbert space with scalar product: ω, θdμ.
(ω, θ) = M
Recall that Hodge operator ∗ : ∧p T ∗ M → ∧(n−p) T ∗ M is determined by ∗ei1 ∧ · · · ∧ eip = sgn σ(i1 . . . ip , ip+1 . . . in )eip+1 ∧ · · · ∧ ein , where σ(i1 . . . ip , ip+1 . . . in ) denotes a permutation of the set (i1 . . . ip , ip+1 . . . in ) and sgn σ is the sign of σ. The operator d∗ : ∧p T ∗ M → ∧(p−1) T ∗ M is defined by d∗ ω = (−1)np+n+1 ∗ d ∗ ω. The Laplacian operator on M is defined by ω = −dd∗ − d∗ d. A p-form ω is called harmonic if ω = 0. A p-form ω is called L2 harmonic if ω = 0 is harmonic and ω ∈ L2 (∧p T ∗ M ). A p-form ω is called parallel if ∇ω = 0, where ∇ is the Levi-Civita connection of (M, g). Each L2 harmonic p-form is closed and coclosed [14]. We denote by H p (L2 (M )) the space of all L2 harmonic p-forms. Let Z2p (M ) = {α ∈ L2 (∧p (T ∗ M )) : dα = 0} and Dp (d) = {α ∈ L2 (∧p (T ∗ M )) : dα ∈ L2 (∧p+1 (T ∗ M ))}. We define the p-th space of reduced L2 cohomology by H2p (M ) =
Z2p (M ) dDp−1 (d)
.
When (M, g) is a complete Riemannian manifold, Carron [1, Corollary 1.6] proved that the space of L2 harmonic p-forms H p (L2 (M )) is isomorphic to the p-th reduced L2 cohomology H2p (M ), for 0 ≤ p ≤ n. Yau [14] showed that if a complete noncompact manifold M has non-negative Ricci curvature, then there is no non-trivial L2 harmonic 1-form and volume of M is infinite. Palmer [10] obtained that there exists no non-trivial L2 harmonic 1-form on a complete noncompact stable minimal hypersurface in Rn+1. This result has been generalized by Tanno [12] and Cheng [3] when the ambient space is replaced by a manifold with bi-Ricci curvature having a low bound. In [4,11], it is showed that a complete minimal stable (or weakly stable) hypersurface in Rn+1 (n ≥ 3) with finite total curvature is a hyperplane. Cavalcante, Mirandola and
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Vitório [2] considered a complete noncompact submanifold M n (n ≥ 3) isometrically immersed in Rn+p and obtained that if the total curvature is less than a positive real number depending only on n, there exists no non-trivial L2 harmonic 1-form on M . Fang and the author [17] proved that a complete noncompact submanifold M n in a sphere Sn+p admits no non-trivial L2 harmonic 1-form if the total curvature is bounded above by a constant depending only on n. Tanno [12] proved that if M n is a complete stable minimal hypersurface in Rn+1 (n ≤ 4), there exist no non-trivial L2 harmonic 2-forms on M . The author [15] showed that Tanno’s result still holds when ambient manifold is replaced by a sphere. On the other hand, the author [16] proved that if a complete manifold has positive isotropic curvature, then there is no non-trivial L2 harmonic 2-form. From the viewpoint of symplectic structure, it is also interesting and necessary to discuss the space of L2 harmonic 2-forms on complete manifolds [16]. Motivated by the above results, we study the space of L2 -harmonic 2-forms on hypersurfaces in spheres in this paper. Suppose that f : M n → Sn+1 is an isometric immersion of an n-dimensional complete hypersurface M in an (n + 1)-dimensional sphere Sn+1 . Let A denote the second fundamental form and H the mean curvature of the immersion f . Let Φ(X, Y ) = A(X, Y ) − HX, Y , for all vector fields X and Y , where , is the induced metric of M . Obviously, |Φ|2 = |A|2 − nH 2 . Moreover, Φ = 0 if and only if M is totally umbilical. We call Φ Ln (M ) =
|Φ|n
n1
M
the scale-invariant total tracefree curvature of M . We obtain two gap theorems for hypersurfaces in spheres (Theorems 1.1 and 1.4). Theorem 1.1. Suppose f : M n → Sn+1 is an immersion of an n-dimensional complete noncompact hypersurface M into an (n + 1)-dimensional sphere Sn+1 for n ≥ 3. Let ω be a harmonic 2-form on M satisfying 1 R→+∞ R2
|ω|2 dμ = 0,
lim
B2R
where B2R is the geodesic ball of radius 2R in M . If the length of the second fundamental form is no more √ than a constant C (C = 2 when n = 3 and C = 2 when n ≥ 4), then ω is parallel. Furthermore, if the length of the second fundamental form is less than C, then ω is trivial (i.e., ω = 0). As an application of Theorem 1.1, we get that Corollary 1.2. Suppose f : M n → Sn+1 is an immersion of an n-dimensional complete noncompact hypersurface M into an (n + 1)-dimensional sphere Sn+1 for n ≥ 3. If the length of the second fundamental form √ is less than C (C = 2 when n = 3 and C = 2 when n ≥ 4), then there is admitted no non-trivial L2 harmonic 2-form (that is, H 2 (L2 (M )) = {0}) and the second space of reduced L2 cohomology of M is trivial (that is, H2p (M ) = {0}).
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n+1 Remark 1.3. Cheng and Zhou [5] showed that if M is an n−2 (n ≥ 3) n -stable minimal hypersurface in R and the length of the second fundamental form is bounded, then M either has one end or must be a catenoid. Here, a minimal hypersurface M in Rn+1 is n−2 n -stable, if
|∇f |2 −
n−2 2 2 |A| f n
≥ 0,
M
for each compactly supported piecewise smooth function f . They proved this result by using of bounded harmonic functions and Bochner formula. Corollary 1.2 can be regarded as the result of the case of L2 harmonic 2-forms. Denote by ωn the volume of the unit ball in Rn . We have: Theorem 1.4. Let M n (n ≥ 3) be a complete noncompact hypersurface isometrically immersed in Sn+1 . Let ω be a harmonic 2-form on M satisfying 1 R→+∞ R2
|ω|2 dμ = 0,
lim
BR
where BR is the geodesic ball of radius R in M . There exists a positive constant δ(n) depending only on n such that if the scale-invariant total tracefree curvature Φ Ln (M ) is less than δ(n), then ω is trivial. More precisely, δ(n) can be expressed as follows: δ(n) =
√ 3ω √ 3 3· 2·44 √ (n−2) n ωn 2n(n−1)·4n+1
for n = 3, for n ≥ 4.
By Theorem 1.4, we obtain that: Corollary 1.5. Let M n (n ≥ 3) be a complete noncompact hypersurface isometrically immersed in Sn+1 . If the scale-invariant total tracefree curvature Φ Ln (M ) is less than δ(n) defined in Theorem 1.4, then there is admitted no non-trivial L2 harmonic 2-form and the second space of reduced L2 cohomology of M is trivial. Remark 1.6. The strategy of the proof of Theorems 1.1 and 1.4 is arranged as follows: we compute the zeroth term in Bochner formula (see Proposition 2.3) and get its estimate. Then, we apply to the localisation of the square of the harmonic form by divergence theorem. Finally, we choose test functions and obtain decay estimates which are used to conclude gap Theorems 1.1 and 1.4. Remark 1.7. Corollaries 1.2 and 1.5 can be regarded as applications of Theorems 1.1 and 1.4, respectively. In fact, we can make use of Carron’s result [1, Corollary 1.6] that the space of L2 harmonic 2-forms is isomorphic to the second reduced L2 cohomology and obtain Corollaries 1.2 and 1.5, respectively. We do not study the second reduced L2 cohomology directly in the paper. Remark 1.8. In this paper, we only consider the case of 2-forms on hypersurfaces. We conjecture that Theorems 1.1 and 1.4 still hold for p-forms (3 ≤ p ≤ n). It is also interesting to consider more general ambient spaces. But there are at least two obstructions if one still uses the methods of this paper: one thing is to estimate the zeroth order term in Bochner formula, the other is the conditions for the Sobolev inequality to be valid, including conditions on test functions and on the injectivity radius of ambient spaces restricted to hypersurfaces.
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2. Computation of the zeroth order term in Bochner formula There is the following version of Bochner formula [8]: Lemma 2.1. If (M n , g) is a Riemannian manifold and ω = aI ωI ∈ ∧p (M ), then |ω|2 = 2 ω, ω + 2|∇ω|2 + 2E(ω), ω, where E(ω) = Rkβ iβ jα iα ai1 ···kβ ···ip eip ∧ . . . ∧ ejα ∧ . . . ∧ ei1 . Denote by ωn the volume of the unit ball in Rn . When the ambient manifold is a nonpositive curved manifold, the Hoffmann–Spruck’s Sobolev inequality holds without the restriction of the test function and the injectivity radius of the hypersurface as a subspace of the ambient manifold [7,9]. Since there is an isometric immersion: M n → Sn+1 → Rn+2 , Fang and the author [17, Proposition 2.1] showed the following property by use of the Hoffmann–Spruck’s Sobolev inequality: Proposition 2.2. (See [17].) Let M n (n ≥ 3) be an n-dimensional complete noncompact hypersurface isometrically immersed in an (n + 1)-dimensional sphere Sn+1 . Then |f |
2n n−2
n−2 n
M
≤ C0 M
|∇f |2 + n2
(H 2 + 1)f 2 M
for each f ∈ C01 (M ), where H is the mean curvature of M in Sn+1 and C0 =
2(n−1)2 ·42n+3 . n ω2 n
Now, we compute the term E(ω), ω in the case of p = 2 and obtain that: Proposition 2.3. Let M n (n ≥ 3) be a complete noncompact hypersurface isometrically immersed in Sn+1 . If ω = aij ej ∧ ei , (aij = −aji ), then E(ω), ω = 2
(n − 2) + (λ1 + · · · + λn )λi − λ2i − λi λj (aij )2 ,
i=j
where λ1 , · · · , λn are the eigenvalues of the second fundamental form A = (hij ). Proof. Let ω1 = bi1 i2 ei2 ∧ ei1 ∈ ∧2 (M ) and ω2 = ci1 i2 ei2 ∧ ei1 ∈ ∧2 (M ), where bi1 i2 = −bi2 i1 and ci1 i2 = −ci2 i1 . By Lemma 2.1, we get that E(ω1 ) = Rk1 i1 j1 i1 bk1 i2 ei2 ∧ ej1 + Rk2 i2 j2 i2 bi1 k2 ej2 ∧ ei1 + Rk2 i2 j1 i1 bi1 k2 ei2 ∧ ej1 + Rk1 i1 j2 i2 bk1 i2 ej2 ∧ ei1 = Ric k1 j1 bk1 i2 ei2 ∧ ej1 + Ric k2 j2 bi1 k2 ej2 ∧ ei1 + Rk2 i2 j1 i1 bi1 k2 ei2 ∧ ej1 + Rk1 i1 j2 i2 bk1 i2 ej2 ∧ ei1 . So, we have
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E(ω1 ), ω2 = Ric k1 j1 bk1 i2 cj1 i2 + Ric k2 j2 bi1 k2 ci1 j2 + Rk2 i2 j1 i1 bi1 k2 cj1 i2 + Rk1 i1 j2 i2 bk1 i2 ci1 j2 . It implies that E(ω), ω = Ric k1 j1 ak1 i2 aj1 i2 + Ric k2 j2 ai1 k2 ai1 j2 + Rk2 i2 j1 i1 ai1 k2 aj1 i2 + Rk1 i1 j2 i2 ak1 i2 ai1 j2 .
(2.1)
By Gauss equation, we obtain that Rijkl = (δik δjl − δil δjk ) + hik hjl − hil hjk . A direct computation shows that Ric k1 j1 = (n − 1)δk1 j1 + nHhk1 j1 − hk1 i hij1 ;
(2.2)
Ric k2 j2 = (n − 1)δk2 j2 + nHhk2 j2 − hk2 i hij2 ;
(2.3)
Rk2 i2 j1 i1 = (δk2 j1 δi2 i1 − δk2 i1 δi2 j1 ) + hk2 j1 hi2 i1 − hk2 i1 hi2 j1
(2.4)
Rk1 i1 j2 i2 = (δk1 j2 δi1 i2 − δk1 i2 δi1 j2 ) + hk1 j2 hi1 i2 − hk1 i2 hi1 j2 .
(2.5)
and
Note that the operator E : ∧p T ∗ M → ∧p T ∗ M ω → E(ω) is linear and the zeroth order differential operator. So it is sufficient to compute E(ω), ω at a point p. We can choose an orthonormal frame {ei } such that hij = λi δij at p. Obviously, nH = λ1 + · · · + λn . By (2.1)–(2.5), we have (aj1 i2 )2 + nHλk1 (ak1 i2 )2 − λ2k1 (ak1 i2 )2 + (n − 1) (ai1 j2 )2 + nHλk2 (ai1 k2 )2 − λ2k2 (ai1 k2 )2 + ai1 j1 aj1 i1 − λk2 λi2 (ak2 i2 )2 + aj2 i2 ai2 j2 − λj2 λi2 (aj2 i2 )2 = 2(n − 2) (aij )2 + 2nHλi (aij )2 −2 λi (aij )2 − 2 λi λj (aij )2 (n − 2) + (λ1 + · · · + λn )λi − λ2i − λi λj (aij )2 . =2
E(ω), ω = (n − 1)
i=j
We obtain the desired result. 2
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3. Proof of main results In this section, we will prove Theorems 1.1 and 1.4. Proof of Theorem 1.1. Let ω be a non-zero harmonic 2-form on M satisfying
1 R→+∞ R2
|ω|2 dμ = 0.
lim
(3.1)
B2R
Denote by ρ(x) the geodesic distance on M from x to a fixed point. Choose φ ∈ C01 (M ) satisfying ⎧ ⎨1 φ(x) =
⎩
on BR , on B2R \ BR , on M \ B2R .
2R−ρ(x) R
0
By Lemma 2.1, we have
φ E(ω), ω + 2
M
1 φ |∇ω| = 2 2
φ2 |ω|2
2
M
M
1 =− 2
∇φ2 , ∇|ω|2 M
|ω|∇φ, φ∇|ω|
= −2
M
1 |ω| |∇φ| + 2
≤2
2
φ2 |∇|ω||2
2
M
M
1 |ω| |∇φ| + 2
≤2
2
φ2 |∇ω|2 ,
2
M
M
where the last inequality holds because of the Kato inequality. It implies that
(2E(ω), ω + |∇ω|2 ) ≤
φ2 (2E(ω), ω + |∇ω|2 ) M
B(R)
≤4
|ω|2 |∇φ|2 M
≤
4 R2
|ω|2 .
(3.2)
B2R
Taking R → +∞ and combining with (3.1), we get the right-hand side of (3.2) tends to zero. Choosing n = 3 in Proposition 2.3, we have E(ω), ω = 2
1≤i=j≤3
=2
1≤i=j≤3
1 + (λ1 + λ2 + λ3 )λi −
λ2i
− λi λj (aij )2
1 + λ1 + . . . + λi + · · · + λj + · · · + λ3 λi (aij )2 .
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If the length of the second fundamental form |A| is no more than that 1 + λi λj ≥ 0 for 1 ≤ i = j ≤ 3. Thus,
√ 2, then λ21 + λ22 + λ23 ≤ 2. This implies
E(ω), ω ≥ 0. By (3.2), we have that ∇ω = 0, that is, ω is parallel. Furthermore, if the length of the second fundamental √ form |A| is less than 2, then λ21 + λ22 + λ23 < 2. This implies that 1 + λi λj > 0 for 1 ≤ i = j ≤ 3. Since ω is non-zero, there exist a point p0 and a constant R0 such that E(ω), ω > 0.
(3.3)
BR0 (p0 )
By (3.2), we get a contradiction. For n ≥ 4, by Proposition 2.3, we have that E(ω), ω = 2
i=j
=
(n − 2) + (λ1 + · · · + λn )λi −
− λi λj (aij )2
2 2 2(n − 2) + (λ1 + · · · + λn )(λi + λj ) − λi + λj − 2λi λj (aij )2
i=j
=
λ2i
2(n − 2) + λ1 + · · · + λi + · · · + λj + · · · + λn (λi + λj ) (aij )2
i=j
=
i=j
1 1 2(n − 2) + (nH)2 − 2 2
≥
i=j
λk
k=1,k=i,j
1 n−2 2(n − 2) + (nH)2 − 2 2
n
1 2 − (λi + λj ) (aij )2 2
λ2k
− (λ2i + λ2j ) (aij )2
k=1,k=i,j
1 n−2 2 2(n − 2) + (nH)2 − |A| (aij )2 2 2 i=j 1 n−2 2 2 |A| |ω|2 . = 2(n − 2) + (nH) − 2 2 ≥
2
n
(3.4)
If the length of the second fundamental form is no more than 2, then E(ω), ω ≥ 0. By (3.2), we obtain that ∇ω = 0, that is, ω is parallel. Furthermore, suppose that the length of the second fundamental form |A| is less than 2. Combining with (3.4), we obtain that there exist a point p0 and a positive constant R0 such that E(ω), ω > 0.
(3.5)
BR0 (p0 )
By (3.2), we get a contradiction. 2 Proof of Theorem 1.4. Let ω be a non-zero harmonic 2-form on M satisfying 1 R→+∞ R2
|ω|2 dμ = 0.
lim
BR
(3.6)
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Then we get |ω|2 = 2|∇|ω||2 + 2|ω| |ω|.
(3.7)
Combining (3.7) with Lemma 2.1, we have that |ω| |ω| = |∇ω|2 − |∇|ω||2 + E(ω), ω.
(3.8)
There exists the Kato inequality [6,13]: |∇ω|2 ≥
n−1 |∇|ω||2 . n−2
(3.9)
Combining (3.8) with (3.9), we have that |ω| |ω| ≥
1 |∇|ω||2 + E(ω), ω. n−2
(3.10)
Set h = |ω|. Let η ∈ C01 (M ). Then div(η 2 h∇h) = η 2 h h + ∇(η 2 h), ∇h = η 2 h h + η 2 |∇h|2 + 2ηh∇η, ∇h. By divergence theorem, we obtain that
M
η 2 |∇h|2 + 2
η 2 h h + M
ηh∇η, ∇h = 0.
(3.11)
M
For n = 3, we get an estimate for the term E(ω), ω: E(ω), ω = 2
1 + (λ1 + λ2 + λ3 )λi − λ2i − λi λj (aij )2
i=j
=
i=j
=
2 + (λ1 + λ2 + λ3 )(λi + λj ) − λ2i + λ2j − 2λi λj (aij )2
2 + (λ1 + · · · + λi + · · · + λj + · · · + λ3 )(λi + λj ) (aij )2
i=j
1 2 2 2
4 + (3H) − (λ1 + · · · + λi + · · · + λj + · · · + λ3 ) − (λi + λj ) (aij )2 = 2 i=j 1 2 2 2 2
4 + (3H) − (λ1 + · · · + λi + · · · + λj + · · · + λ3 ) − 2(λi + λj ) (aij )2 ≥ 2 i=j |A|2 2 9H 2 3|Φ|2 − |A|2 h2 = 2 − + h . ≥ 2+ 2 2 2
(3.12)
Combining (3.10) with (3.12), we get that |A|2 2 3|Φ|2 + h . h h ≥ |∇h| + 2 − 2 2
2
(3.13)
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By (3.11) and (3.13), we obtain that −2
ηh∇η, ∇h − 2
M
−
|A| 2
2+
M
2
η 2 |∇h|2
3 2
η 2 h2 +
M
|Φ|2 η 2 h2 ≥ 0.
(3.14)
M
Note that |A|2 ≥ 3H 2 and
1 η |∇h| + a1 2
a1
h |∇η| ≥ −2
2
M
2
ηh∇η, ∇h,
2
M
M
for each positive constant a1 . By (3.14), we have (a1 − 2) −
1 a1
η 2 |∇h|2 + M
3H 2+ 2
2
h2 |∇η|2 M
3 η h + 2
|Φ|2 η 2 h2 ≥ 0.
2 2
M
(3.15)
M
Set
|Φ|
3
φ1 (η) =
13 .
supp(η)
By Hölder inequality and Proposition 2.2, we get
23 13 6 |Φ| · (ηh)
|Φ| η h ≤ 2 2 2
M
3
M
supp(η)
≤ C0 φ21 (η) M
≤
C0 φ21 (η)
|∇(ηh)|2 + 9
(H 2 + 1)η 2 h2 M
1 η |∇h| + 1 + b1 2
(1 + b1 )
M
h |∇η| + 9
2
2
2
M
2 2 H +1 η h , 2
(3.16)
M
for each positive constant b1 . Combining (3.15) with (3.16), we have A1 M
where
η 2 |∇h|2 + B1
H 2 η 2 h2 + C 1
M
η 2 h2 ≤ D1
M
|∇η|2 h2 , M
(3.17)
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3C0 φ21 (η)(1 + b1 ) , 2 3 27C0 φ21 (η) , B1 = − 2 2 27C0 φ21 (η) C1 = 2 − 2
A 1 = 2 − a1 −
and 1 1 3C0 φ21 (η) . 1+ D1 = + a1 2 b1 Obviously, D1 is positive. If the scale-invariant total tracefree curvature Φ L3 (M ) is less than δ(3), then B1 and C1 are positive. Choose a1 and b1 small enough such that A1 is positive. Let BR be a geodesic ball of radius R on M centered at a fixed point. Choose η ∈ C01 (M ) such that ⎧ 0 ≤ η ≤ 1, ⎪ ⎪ ⎨ η ≡ 1 on B R , 2
η ≡ 0 on M \ BR , ⎪ ⎪ ⎩ |∇η| ≤ R2 . Let R → +∞ in (3.17). We obtain that h = 0. That is, ω is trivial. For n ≥ 4, by (3.4), we get an estimate for the term E(ω), ω: 1 n−2 2 |A| |ω|2 2(n − 2) + (nH)2 − 2 2 n = 2(n − 2) − |Φ|2 + |A|2 h2 . 2
E(ω), ω ≥
(3.18)
By (3.10) and (3.18), we obtain that 1 n 2 2 2 |∇h| + 2(n − 2) − |Φ| + |A| h2 . h h ≥ n−2 2
(3.19)
Combining (3.11) with (3.19), we get that −2
ηh∇η, ∇h − M
n−1 n−2
−
η 2 |∇h|2 M
(2(n − 2) + |A|2 )η 2 h2 +
n 2
M
|Φ|2 η 2 h2 ≥ 0. M
Note that |A|2 ≥ nH 2 and
1 η |∇h| + a2 2
a2 M
h |∇η| ≥ −2
2
2
ηh∇η, ∇h,
2
M
for each positive constant a2 . Combining with (3.20), we have
M
(3.20)
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n−1 a2 − n−2
−
η 2 |∇h|2 +
1 a2
M
h2 |∇η|2 M
2
n 2(n − 2) + nH η 2 h2 + 2
M
|Φ|2 η 2 h2 ≥ 0.
(3.21)
M
Set |Φ|n
φ2 (η) =
n1 .
supp(η)
By Hölder inequality and Proposition 2.2, we get
n2 n−2 n 2n |Φ| · (ηh) n−2
|Φ| η h ≤ 2 2 2
M
n
M
supp(η)
≤
|∇(ηh)| + n
C0 φ22 (η)
2
M
2
2 H + 1 η 2 h2
M
2 2 2 1 2 2 2 2 2 2 ≤ C0 φ2 (η) (1 + b2 ) η |∇h| + 1 + h |∇η| + n H +1 η h , b2 M
M
(3.22)
M
for each positive constant b2 . By (3.21) and (3.22), we have A2
η 2 |∇h|2 + B2
M
H 2 η 2 h2 + C 2
M
η 2 h2 ≤ D2
M
|∇η|2 h2 ,
(3.23)
M
where n−1 nC0 φ22 (η)(1 + b2 ) − a2 − , n−2 2 n3 C0 φ22 (η) B2 = n − , 2 C0 n3 φ22 (η) C2 = 2(n − 2) − 2
A2 =
and 1 1 nC0 φ22 (η) . 1+ D2 = + a2 2 b2 Obviously, D2 is positive. If the scale-invariant total tracefree curvature Φ Ln (M ) is less than δ(n), then B2 and C2 are positive. Choose a2 and b2 small enough such that A2 is positive. Let BR be a geodesic ball of radius R on M centered at a fixed point. Choose η ∈ C01 (M ) such that ⎧ 0 ≤ η ≤ 1, ⎪ ⎪ ⎨ η ≡ 1 on B R , 2 η ≡ 0 on M \ BR , ⎪ ⎪ ⎩ |∇η| ≤ R2 . Let R → +∞ in (3.23). We obtain that h = 0. That is, ω = 0. 2
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Acknowledgments The author would like to thank professors Detang Zhou and Hongyu Wang for many invaluable suggestions. The author is also grateful to the referees for their valuable comments. References [1] G. Carron, L2 harmonic forms on non compact manifolds, arXiv:0704.3194v1, 2007. [2] M.P. Cavalcante, H. Mirandola, F. Vitório, L2 -harmonic 1-forms on submanifolds with finite total curvature, J. Geom. Anal. 24 (2014) 205–222. [3] X. Cheng, L2 harmonic forms and stability of hypersurfaces with constant mean curvature, Bol. Soc. Bras. Mat. 31 (2000) 225–239. [4] X. Cheng, L.F. Cheung, D.T. Zhou, The structure of weakly stable constant mean curvature hypersurfaces, Tohoku Math. J. 60 (2008) 101–121. [5] X. Cheng, D.T. Zhou, Manifolds with weighted Poincaré inequality and uniqueness of minimal hypersurfaces, Comm. Anal. Geom. 17 (2009) 139–154. [6] D. Cibotaru, P. Zhu, Refined Kato inequalities for harmonic fields on Kähler manifolds, Pacific J. Math. 256 (2012) 51–66. [7] D. Hoffman, J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math. 27 (1974) 715–727. [8] P. Li, Lecture Notes on Geometric Analysis, Lecture Notes Ser., vol. 6, National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1993. [9] J.H. Michael, L.M. Simon, Sobolev and mean-value inequalities on generalized submanifolds of Rn , Comm. Pure Appl. Math. 26 (1973) 361–379. [10] B. Palmer, Stability of minimal hypersurfaces, Comment. Math. Helv. 66 (1991) 185–188. [11] Y. Shen, X.H. Zhu, On stable complete minimal hypersurfaces in Rn+1 , Amer. J. Math. 120 (1998) 103–116. [12] S. Tanno, L2 harmonic forms and stability of minimal hypersurfaces, J. Math. Soc. Japan 48 (1996) 761–768. [13] X.D. Wang, On the L2 -cohomology of a convex cocompact hyperbolic manifold, Duke Math. J. 115 (2002) 311–327. [14] S.T. Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976) 659–670. [15] P. Zhu, L2 -harmonic forms and stable hypersurfaces in space forms, Arch. Math. 97 (2011) 271–279. [16] P. Zhu, Harmonic two-forms on manifolds with nonnegative isotropic curvature, Ann. Global Anal. Geom. 40 (2011) 427–434. [17] P. Zhu, S.W. Fang, A gap theorem on submanifolds with finite total curvature in spheres, J. Math. Anal. Appl. 413 (2014) 195–201.
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J. Math. Anal. Appl. ••• (••••) •••–•••
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Gap theorems on hypersurfaces in spheres ✩ Peng Zhu School of Mathematics and Physics, Jiangsu University of Technology, Changzhou, Jiangsu, 213001, PR China
a r t i c l e
i n f o
Article history: Received 28 January 2015 Available online xxxx Submitted by Y. Du Keywords: Gap theorem Scale-invariant total tracefree curvature The second fundamental form
a b s t r a c t We study a complete noncompact hypersurface M n isometrically immersed in an (n + 1)-dimensional sphere Sn+1 (n ≥ 3). We prove that there is no non-trivial L2 -harmonic 2-form on M , if the length of the second fundamental form is less than a fixed constant. We also showed that the same conclusion holds if the scale-invariant total tracefree curvature is bounded above by a small constant depending only on n. These results are generalized versions of the result of Cheng and Zhou on bounded harmonic functions with finite Dirichlet integral and the one of Fang and the author on L2 harmonic 1-forms. © 2015 Elsevier Inc. All rights reserved.
1. Introduction It is well known that harmonic forms give canonical representation to de Rham cohomology on compact manifolds. Suppose that (M n , g) is a complete Riemannian manifold of dimension n (n ≥ 3). Let {e1 , . . . , en } be locally defined orthogonal frame fields of tangent bundle T M . Denote the dual coframe fields by {e1 , . . . , en }. Suppose ω = ai1 ...ip eip ∧ · · · ∧ ei1 = aI ωI , θ = bi1 ...ip eip ∧ · · · ∧ ei1 = bI ωI , where the summation is being performed over the multi-index I = (i1 , . . . , ip ), ai1 ...ik ...il ...ip = −ai1 ...il ...ik ...ip and bi1 ...ik ...il ...ip = −bi1 ...il ...ik ...ip . Set |ω|2 =
a2I ,
I
✩
This work was partially supported by NSFC Grant 11471145 and Qing Lan Project. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.jmaa.2015.05.018 0022-247X/© 2015 Elsevier Inc. All rights reserved.
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|∇ω|2 =
n
|∇ei ω|2
i=1
and ω, θ =
aI bI .
I
The space L2 (∧p T ∗ M ) is a Hilbert space with scalar product: ω, θdμ.
(ω, θ) = M
Recall that Hodge operator ∗ : ∧p T ∗ M → ∧(n−p) T ∗ M is determined by ∗ei1 ∧ · · · ∧ eip = sgn σ(i1 . . . ip , ip+1 . . . in )eip+1 ∧ · · · ∧ ein , where σ(i1 . . . ip , ip+1 . . . in ) denotes a permutation of the set (i1 . . . ip , ip+1 . . . in ) and sgn σ is the sign of σ. The operator d∗ : ∧p T ∗ M → ∧(p−1) T ∗ M is defined by d∗ ω = (−1)np+n+1 ∗ d ∗ ω. The Laplacian operator on M is defined by ω = −dd∗ − d∗ d. A p-form ω is called harmonic if ω = 0. A p-form ω is called L2 harmonic if ω = 0 is harmonic and ω ∈ L2 (∧p T ∗ M ). A p-form ω is called parallel if ∇ω = 0, where ∇ is the Levi-Civita connection of (M, g). Each L2 harmonic p-form is closed and coclosed [14]. We denote by H p (L2 (M )) the space of all L2 harmonic p-forms. Let Z2p (M ) = {α ∈ L2 (∧p (T ∗ M )) : dα = 0} and Dp (d) = {α ∈ L2 (∧p (T ∗ M )) : dα ∈ L2 (∧p+1 (T ∗ M ))}. We define the p-th space of reduced L2 cohomology by H2p (M ) =
Z2p (M ) dDp−1 (d)
.
When (M, g) is a complete Riemannian manifold, Carron [1, Corollary 1.6] proved that the space of L2 harmonic p-forms H p (L2 (M )) is isomorphic to the p-th reduced L2 cohomology H2p (M ), for 0 ≤ p ≤ n. Yau [14] showed that if a complete noncompact manifold M has non-negative Ricci curvature, then there is no non-trivial L2 harmonic 1-form and volume of M is infinite. Palmer [10] obtained that there exists no non-trivial L2 harmonic 1-form on a complete noncompact stable minimal hypersurface in Rn+1. This result has been generalized by Tanno [12] and Cheng [3] when the ambient space is replaced by a manifold with bi-Ricci curvature having a low bound. In [4,11], it is showed that a complete minimal stable (or weakly stable) hypersurface in Rn+1 (n ≥ 3) with finite total curvature is a hyperplane. Cavalcante, Mirandola and
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Vitório [2] considered a complete noncompact submanifold M n (n ≥ 3) isometrically immersed in Rn+p and obtained that if the total curvature is less than a positive real number depending only on n, there exists no non-trivial L2 harmonic 1-form on M . Fang and the author [17] proved that a complete noncompact submanifold M n in a sphere Sn+p admits no non-trivial L2 harmonic 1-form if the total curvature is bounded above by a constant depending only on n. Tanno [12] proved that if M n is a complete stable minimal hypersurface in Rn+1 (n ≤ 4), there exist no non-trivial L2 harmonic 2-forms on M . The author [15] showed that Tanno’s result still holds when ambient manifold is replaced by a sphere. On the other hand, the author [16] proved that if a complete manifold has positive isotropic curvature, then there is no non-trivial L2 harmonic 2-form. From the viewpoint of symplectic structure, it is also interesting and necessary to discuss the space of L2 harmonic 2-forms on complete manifolds [16]. Motivated by the above results, we study the space of L2 -harmonic 2-forms on hypersurfaces in spheres in this paper. Suppose that f : M n → Sn+1 is an isometric immersion of an n-dimensional complete hypersurface M in an (n + 1)-dimensional sphere Sn+1 . Let A denote the second fundamental form and H the mean curvature of the immersion f . Let Φ(X, Y ) = A(X, Y ) − HX, Y , for all vector fields X and Y , where , is the induced metric of M . Obviously, |Φ|2 = |A|2 − nH 2 . Moreover, Φ = 0 if and only if M is totally umbilical. We call Φ Ln (M ) =
|Φ|n
n1
M
the scale-invariant total tracefree curvature of M . We obtain two gap theorems for hypersurfaces in spheres (Theorems 1.1 and 1.4). Theorem 1.1. Suppose f : M n → Sn+1 is an immersion of an n-dimensional complete noncompact hypersurface M into an (n + 1)-dimensional sphere Sn+1 for n ≥ 3. Let ω be a harmonic 2-form on M satisfying 1 R→+∞ R2
|ω|2 dμ = 0,
lim
B2R
where B2R is the geodesic ball of radius 2R in M . If the length of the second fundamental form is no more √ than a constant C (C = 2 when n = 3 and C = 2 when n ≥ 4), then ω is parallel. Furthermore, if the length of the second fundamental form is less than C, then ω is trivial (i.e., ω = 0). As an application of Theorem 1.1, we get that Corollary 1.2. Suppose f : M n → Sn+1 is an immersion of an n-dimensional complete noncompact hypersurface M into an (n + 1)-dimensional sphere Sn+1 for n ≥ 3. If the length of the second fundamental form √ is less than C (C = 2 when n = 3 and C = 2 when n ≥ 4), then there is admitted no non-trivial L2 harmonic 2-form (that is, H 2 (L2 (M )) = {0}) and the second space of reduced L2 cohomology of M is trivial (that is, H2p (M ) = {0}).
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n+1 Remark 1.3. Cheng and Zhou [5] showed that if M is an n−2 (n ≥ 3) n -stable minimal hypersurface in R and the length of the second fundamental form is bounded, then M either has one end or must be a catenoid. Here, a minimal hypersurface M in Rn+1 is n−2 n -stable, if
|∇f |2 −
n−2 2 2 |A| f n
≥ 0,
M
for each compactly supported piecewise smooth function f . They proved this result by using of bounded harmonic functions and Bochner formula. Corollary 1.2 can be regarded as the result of the case of L2 harmonic 2-forms. Denote by ωn the volume of the unit ball in Rn . We have: Theorem 1.4. Let M n (n ≥ 3) be a complete noncompact hypersurface isometrically immersed in Sn+1 . Let ω be a harmonic 2-form on M satisfying 1 R→+∞ R2
|ω|2 dμ = 0,
lim
BR
where BR is the geodesic ball of radius R in M . There exists a positive constant δ(n) depending only on n such that if the scale-invariant total tracefree curvature Φ Ln (M ) is less than δ(n), then ω is trivial. More precisely, δ(n) can be expressed as follows: δ(n) =
√ 3ω √ 3 3· 2·44 √ (n−2) n ωn 2n(n−1)·4n+1
for n = 3, for n ≥ 4.
By Theorem 1.4, we obtain that: Corollary 1.5. Let M n (n ≥ 3) be a complete noncompact hypersurface isometrically immersed in Sn+1 . If the scale-invariant total tracefree curvature Φ Ln (M ) is less than δ(n) defined in Theorem 1.4, then there is admitted no non-trivial L2 harmonic 2-form and the second space of reduced L2 cohomology of M is trivial. Remark 1.6. The strategy of the proof of Theorems 1.1 and 1.4 is arranged as follows: we compute the zeroth term in Bochner formula (see Proposition 2.3) and get its estimate. Then, we apply to the localisation of the square of the harmonic form by divergence theorem. Finally, we choose test functions and obtain decay estimates which are used to conclude gap Theorems 1.1 and 1.4. Remark 1.7. Corollaries 1.2 and 1.5 can be regarded as applications of Theorems 1.1 and 1.4, respectively. In fact, we can make use of Carron’s result [1, Corollary 1.6] that the space of L2 harmonic 2-forms is isomorphic to the second reduced L2 cohomology and obtain Corollaries 1.2 and 1.5, respectively. We do not study the second reduced L2 cohomology directly in the paper. Remark 1.8. In this paper, we only consider the case of 2-forms on hypersurfaces. We conjecture that Theorems 1.1 and 1.4 still hold for p-forms (3 ≤ p ≤ n). It is also interesting to consider more general ambient spaces. But there are at least two obstructions if one still uses the methods of this paper: one thing is to estimate the zeroth order term in Bochner formula, the other is the conditions for the Sobolev inequality to be valid, including conditions on test functions and on the injectivity radius of ambient spaces restricted to hypersurfaces.
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2. Computation of the zeroth order term in Bochner formula There is the following version of Bochner formula [8]: Lemma 2.1. If (M n , g) is a Riemannian manifold and ω = aI ωI ∈ ∧p (M ), then |ω|2 = 2 ω, ω + 2|∇ω|2 + 2E(ω), ω, where E(ω) = Rkβ iβ jα iα ai1 ···kβ ···ip eip ∧ . . . ∧ ejα ∧ . . . ∧ ei1 . Denote by ωn the volume of the unit ball in Rn . When the ambient manifold is a nonpositive curved manifold, the Hoffmann–Spruck’s Sobolev inequality holds without the restriction of the test function and the injectivity radius of the hypersurface as a subspace of the ambient manifold [7,9]. Since there is an isometric immersion: M n → Sn+1 → Rn+2 , Fang and the author [17, Proposition 2.1] showed the following property by use of the Hoffmann–Spruck’s Sobolev inequality: Proposition 2.2. (See [17].) Let M n (n ≥ 3) be an n-dimensional complete noncompact hypersurface isometrically immersed in an (n + 1)-dimensional sphere Sn+1 . Then |f |
2n n−2
n−2 n
M
≤ C0 M
|∇f |2 + n2
(H 2 + 1)f 2 M
for each f ∈ C01 (M ), where H is the mean curvature of M in Sn+1 and C0 =
2(n−1)2 ·42n+3 . n ω2 n
Now, we compute the term E(ω), ω in the case of p = 2 and obtain that: Proposition 2.3. Let M n (n ≥ 3) be a complete noncompact hypersurface isometrically immersed in Sn+1 . If ω = aij ej ∧ ei , (aij = −aji ), then E(ω), ω = 2
(n − 2) + (λ1 + · · · + λn )λi − λ2i − λi λj (aij )2 ,
i=j
where λ1 , · · · , λn are the eigenvalues of the second fundamental form A = (hij ). Proof. Let ω1 = bi1 i2 ei2 ∧ ei1 ∈ ∧2 (M ) and ω2 = ci1 i2 ei2 ∧ ei1 ∈ ∧2 (M ), where bi1 i2 = −bi2 i1 and ci1 i2 = −ci2 i1 . By Lemma 2.1, we get that E(ω1 ) = Rk1 i1 j1 i1 bk1 i2 ei2 ∧ ej1 + Rk2 i2 j2 i2 bi1 k2 ej2 ∧ ei1 + Rk2 i2 j1 i1 bi1 k2 ei2 ∧ ej1 + Rk1 i1 j2 i2 bk1 i2 ej2 ∧ ei1 = Ric k1 j1 bk1 i2 ei2 ∧ ej1 + Ric k2 j2 bi1 k2 ej2 ∧ ei1 + Rk2 i2 j1 i1 bi1 k2 ei2 ∧ ej1 + Rk1 i1 j2 i2 bk1 i2 ej2 ∧ ei1 . So, we have
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E(ω1 ), ω2 = Ric k1 j1 bk1 i2 cj1 i2 + Ric k2 j2 bi1 k2 ci1 j2 + Rk2 i2 j1 i1 bi1 k2 cj1 i2 + Rk1 i1 j2 i2 bk1 i2 ci1 j2 . It implies that E(ω), ω = Ric k1 j1 ak1 i2 aj1 i2 + Ric k2 j2 ai1 k2 ai1 j2 + Rk2 i2 j1 i1 ai1 k2 aj1 i2 + Rk1 i1 j2 i2 ak1 i2 ai1 j2 .
(2.1)
By Gauss equation, we obtain that Rijkl = (δik δjl − δil δjk ) + hik hjl − hil hjk . A direct computation shows that Ric k1 j1 = (n − 1)δk1 j1 + nHhk1 j1 − hk1 i hij1 ;
(2.2)
Ric k2 j2 = (n − 1)δk2 j2 + nHhk2 j2 − hk2 i hij2 ;
(2.3)
Rk2 i2 j1 i1 = (δk2 j1 δi2 i1 − δk2 i1 δi2 j1 ) + hk2 j1 hi2 i1 − hk2 i1 hi2 j1
(2.4)
Rk1 i1 j2 i2 = (δk1 j2 δi1 i2 − δk1 i2 δi1 j2 ) + hk1 j2 hi1 i2 − hk1 i2 hi1 j2 .
(2.5)
and
Note that the operator E : ∧p T ∗ M → ∧p T ∗ M ω → E(ω) is linear and the zeroth order differential operator. So it is sufficient to compute E(ω), ω at a point p. We can choose an orthonormal frame {ei } such that hij = λi δij at p. Obviously, nH = λ1 + · · · + λn . By (2.1)–(2.5), we have (aj1 i2 )2 + nHλk1 (ak1 i2 )2 − λ2k1 (ak1 i2 )2 + (n − 1) (ai1 j2 )2 + nHλk2 (ai1 k2 )2 − λ2k2 (ai1 k2 )2 + ai1 j1 aj1 i1 − λk2 λi2 (ak2 i2 )2 + aj2 i2 ai2 j2 − λj2 λi2 (aj2 i2 )2 = 2(n − 2) (aij )2 + 2nHλi (aij )2 −2 λi (aij )2 − 2 λi λj (aij )2 (n − 2) + (λ1 + · · · + λn )λi − λ2i − λi λj (aij )2 . =2
E(ω), ω = (n − 1)
i=j
We obtain the desired result. 2
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3. Proof of main results In this section, we will prove Theorems 1.1 and 1.4. Proof of Theorem 1.1. Let ω be a non-zero harmonic 2-form on M satisfying
1 R→+∞ R2
|ω|2 dμ = 0.
lim
(3.1)
B2R
Denote by ρ(x) the geodesic distance on M from x to a fixed point. Choose φ ∈ C01 (M ) satisfying ⎧ ⎨1 φ(x) =
⎩
on BR , on B2R \ BR , on M \ B2R .
2R−ρ(x) R
0
By Lemma 2.1, we have
φ E(ω), ω + 2
M
1 φ |∇ω| = 2 2
φ2 |ω|2
2
M
M
1 =− 2
∇φ2 , ∇|ω|2 M
|ω|∇φ, φ∇|ω|
= −2
M
1 |ω| |∇φ| + 2
≤2
2
φ2 |∇|ω||2
2
M
M
1 |ω| |∇φ| + 2
≤2
2
φ2 |∇ω|2 ,
2
M
M
where the last inequality holds because of the Kato inequality. It implies that
(2E(ω), ω + |∇ω|2 ) ≤
φ2 (2E(ω), ω + |∇ω|2 ) M
B(R)
≤4
|ω|2 |∇φ|2 M
≤
4 R2
|ω|2 .
(3.2)
B2R
Taking R → +∞ and combining with (3.1), we get the right-hand side of (3.2) tends to zero. Choosing n = 3 in Proposition 2.3, we have E(ω), ω = 2
1≤i=j≤3
=2
1≤i=j≤3
1 + (λ1 + λ2 + λ3 )λi −
λ2i
− λi λj (aij )2
1 + λ1 + . . . + λi + · · · + λj + · · · + λ3 λi (aij )2 .
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If the length of the second fundamental form |A| is no more than that 1 + λi λj ≥ 0 for 1 ≤ i = j ≤ 3. Thus,
√ 2, then λ21 + λ22 + λ23 ≤ 2. This implies
E(ω), ω ≥ 0. By (3.2), we have that ∇ω = 0, that is, ω is parallel. Furthermore, if the length of the second fundamental √ form |A| is less than 2, then λ21 + λ22 + λ23 < 2. This implies that 1 + λi λj > 0 for 1 ≤ i = j ≤ 3. Since ω is non-zero, there exist a point p0 and a constant R0 such that E(ω), ω > 0.
(3.3)
BR0 (p0 )
By (3.2), we get a contradiction. For n ≥ 4, by Proposition 2.3, we have that E(ω), ω = 2
i=j
=
(n − 2) + (λ1 + · · · + λn )λi −
− λi λj (aij )2
2 2 2(n − 2) + (λ1 + · · · + λn )(λi + λj ) − λi + λj − 2λi λj (aij )2
i=j
=
λ2i
2(n − 2) + λ1 + · · · + λi + · · · + λj + · · · + λn (λi + λj ) (aij )2
i=j
=
i=j
1 1 2(n − 2) + (nH)2 − 2 2
≥
i=j
λk
k=1,k=i,j
1 n−2 2(n − 2) + (nH)2 − 2 2
n
1 2 − (λi + λj ) (aij )2 2
λ2k
− (λ2i + λ2j ) (aij )2
k=1,k=i,j
1 n−2 2 2(n − 2) + (nH)2 − |A| (aij )2 2 2 i=j 1 n−2 2 2 |A| |ω|2 . = 2(n − 2) + (nH) − 2 2 ≥
2
n
(3.4)
If the length of the second fundamental form is no more than 2, then E(ω), ω ≥ 0. By (3.2), we obtain that ∇ω = 0, that is, ω is parallel. Furthermore, suppose that the length of the second fundamental form |A| is less than 2. Combining with (3.4), we obtain that there exist a point p0 and a positive constant R0 such that E(ω), ω > 0.
(3.5)
BR0 (p0 )
By (3.2), we get a contradiction. 2 Proof of Theorem 1.4. Let ω be a non-zero harmonic 2-form on M satisfying 1 R→+∞ R2
|ω|2 dμ = 0.
lim
BR
(3.6)
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Then we get |ω|2 = 2|∇|ω||2 + 2|ω| |ω|.
(3.7)
Combining (3.7) with Lemma 2.1, we have that |ω| |ω| = |∇ω|2 − |∇|ω||2 + E(ω), ω.
(3.8)
There exists the Kato inequality [6,13]: |∇ω|2 ≥
n−1 |∇|ω||2 . n−2
(3.9)
Combining (3.8) with (3.9), we have that |ω| |ω| ≥
1 |∇|ω||2 + E(ω), ω. n−2
(3.10)
Set h = |ω|. Let η ∈ C01 (M ). Then div(η 2 h∇h) = η 2 h h + ∇(η 2 h), ∇h = η 2 h h + η 2 |∇h|2 + 2ηh∇η, ∇h. By divergence theorem, we obtain that
M
η 2 |∇h|2 + 2
η 2 h h + M
ηh∇η, ∇h = 0.
(3.11)
M
For n = 3, we get an estimate for the term E(ω), ω: E(ω), ω = 2
1 + (λ1 + λ2 + λ3 )λi − λ2i − λi λj (aij )2
i=j
=
i=j
=
2 + (λ1 + λ2 + λ3 )(λi + λj ) − λ2i + λ2j − 2λi λj (aij )2
2 + (λ1 + · · · + λi + · · · + λj + · · · + λ3 )(λi + λj ) (aij )2
i=j
1 2 2 2
4 + (3H) − (λ1 + · · · + λi + · · · + λj + · · · + λ3 ) − (λi + λj ) (aij )2 = 2 i=j 1 2 2 2 2
4 + (3H) − (λ1 + · · · + λi + · · · + λj + · · · + λ3 ) − 2(λi + λj ) (aij )2 ≥ 2 i=j |A|2 2 9H 2 3|Φ|2 − |A|2 h2 = 2 − + h . ≥ 2+ 2 2 2
(3.12)
Combining (3.10) with (3.12), we get that |A|2 2 3|Φ|2 + h . h h ≥ |∇h| + 2 − 2 2
2
(3.13)
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By (3.11) and (3.13), we obtain that −2
ηh∇η, ∇h − 2
M
−
|A| 2
2+
M
2
η 2 |∇h|2
3 2
η 2 h2 +
M
|Φ|2 η 2 h2 ≥ 0.
(3.14)
M
Note that |A|2 ≥ 3H 2 and
1 η |∇h| + a1 2
a1
h |∇η| ≥ −2
2
M
2
ηh∇η, ∇h,
2
M
M
for each positive constant a1 . By (3.14), we have (a1 − 2) −
1 a1
η 2 |∇h|2 + M
3H 2+ 2
2
h2 |∇η|2 M
3 η h + 2
|Φ|2 η 2 h2 ≥ 0.
2 2
M
(3.15)
M
Set
|Φ|
3
φ1 (η) =
13 .
supp(η)
By Hölder inequality and Proposition 2.2, we get
23 13 6 |Φ| · (ηh)
|Φ| η h ≤ 2 2 2
M
3
M
supp(η)
≤ C0 φ21 (η) M
≤
C0 φ21 (η)
|∇(ηh)|2 + 9
(H 2 + 1)η 2 h2 M
1 η |∇h| + 1 + b1 2
(1 + b1 )
M
h |∇η| + 9
2
2
2
M
2 2 H +1 η h , 2
(3.16)
M
for each positive constant b1 . Combining (3.15) with (3.16), we have A1 M
where
η 2 |∇h|2 + B1
H 2 η 2 h2 + C 1
M
η 2 h2 ≤ D1
M
|∇η|2 h2 , M
(3.17)
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3C0 φ21 (η)(1 + b1 ) , 2 3 27C0 φ21 (η) , B1 = − 2 2 27C0 φ21 (η) C1 = 2 − 2
A 1 = 2 − a1 −
and 1 1 3C0 φ21 (η) . 1+ D1 = + a1 2 b1 Obviously, D1 is positive. If the scale-invariant total tracefree curvature Φ L3 (M ) is less than δ(3), then B1 and C1 are positive. Choose a1 and b1 small enough such that A1 is positive. Let BR be a geodesic ball of radius R on M centered at a fixed point. Choose η ∈ C01 (M ) such that ⎧ 0 ≤ η ≤ 1, ⎪ ⎪ ⎨ η ≡ 1 on B R , 2
η ≡ 0 on M \ BR , ⎪ ⎪ ⎩ |∇η| ≤ R2 . Let R → +∞ in (3.17). We obtain that h = 0. That is, ω is trivial. For n ≥ 4, by (3.4), we get an estimate for the term E(ω), ω: 1 n−2 2 |A| |ω|2 2(n − 2) + (nH)2 − 2 2 n = 2(n − 2) − |Φ|2 + |A|2 h2 . 2
E(ω), ω ≥
(3.18)
By (3.10) and (3.18), we obtain that 1 n 2 2 2 |∇h| + 2(n − 2) − |Φ| + |A| h2 . h h ≥ n−2 2
(3.19)
Combining (3.11) with (3.19), we get that −2
ηh∇η, ∇h − M
n−1 n−2
−
η 2 |∇h|2 M
(2(n − 2) + |A|2 )η 2 h2 +
n 2
M
|Φ|2 η 2 h2 ≥ 0. M
Note that |A|2 ≥ nH 2 and
1 η |∇h| + a2 2
a2 M
h |∇η| ≥ −2
2
2
ηh∇η, ∇h,
2
M
for each positive constant a2 . Combining with (3.20), we have
M
(3.20)
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n−1 a2 − n−2
−
η 2 |∇h|2 +
1 a2
M
h2 |∇η|2 M
2
n 2(n − 2) + nH η 2 h2 + 2
M
|Φ|2 η 2 h2 ≥ 0.
(3.21)
M
Set |Φ|n
φ2 (η) =
n1 .
supp(η)
By Hölder inequality and Proposition 2.2, we get
n2 n−2 n 2n |Φ| · (ηh) n−2
|Φ| η h ≤ 2 2 2
M
n
M
supp(η)
≤
|∇(ηh)| + n
C0 φ22 (η)
2
M
2
2 H + 1 η 2 h2
M
2 2 2 1 2 2 2 2 2 2 ≤ C0 φ2 (η) (1 + b2 ) η |∇h| + 1 + h |∇η| + n H +1 η h , b2 M
M
(3.22)
M
for each positive constant b2 . By (3.21) and (3.22), we have A2
η 2 |∇h|2 + B2
M
H 2 η 2 h2 + C 2
M
η 2 h2 ≤ D2
M
|∇η|2 h2 ,
(3.23)
M
where n−1 nC0 φ22 (η)(1 + b2 ) − a2 − , n−2 2 n3 C0 φ22 (η) B2 = n − , 2 C0 n3 φ22 (η) C2 = 2(n − 2) − 2
A2 =
and 1 1 nC0 φ22 (η) . 1+ D2 = + a2 2 b2 Obviously, D2 is positive. If the scale-invariant total tracefree curvature Φ Ln (M ) is less than δ(n), then B2 and C2 are positive. Choose a2 and b2 small enough such that A2 is positive. Let BR be a geodesic ball of radius R on M centered at a fixed point. Choose η ∈ C01 (M ) such that ⎧ 0 ≤ η ≤ 1, ⎪ ⎪ ⎨ η ≡ 1 on B R , 2 η ≡ 0 on M \ BR , ⎪ ⎪ ⎩ |∇η| ≤ R2 . Let R → +∞ in (3.23). We obtain that h = 0. That is, ω = 0. 2
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Acknowledgments The author would like to thank professors Detang Zhou and Hongyu Wang for many invaluable suggestions. The author is also grateful to the referees for their valuable comments. References [1] G. Carron, L2 harmonic forms on non compact manifolds, arXiv:0704.3194v1, 2007. [2] M.P. Cavalcante, H. Mirandola, F. Vitório, L2 -harmonic 1-forms on submanifolds with finite total curvature, J. Geom. Anal. 24 (2014) 205–222. [3] X. Cheng, L2 harmonic forms and stability of hypersurfaces with constant mean curvature, Bol. Soc. Bras. Mat. 31 (2000) 225–239. [4] X. Cheng, L.F. Cheung, D.T. Zhou, The structure of weakly stable constant mean curvature hypersurfaces, Tohoku Math. J. 60 (2008) 101–121. [5] X. Cheng, D.T. Zhou, Manifolds with weighted Poincaré inequality and uniqueness of minimal hypersurfaces, Comm. Anal. Geom. 17 (2009) 139–154. [6] D. Cibotaru, P. Zhu, Refined Kato inequalities for harmonic fields on Kähler manifolds, Pacific J. Math. 256 (2012) 51–66. [7] D. Hoffman, J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math. 27 (1974) 715–727. [8] P. Li, Lecture Notes on Geometric Analysis, Lecture Notes Ser., vol. 6, National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1993. [9] J.H. Michael, L.M. Simon, Sobolev and mean-value inequalities on generalized submanifolds of Rn , Comm. Pure Appl. Math. 26 (1973) 361–379. [10] B. Palmer, Stability of minimal hypersurfaces, Comment. Math. Helv. 66 (1991) 185–188. [11] Y. Shen, X.H. Zhu, On stable complete minimal hypersurfaces in Rn+1 , Amer. J. Math. 120 (1998) 103–116. [12] S. Tanno, L2 harmonic forms and stability of minimal hypersurfaces, J. Math. Soc. Japan 48 (1996) 761–768. [13] X.D. Wang, On the L2 -cohomology of a convex cocompact hyperbolic manifold, Duke Math. J. 115 (2002) 311–327. [14] S.T. Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976) 659–670. [15] P. Zhu, L2 -harmonic forms and stable hypersurfaces in space forms, Arch. Math. 97 (2011) 271–279. [16] P. Zhu, Harmonic two-forms on manifolds with nonnegative isotropic curvature, Ann. Global Anal. Geom. 40 (2011) 427–434. [17] P. Zhu, S.W. Fang, A gap theorem on submanifolds with finite total curvature in spheres, J. Math. Anal. Appl. 413 (2014) 195–201.