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Eigenvalue estimate and gap theorems for submanifolds in the hyperbolic space
Nonlinear Analysis 148 (2017) 126–137
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Eigenvalue estimate and gap theorems for submanifolds in the hyperbolic space✩ Hezi Lin ∗ School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, Fujian 350117, China
article
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Article history: Received 14 April 2016 Accepted 20 September 2016 Communicated by Enzo Mitidieri MSC: 53C21 53C42 Keywords: Complete submanifolds First eigenvalue Gap theorems L2 -harmonic p-forms Ends
abstract Let M n be a complete non-compact submanifold in the hyperbolic space Hn+m . We first give an estimate for the bottom of the spectral of the Laplace operator on M n , under an integral pinching condition on the mean curvature. As a consequence of this estimation, we show some vanishing theorems for L2 harmonic forms in certain degrees if the total mean curvature of M n is less than an explicit constant and its total curvature is less than a suitable related constant. In addition, we obtain some vanishing results under certain pointwise restrictions on the traceless second fundamental form. Moreover, according to the nonexistence of nontrivial L2 harmonic 1-forms, we can further prove some one-end theorems. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Let M n be a submanifold in a Riemannian manifold N n+m . Fix a point x ∈ M and a local orthonormal frame {e1 , . . . , en+m } of N n+m such that {e1 , . . . , en } are tangent fields of M n at x. In the following we shall use the following convention on the ranges of indices: 1 ≤ i, j, k, . . . ≤ n and n + 1 ≤ α ≤ n + m. The second fundamental form A is defined by A(X, Y ) = ⟨∇X Y, eα ⟩eα , ∀X, Y ∈ Tx M, α 2 where ∇ is the Riemannian connection of N n+m . Denote hα ij = ⟨∇ei ej , eα ⟩, then |A| = and the mean curvature vector H is defined by 1 α H= H α eα = hii eα . n α α i ✩
Supported by NSFC Grant No. 11401099.. ∗ Corresponding author. Fax: +86 059122868115. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.na.2016.09.015 0362-546X/© 2016 Elsevier Ltd. All rights reserved.
α
α 2 i,j (hij ) ,
H. Lin / Nonlinear Analysis 148 (2017) 126–137
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The traceless second fundamental form φ is defined by φ(X, Y ) = A(X, Y ) − ⟨X, Y ⟩H,
∀X, Y ∈ Tx M.
It is easy to see that |φ|2 = |A|2 − n|H|2 , which measures how much the immersion deviates from being totally umbilical. We say M |φ|n dv to be the total curvature of M n . The aim of this note is to explore some vanishing theorems for L2 harmonic forms on complete noncompact submanifolds in the hyperbolic space. There are some relations between the geometry and topology of a manifold and the space of L2 harmonic forms. According to the decomposition theorem by Hodge–de Rham–Kodaira, L2 harmonic forms completely represent the (reduced) L2 cohomology of the underlying manifold. The nonexistence of nontrivial L2 harmonic 1-forms on M implies that any codimension one cycle on M must disconnect M (by Corollary 1 in [7]), and also implies the uniqueness of the non-parabolic ends of the underlying manifold (by the important result in [11]). Vanishing theorems for L2 harmonic forms (including harmonic mappings for example) on complete noncompact submanifolds have been investigated extensively by many authors from various points of views. Let H p (L2 (M )) denote the space of all L2 harmonic p-forms on M . It is clear that H p (L2 (M )) is naturally isomorphic to H n−p (L2 (M )) under the Hodge ∗ operator. By using the lower bound estimate of the first eigenvalue due to Cheung–Leung [5], Fu–Xu [8] proved that if an oriented complete submanifold M n (n ≥ 3) in Hn+m with finite total curvature satisfies |H| < 1 − √2n , then dim H 1 (L2 (M )) < ∞ and M must have finitely many ends. Later, Seo [17] showed that if a minimal submanifold M n (n ≥ 5) in Hn+m satisfies |A|n dv < c(n) for some positive constant c(n), then H 1 (L2 (M )) = {0}, which implies that M has M only one end. In [16], Seo also proved that a complete super stable minimal submanifold M n in Hn+m with λ1 (M ) > (2n − 1)(n − 1) must satisfy H 1 (L2 (M )) = {0}. On the other hand, Wang–Xia [18] used the eigenvalue estimate due to Bessa–Montenegro [1] to obtain some vanishing theorems for L2 harmonic 1-forms on a minimal submanifold of Hn+m (n ≥ 5), by assuming that the squared norm or Ln -norm of its second fundamental form is less than an explicit constant. Recently, Cavalcante–Mirandola–Vit´orio [3] proved some finiteness and vanishing theorems for L2 harmonic 1-forms on submanifolds in a nonpositive curved pinching manifold, by adding some conditions on the first eigenvalue and total curvature. In this paper, we mainly consider the relation between the extrinsic geometric restrictions of submanifolds and the existence of L2 harmonic forms. In order to derive the vanishing theorems for L2 harmonic forms, we first need the following lower bound estimate for the first eigenvalue, which will be used to prove Theorem 1.2 and may be of independent interest. Theorem 1.1. Let N be a complete simply connected Riemannian manifold with sectional curvature KN satisfying KN ≤ −a2 for a positive constant a > 0. Let i : M n → N be a complete non-compact submanifold 1 in N satisfying ∥H∥n < nD(n) . Then λ1 (M ) ≥
2 (n − 1)2 a2 1 − nD(n)∥H∥n , 4
where D(n) is the constant in (2.11). As a consequence of this estimation, we get the following vanishing theorems for L2 harmonic forms under certain integral pinching conditions on the mean curvature and the traceless second fundamental form. Furthermore, by the results in [11,12], we show that the underlying submanifold has only one end. This result generalizes Theorem 1.2 of [18].
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Theorem 1.2. Let M n , n ≥ 5, be a complete non-compact submanifold in Hn+m . There exist positive constants c1 and c2 depending only on n such that if |H|n dv < c1 and |φ|n dv < c2 , M 1
M
2
then H (L (M )) = {0} and M has only one end. By assuming a more restrictive integral curvature condition, we can deduce the following nonexistence result for more general L2 harmonic p-forms. Theorem 1.3. Let M n , n ≥ 5, be a complete non-compact submanifold in Hn+m with flat normal bundle. There exists a positive constant C0 such that if |A|n dv < C0 , M
then H (L (M )) = {0} for all integers p ∈ [0, n2 − 1) ∪ ( n2 + 1, n]. p
2
Finally, using the estimate for the bottom of the spectrum due to Cheung–Leung [5], we have the following vanishing theorem under pointwise pinching restrictions on the mean curvature and the traceless second fundamental form. Theorem 1.4. Let M n , n ≥ 5, be a complete non-compact submanifold in Hn+m with flat normal bundle. There exist constants c1 > 0 and c2 > 0 such that if n|H| < c1
and
sup |φ|2 < c2 , x∈M
then H p (L2 (M )) = {0} for all integers p ∈ [0, n2 − 1) ∪ ( n2 + 1, n]. Moreover, M has only one end.
2. Preliminaries Let M n be an n-dimensional complete submanifold in Hn+m , and let △ be the Hodge Laplace–Beltrami operator of M n acting on the space of differential p-forms. Given two p-forms ω and θ, we define a pointwise inner product ⟨ω, θ⟩ =
n
ω(ei1 , . . . , eip )θ(ei1 , . . . , eip ).
i1 ,...,ip =1
Here we omit the normalizing factor 1/p!. The Weitzenb¨ock formula [19] gives △ = ∇2 − W p ,
(2.1)
where ∇2 is the Bochner Laplacian and Wp is an endomorphism depending upon the curvature tensor of M n . Let {θ1 , . . . , θn } be an orthonormal basis dual to {e1 , . . . , en }, then n k ⟨Wp (ω), ω⟩ = θ ∧ iej R(ek , ej )ω, ω j,k=1
for any p-form ω. In particular, if ω is a 1-form, then ⟨W1 (ω), ω⟩ = Ric(ω ♯ , ω ♯ ),
H. Lin / Nonlinear Analysis 148 (2017) 126–137
where ω ♯ is the vector field dual to ω. For any ω ∈ H p (L2 (M )), by (2.1) we obtain n 1 2 2 k △|ω| = |∇ω| + θ ∧ iej R(ek , ej )ω, ω 2 j,k=1 p−1 Rijkl ω iji3 ···ip ωikl3 ···ip . = |∇ω|2 + p Rij ω ii2 ···ip ωij2 ···ip − 2
129
(2.2)
Since ω is closed and co-closed [20], we have the Kato’s inequality [2] |∇ω|2 ≥ (1 + Kp )|∇|ω| |2 , where Kp =
1 n−p
if 1 ≤ p ≤ n/2, and Kp =
1 p
(2.3)
if n/2 ≤ p ≤ n − 1. On the other hand, we have
1 △|ω|2 = |ω|△|ω| + |∇|ω| |2 . 2 Substituting (2.3) and (2.4) into (2.2), it follows that p−1 iji3 ···ip kl 2 ii2 ···ip j Rijkl ω ωi3 ···ip . |ω|△|ω| ≥ Kp |∇|ω| | + p Rij ω ωi2 ···ip − 2
(2.4)
(2.5)
When p = 1, using the Ricci curvature estimate in [10], we get 1 |∇|ω| |2 + Ric(ω ♯ , ω ♯ ) n−1 1 n−1 2 n−1 2 2 ≥ |∇|ω| | + −(n − 1) − |φ| + (n − 1)|H| − (n − 2) |H| |φ| |ω|2 . (2.6) n−1 n n When 2 ≤ p ≤ n − 2, assume further that M has flat normal bundle. By the Gauss equation, we have |ω|△|ω| ≥
α α α Rijkl = −(δik δjl − δil δjk ) + hα ik hjl − hil hjk ,
and α α Rij = −(n − 1)δij + nH α hα ij − hik hjk .
Thus, Rij ω ii2 ···ip ωij2 ···ip −
p−1 Rijkl ω iji3 ···ip ωikl3 ···ip = F1 (ω) + F2 (ω), 2
where F1 (ω) = −(n − 1)
δij ω ii2 ···ip ωij2 ···ip +
i,j,i2 ,...,ip
p−1 (δik δjl − δil δjk )ω iji3 ···ip ωikl3 ···ip 2
= −(n − p)|ω|2 ,
(2.7)
and p−1 α α α iji3 ···ip kl (hik hjl − hα ωi3 ···ip il hjk )ω 2 α ii2 ···ip j α iji3 ···ip kl − hα ωi2 ···ip − (p − 1)hα ωi3 ···ip . ik hjk ω ik hjl ω
α α ii2 ···ip j F2 (ω) = (nH α hα ωi2 ···ip − ij − hik hjk )ω ii2 ···ip j = nH α hα ωi2 ···ip ij ω
From the computation in [14,13], it follows that 1 2 F2 (ω) ≥ n |H|2 − max{p, n − p}|A|2 . 2 Substituting (2.7), (2.8) into (2.5), we conclude that 1 2 |ω|△|ω| ≥ Kp |∇|ω| |2 − p(n − p)|ω|2 + n |H|2 − max{p, n − p}|A|2 |ω|2 2 1 n 2 2 ≥ Kp |∇|ω| | − p(n − p)|ω| − (n − 1)|φ|2 |ω|2 + |H|2 |ω|2 . 2 2
(2.8)
(2.9)
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Recall that an end E of a complete manifold M is non-parabolic if E admits a positive Green’s function with Neumann boundary condition. To characterize the non-parabolicity of ends, we recall the following result due to Li and Wang: Lemma 2.1 ([12]). Let E be an end of a complete Riemannian manifold. Suppose for some µ ≥ 1, E satisfies a Sobolev type inequality of the form µ1 2µ ≤C f dv |∇f |2 dv (2.10) E
E
for all compactly supported function f ∈ W be non-parabolic.
1,2
(E) defined on E, then E must either have finite volume or
If the ambient manifold has nonpositive sectional curvature, then the L1 -Sobolev inequality due to Hoffman and Spruck [9]
n
n−1 n
g n−1 dv
≤ D(n)
(|∇g| + n|H|g) dv
(2.11)
M
M
−1/n
with σn = volume of the unit ball holds for any g ∈ C01 (M ), where D(n) = 2n (1 + n)(n+1)/n (n − 1)−1 σn n in R . In order to estimate the number of ends of complete manifolds, we also need the following result due to Li and Tam: 0 (M ) be the space of bounded Lemma 2.2 ([11]). Let M be a complete Riemannian manifold and let HD harmonic functions with finite energy. Then the number of non-parabolic ends of M is bounded from above 0 (M ) ≤ dim H 1 (L2 (M )) + 1. by dim HD
3. Estimate of the first eigenvalue Let M be a complete noncompact Riemannian manifold and let Ω be a compact domain in M . Denote by λ1 (Ω ) > 0 the first eigenvalue of the Dirichlet boundary value problem △f + λf = 0 in Ω f =0 on ∂Ω where △ is the Laplace operator on M . If Ω1 ⊂ Ω2 are bounded domains, then λ1 (Ω1 ) ≥ λ1 (Ω2 ). Thus, the first eigenvalue λ1 (M ) of a complete noncompact manifold M can be defined by λ1 (M ) = inf λ1 (Ω ), Ω
where the infimum is taken over all compact domains in M . In this section, we attempt to estimate the first eigenvalue of a complete noncompact submanifold in terms of its mean curvature. Let us recall the previous results in this direction. In [5], Cheung and Leung showed Theorem A ([5]). Let M n be a complete non-compact submanifold in Hn+m . If n|H| ≤ α for some constant 0 ≤ α < n − 1, then λ1 (M ) ≥
(n − 1 − α)2 . 4
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Later, Bessa and Montenegro [1] extended this estimate to complete noncompact submanifolds in a complete simply connected Riemannian manifold with sectional curvature bounded above by a negative constant. More precisely, they proved Theorem B ([1]). Let N n+m be a complete simply connected Riemannian manifold with sectional curvature KN satisfying KN ≤ −a2 for a positive constant a > 0. Let M n be a complete noncompact submanifold in N with n|H| ≤ b < (n − 1)a. Then λ1 (M ) ≥
[(n − 1)a − b]2 . 4
When the ambient space N n+m is of nonpositive curvature, using the L1 -Sobolev inequality (2.11), we establish an estimate of the first eigenvalue in terms of M |H|n dv and prove Theorem 1.1. Proof of Theorem 1.1. Let ρ(x) be the geodesic distance on N from a fixed point x0 ∈ N \ M to x. From the proof of [1], we have △ρ ◦ i ≥ (n − 1)a − n|H|. Denote r = ρ ◦ i. For any f ∈ C0∞ (M ), we have div(f 2 ∇r) = ⟨∇f 2 , ∇r⟩ + f 2 △r ≥ −|∇f 2 | + (n − 1)af 2 − n|H|f 2 . Integrating over M , we have 0≥−
|∇f 2 |dv + (n − 1)a
M
f 2 dv − n
M
|H|f 2 dv.
(3.1)
M
On the other hand, using the Sobolev inequality (2.11), we infer n1 n−1 2n n f n−1 dv |H|n dv n |H|f 2 dv ≤ n M M M ≤ nD(n)∥H∥n |∇f 2 | + n|H|f 2 dv. M
Hence, n
|H|f 2 dv ≤
M
nD(n)∥H∥n 1 − nD(n)∥H∥n
|∇f 2 |dv.
M
Substituting into (3.1) yields 1 |∇f 2 |dv + (n − 1)a f 2 dv 1 − nD(n)∥H∥n M M 2 =− |f | |∇f |dv + (n − 1)a f 2 dv. 1 − nD(n)∥H∥n M M
0≥−
(3.2)
Using the Cauchy–Schwarz inequality, we have 1 −2|f | |∇f | ≥ − |∇f |2 − ϵf 2 ϵ for any ϵ > 0. Hence it follows from (3.2) that 1 ϵ 0≥− |∇f |2 dv + (n − 1)a − f 2 dv, 1 − nD(n)∥H∥n M ϵ 1 − nD(n)∥H∥n M which implies that M
|∇f |2 dv ≥ ϵ 1 − nD(n)∥H∥n (n − 1)a −
ϵ 1 − nD(n)∥H∥n
M
f 2 dv.
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Let Q(ϵ) = ϵ 1 − nD(n)∥H∥n (n − 1)a −
ϵ . 1 − nD(n)∥H∥n
Then it is easy to compute that max Q(ϵ) = ϵ>0
2 (n − 1)2 a2 1 − nD(n)∥H∥n . 4
Therefore,
2 |∇f |2 dv (n − 1)2 a2 ≥ 1 − nD(n)∥H∥ , n 4 f 2 dv M
M
λ1 (M ) = inf∞ f ∈C0
which completes the proof of Theorem 1.1.
Remark 3.1. In particular, if M is a complete non-compact minimal submanifold in N n+m , we have 2 2 a . λ1 (M ) ≥ (n−1) 4 4. Vanishing theorems under global conditions Let f ∈ C01 (M ). Substituting g = f |f |
2n n−2
n−1 n dv
M
2(n−1) n−2
, n ≥ 3 in (2.11), and using the H¨older inequality, we have
2(n − 1)D(n) ≤ n−2 ≤
|f |
n n−2
|∇f |dv + nD(n)
M
|H|f
2(n−1) n−2
dv
M
12 12 2n 2(n − 1)D(n) |f | n−2 dv |∇f |2 dv n−2 M M 12 21 2n + nD(n) |f | n−2 dv , |H|2 f 2 dv M
M
2
which implies the following L -Sobolev inequality
2n
n−2 n
|f | n−2 dv
≤ c(n)
M
(|∇f |2 + |H|2 f 2 )dv,
∀f ∈ C01 (M ),
(4.1)
M
where c(n) = 2n2 D(n)2 . Using the above L2 -Sobolev inequality, we deduce the following general global vanishing result, from which we immediately have Theorem 1.2. n Proposition 4.1. , n ≥ 5,be a complete non-compact submanifold in Hn+m with ∥H∥n ≤ α, where Let M p(n−p) 1 2 0 < α < nD(n) 1 − n−1 . Assume that 1+Kp
M
|φ|n dv
n2
<
2 4p(n − p) 1 + Kp − . (n − 1)c(n) (n − 1)2 (1 − nαD(n))2
(i) When 2 ≤ p ≤ n − 2, assume further that M has flat normal bundle. Then H p (L2 (M )) = {0}. (ii) When p = 1, then H 1 (L2 (M )) = {0} and M has only one end.
(4.2)
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133
Proof. (i) Given ω ∈ H p (L2 (M )) for 2 ≤ p ≤ n − 2. Let η ∈ C0∞ (M ). Multiplying (2.9) by η 2 and integrating over M , we obtain n−1 |φ|2 η 2 |ω|2 dv η 2 |ω|△|ω|dv ≥ Kp η 2 |∇|ω| |2 dv − p(n − p) η 2 |ω|2 dv − 2 M M M M n + |H|2 η 2 |ω|2 dv. (4.3) 2 M For the term at the left hand side, integrating by parts over M , we have η 2 |ω|△|ω|dv = −2 η|ω|⟨∇η, ∇|ω|⟩dv − η 2 |∇|ω| |2 dv. M
M
(4.4)
M
Using the H¨ older inequality together with (4.1), it follows that
2 2
2
n
|φ| η |ω| dv ≤
n2
|φ| dv
M
η|ω|
2n n−2
n−2 n dv
M
supp(η)
n
≤ c(n)
n2
|φ| dv 2
|∇(η|ω|)|2 + |H|2 η 2 |ω|2 dv
M
supp(η)
2
2
2
2 2
2
(η |∇|ω| | + |ω| |∇η| + |H| η |ω| )dv + 2S(η)
= S(η) M
η|ω|⟨∇η, ∇|ω|⟩dv, M
(4.5) where S(η) = c(n)( that
2
supp(η)
|φ|n dv) n . By the variational characterization of λ1 (M ), we have from Theorem 1.1
4 η |ω| dv ≤ |∇(η|ω|)|2 dv (n − 1)2 (1 − nD(n)∥H∥n )2 M M 4 (η 2 |∇|ω| |2 + |ω|2 |∇η|2 )dv = (n − 1)2 (1 − nD(n)∥H∥n )2 M 8 + η|ω|⟨∇η, ∇|ω|⟩dv. (n − 1)2 (1 − nD(n)∥H∥n )2 M 2
2
Combining with (4.3)–(4.5), and using the Cauchy–Schwarz inequality, we infer (1 + Kp − E) η 2 |∇|ω| |2 dv M ≤ 2(E − 1) η|ω|⟨∇η, ∇|ω|⟩dv + E |ω|2 |∇η|2 dv − F |H|2 η 2 |ω|2 dv M M M |E − 1| 2 2 2 2 |ω| |∇η| dv − F |H|2 η 2 |ω|2 dv ≤ |E − 1|ϵ η |∇|ω| | dv + E + ϵ M M M for all ϵ > 0, where n−1 4p(n − p) S(η) + , 2 (n − 1)2 (1 − nD(n)∥H∥n )2 n n−1 F = − S(η). 2 2
E=
Hence, |E − 1| |ω|2 |∇η|2 dv − F |H|2 η 2 |ω|2 dv. (4.6) η 2 |∇|ω| |2 dv ≤ E + ϵ M M M
(1 + Kp − E − |E − 1|ϵ)
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Fix a point x0 ∈ M and let ρ(x) be the geodesic distance on M from x0 to x. Let us choose η ∈ C0∞ (M ) satisfying 1 if ρ(x) ≤ r, η(x) = 0 if 2r ≤ ρ(x) and |∇η|(x) ≤
2 r
if r ≤ ρ(x) ≤ 2r
for r > 0. The hypothesis (4.2) implies that 1 + Kp − E > 0 and F ≥ 0. Choose ϵ small enough such that 1 + Kp − E − |E − 1|ϵ > 0. Then it follows from (4.6) that (1 + Kp − E − |E − 1|ϵ)
2
|∇|ω| | dv ≤ (1 + Kp − E − |E − 1|ϵ)
Bx0 (r)
≤ Since
M
η 2 |∇|ω| |2 dv
M
|E − 1| 4 E + r2 ϵ
|ω|2 dv − F
M
|H|2 |ω|2 dv.
Bx0 (r)
2
|ω| dv < ∞, letting r → ∞ gives ∇|ω| = 0
and |H| |ω| = 0.
Hence, |ω| = constant. If |ω| is not identically zero, then H = 0, which implies that the volume of M is infinite. Thus we have a contradiction, since M |ω|2 dv < ∞. Therefore, H p (L2 (M )) = {0}. (ii) Using the Cauchy–Schwarz inequality, we deduce from (2.6) that (n − 1)(n − 2)2 2 2 n − 1 1 |∇|ω| |2 − (n − 1)|ω|2 + (n − 1) − H |ω| − + ϵ |φ|2 |ω|2 |ω|△|ω| ≥ n−1 4nϵ n (4.7) for any ϵ > 0. Let ϵ =
(n−1)(n−2) , 2n
|ω|△|ω| ≥
then
1 1 n |∇|ω| |2 − (n − 1)|ω|2 − (n − 1)|φ|2 |ω|2 + |H|2 |ω|2 . n−1 2 2
Using the same arguments as in the above case, we conclude that H 1 (L2 (M )) = {0}. From the discussion of Fu–Xu in [8], the assumption M |H|n dv ≤ α < ∞ implies that each end of M is non-parabolic. Hence according to Lemma 2.2, M must have only one end. p(n−p) 2 ± 1, we have 1 − n−1 1+Kp < 0. p(n−p) 2 is odd, then for p = n±1 2 , we also have 1 − n−1 1+Kp < 0. Hence for these
Remark 4.1. When the dimension of M n is even, then for p = When the dimension of M n
n 2
or
n 2
integers p, the above vanishing result for L2 harmonic p-forms does not hold. We are now in the position to prove Theorem 1.3. Proof of Theorem 1.3. Let ω ∈ H p (L2 (M )). When 1 ≤ p < p(n−p) 2 that 1 − n−1 1+Kp > 0. Set C0 =
n 2
− 1 and
n 2
+ 1 < p ≤ n − 1, it is easy to check
2 4p(n − p) min 1 + Kp − 2 2 (n − 1)c(n) p (n − 1) (1 − nαD(n))
H. Lin / Nonlinear Analysis 148 (2017) 126–137
with 0 < α <
1 nD(n)
1−
135
, then the condition M |A|n dv < C0 implies that 2 1 p(n − p) 1− , ∥H∥n ≤ n−n/2 |A|n dv < nD(n) n−1 1 + Kp M 2 n−1
p(n−p) 1+Kp
and
|φ|n dv ≤
M
|A|n dv <
M
4p(n − p) 2 1 + Kp − . (n − 1)c(n) (n − 1)2 (1 − nαD(n))2
Hence, it follows from Proposition 4.1 that ω = 0. When p = 0 or n, for any f ∈ H 0 (L2 (M )), applying a theorem of Yau in [20], we conclude that f is constant. On the other hand, by the discussion in [15], the fact ∥H∥n < ∞ implies that the volume of M is infinite. Since M f 2 dv < +∞ and f is constant, we conclude that f = 0, which means H 0 (L2 (M )) = {0}. By Poincar´e duality, we have H n (L2 (M )) = {0}. The proof of the theorem is completed.
5. Vanishing theorems under pointwise conditions In this section, we obtain some vanishing results under pointwise curvature pinching assumptions. n n+m Proposition 5.1. Let with n|H| ≤ α, where M , n ≥ 5, be a complete non-compact submanifold in H p(n−p) 0 < α < n − 1 − 2 1+Kp for 1 ≤ p ≤ n − 1.
(i) When 2 ≤ p ≤ n − 2, assume that M has flat normal bundle, and that sup |φ|2 < x∈M
(n − 1 − α)2 (1 + Kp ) + 2n inf |H|2 − 4p(n − p) . 2(n − 1)
(5.1)
Then H p (L2 (M )) = {0}. (ii) When p = 1, assume that sup |φ|2 < x∈M
n(n − 1 − α)2 − 4(n − 1)2 . n(n − 1)
Then H 1 (L2 (M )) = {0} and M has only one end. Proof. (i) Given ω ∈ H p (L2 (M )) for 2 ≤ p ≤ n − 2. Combining the relations (4.3) and (4.4), and using Theorem A, we have 2 2 2 2 Kp η |∇|ω| | dv ≤ −2 η|ω|⟨∇η, ∇|ω|⟩dv − η |∇|ω| | dv + E |∇(η|ω|)|2 dv, M
M
M
M
where 4p(n − p) − 2n inf |H|2 + 2(n − 1) sup |φ|2 x∈M
E=
x∈M
(n − 1 − α)2
Using the Cauchy–Schwarz inequality, a direct computation gives A(ϵ) η 2 |∇|ω| |2 dv ≤ B(ϵ) |ω|2 |∇η|2 dv, M
M
.
(5.2)
H. Lin / Nonlinear Analysis 148 (2017) 126–137
136
where A(ϵ) = Kp + 1 − E − ϵ|1 − E|, 1 B(ϵ) = E + |1 − E| ϵ for all ϵ > 0. The assumption (5.1) is equivalent to Kp + 1 − E > 0, which implies that there exists a sufficiently small ϵ > 0 such that A(ϵ) = Kp + 1 − E − ϵ|1 − E| > 0. Hence it follows from (5.2) that 4B(ϵ) |ω|2 dv. A(ϵ) |∇|ω| |2 dv ≤ A(ϵ) η 2 |∇|ω| |2 dv ≤ 2 r M Bx0 (r) M 2 Letting r → ∞, and noting that M |ω| dv < ∞, we conclude that |ω| = constant. Since the mean curvature of M is bounded from above, according to Corollary 2.1 of [4], the volume of M is infinite. Therefore, we have ω = 0. (ii) Let ϵ =
(n−2)2 4n
in (4.7), then we have |ω|△|ω| ≥
1 n |∇|ω| |2 − (n − 1)|ω|2 − |φ|2 |ω|2 . n−1 4
Arguing as in the proof of case (i), we conclude that H 1 (L2 (M )) = {0}. Since λ1 (M ) > 0 by Theorem A, the Sobolev type inequality (2.10) with µ = 1 holds on M . On the other hand, by Corollary 2.1 of [4], the bounded mean curvature implies that each end of M has infinite volume. According to Lemma 2.1, each end of M is non-parabolic. Therefore, using Lemma 2.2, M has only one end. We now prove Theorem 1.4. Proof of Theorem 1.4. Since the condition n|H| < c1 < ∞ implies that the volume of M is infinite, the cases of p = 0 or n follow the line of the proof of Theorem 1.3. The other cases of p are already proved in Proposition 5.1. Remark 5.1. In particular, if the dimension of M n is even and the second fundamental form of M n identically vanishes, that is, M n is the totally geodesic submanifold Hn in Hn+m , by the main theorem in [6] there are no nontrivial L2 harmonic p-forms of degree p ̸= n2 on Hn . Acknowledgment The author would like to thank the referee for the helpful comments and detailed corrections. References [1] G.P. Bessa, J.F. Montenegro, Eigenvalue estimates for submanifolds with locally bounded mean curvature, Ann. Global Anal. Geom. 24 (2003) 279–290. [2] D.M.J. Calderbank, P. Gauduchon, M. Herzlich, Refined Kato inequalities and conformal weights in Riemannian geometry, J. Funct. Anal. 173 (1) (2000) 214–255. [3] M.P. Cavalcante, H. Mirandola, F. Vit´ orio, L2 -harmonic 1-forms on submanifolds with finite total curvature, J. Geom. Anal. 24 (2014) 205–222. [4] X. Cheng, L.F. Cheung, D.T. Zhou, The structure of weakly stable constant mean curvature hypersurfaces, Tohoku Math. J. 60 (1) (2008) 101–121. [5] L.F. Cheung, P.F. Leung, Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space, Math. Z. 236 (2001) 525–530. [6] J. Dodziuk, L2 harmonic forms on rotationally symmetric Riemannian manifolds, Proc. Amer. Math. Soc. 77 (3) (1979) 395–400.
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[7] J. Dodziuk, L2 harmonic forms on complete manifolds, Ann. Math. Stud. 102 (1982) 291–302. [8] H.P. Fu, H.W. Xu, Total curvature and L2 harmonic 1-forms on complete submanifolds in space forms, Geom. Dedicata 144 (2010) 129–140. [9] D. Hoffman, J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math. 27 (1974) 715–727. [10] P.F. Leung, An estimate on the Ricci curvature of a submanifold and some applications, Proc. Amer. Math. Soc. 114 (1992) 1051–1061. [11] P. Li, L.F. Tam, Harmonic functions and the structure of complete manifolds, J. Differential Geom. 35 (2) (1992) 359–383. [12] P. Li, J.P. Wang, Minimal hypersurfaces with finite index, Math. Res. Lett. 9 (1) (2002) 95–103. [13] H.Z. Lin, On the structure of submanifolds in Euclidean space with flat normal bundle, Results Math. 68 (2015) 313–329. [14] H.Z. Lin, Vanishing theorems for L2 -harmonic forms on complete submanifolds in Euclidean space, J. Math. Anal. Appl. 425 (2) (2015) 774–787. [15] S. Pigola, M. Rigoli, A.G. Setti, Some characterizations of space-forms, Trans. Amer. Math. Soc. 359 (4) (2007) 1817–1828. [16] K. Seo, L2 harmonic 1-forms on minimal submanifolds in hyperbolic space, J. Math. Anal. Appl. 371 (2) (2010) 546–551. [17] K. Seo, Rigidity of minimal submanifolds in hyperbolic space, Arch. Math. (Basel) 94 (2) (2010) 173–181. [18] Q.L. Wang, C.Y. Xia, Complete submanifolds of manifolds of negative curvature, Ann. Global Anal. Geom. 39 (2011) 83–97. [19] H.H. Wu, The Bochner technique in differential geometry, Math. Rep. 3 (2) (1988) i–xii. and 289–538. [20] S.T. Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976) 659–670.
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Nonlinear Analysis www.elsevier.com/locate/na
Eigenvalue estimate and gap theorems for submanifolds in the hyperbolic space✩ Hezi Lin ∗ School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, Fujian 350117, China
article
info
Article history: Received 14 April 2016 Accepted 20 September 2016 Communicated by Enzo Mitidieri MSC: 53C21 53C42 Keywords: Complete submanifolds First eigenvalue Gap theorems L2 -harmonic p-forms Ends
abstract Let M n be a complete non-compact submanifold in the hyperbolic space Hn+m . We first give an estimate for the bottom of the spectral of the Laplace operator on M n , under an integral pinching condition on the mean curvature. As a consequence of this estimation, we show some vanishing theorems for L2 harmonic forms in certain degrees if the total mean curvature of M n is less than an explicit constant and its total curvature is less than a suitable related constant. In addition, we obtain some vanishing results under certain pointwise restrictions on the traceless second fundamental form. Moreover, according to the nonexistence of nontrivial L2 harmonic 1-forms, we can further prove some one-end theorems. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Let M n be a submanifold in a Riemannian manifold N n+m . Fix a point x ∈ M and a local orthonormal frame {e1 , . . . , en+m } of N n+m such that {e1 , . . . , en } are tangent fields of M n at x. In the following we shall use the following convention on the ranges of indices: 1 ≤ i, j, k, . . . ≤ n and n + 1 ≤ α ≤ n + m. The second fundamental form A is defined by A(X, Y ) = ⟨∇X Y, eα ⟩eα , ∀X, Y ∈ Tx M, α 2 where ∇ is the Riemannian connection of N n+m . Denote hα ij = ⟨∇ei ej , eα ⟩, then |A| = and the mean curvature vector H is defined by 1 α H= H α eα = hii eα . n α α i ✩
Supported by NSFC Grant No. 11401099.. ∗ Corresponding author. Fax: +86 059122868115. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.na.2016.09.015 0362-546X/© 2016 Elsevier Ltd. All rights reserved.
α
α 2 i,j (hij ) ,
H. Lin / Nonlinear Analysis 148 (2017) 126–137
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The traceless second fundamental form φ is defined by φ(X, Y ) = A(X, Y ) − ⟨X, Y ⟩H,
∀X, Y ∈ Tx M.
It is easy to see that |φ|2 = |A|2 − n|H|2 , which measures how much the immersion deviates from being totally umbilical. We say M |φ|n dv to be the total curvature of M n . The aim of this note is to explore some vanishing theorems for L2 harmonic forms on complete noncompact submanifolds in the hyperbolic space. There are some relations between the geometry and topology of a manifold and the space of L2 harmonic forms. According to the decomposition theorem by Hodge–de Rham–Kodaira, L2 harmonic forms completely represent the (reduced) L2 cohomology of the underlying manifold. The nonexistence of nontrivial L2 harmonic 1-forms on M implies that any codimension one cycle on M must disconnect M (by Corollary 1 in [7]), and also implies the uniqueness of the non-parabolic ends of the underlying manifold (by the important result in [11]). Vanishing theorems for L2 harmonic forms (including harmonic mappings for example) on complete noncompact submanifolds have been investigated extensively by many authors from various points of views. Let H p (L2 (M )) denote the space of all L2 harmonic p-forms on M . It is clear that H p (L2 (M )) is naturally isomorphic to H n−p (L2 (M )) under the Hodge ∗ operator. By using the lower bound estimate of the first eigenvalue due to Cheung–Leung [5], Fu–Xu [8] proved that if an oriented complete submanifold M n (n ≥ 3) in Hn+m with finite total curvature satisfies |H| < 1 − √2n , then dim H 1 (L2 (M )) < ∞ and M must have finitely many ends. Later, Seo [17] showed that if a minimal submanifold M n (n ≥ 5) in Hn+m satisfies |A|n dv < c(n) for some positive constant c(n), then H 1 (L2 (M )) = {0}, which implies that M has M only one end. In [16], Seo also proved that a complete super stable minimal submanifold M n in Hn+m with λ1 (M ) > (2n − 1)(n − 1) must satisfy H 1 (L2 (M )) = {0}. On the other hand, Wang–Xia [18] used the eigenvalue estimate due to Bessa–Montenegro [1] to obtain some vanishing theorems for L2 harmonic 1-forms on a minimal submanifold of Hn+m (n ≥ 5), by assuming that the squared norm or Ln -norm of its second fundamental form is less than an explicit constant. Recently, Cavalcante–Mirandola–Vit´orio [3] proved some finiteness and vanishing theorems for L2 harmonic 1-forms on submanifolds in a nonpositive curved pinching manifold, by adding some conditions on the first eigenvalue and total curvature. In this paper, we mainly consider the relation between the extrinsic geometric restrictions of submanifolds and the existence of L2 harmonic forms. In order to derive the vanishing theorems for L2 harmonic forms, we first need the following lower bound estimate for the first eigenvalue, which will be used to prove Theorem 1.2 and may be of independent interest. Theorem 1.1. Let N be a complete simply connected Riemannian manifold with sectional curvature KN satisfying KN ≤ −a2 for a positive constant a > 0. Let i : M n → N be a complete non-compact submanifold 1 in N satisfying ∥H∥n < nD(n) . Then λ1 (M ) ≥
2 (n − 1)2 a2 1 − nD(n)∥H∥n , 4
where D(n) is the constant in (2.11). As a consequence of this estimation, we get the following vanishing theorems for L2 harmonic forms under certain integral pinching conditions on the mean curvature and the traceless second fundamental form. Furthermore, by the results in [11,12], we show that the underlying submanifold has only one end. This result generalizes Theorem 1.2 of [18].
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Theorem 1.2. Let M n , n ≥ 5, be a complete non-compact submanifold in Hn+m . There exist positive constants c1 and c2 depending only on n such that if |H|n dv < c1 and |φ|n dv < c2 , M 1
M
2
then H (L (M )) = {0} and M has only one end. By assuming a more restrictive integral curvature condition, we can deduce the following nonexistence result for more general L2 harmonic p-forms. Theorem 1.3. Let M n , n ≥ 5, be a complete non-compact submanifold in Hn+m with flat normal bundle. There exists a positive constant C0 such that if |A|n dv < C0 , M
then H (L (M )) = {0} for all integers p ∈ [0, n2 − 1) ∪ ( n2 + 1, n]. p
2
Finally, using the estimate for the bottom of the spectrum due to Cheung–Leung [5], we have the following vanishing theorem under pointwise pinching restrictions on the mean curvature and the traceless second fundamental form. Theorem 1.4. Let M n , n ≥ 5, be a complete non-compact submanifold in Hn+m with flat normal bundle. There exist constants c1 > 0 and c2 > 0 such that if n|H| < c1
and
sup |φ|2 < c2 , x∈M
then H p (L2 (M )) = {0} for all integers p ∈ [0, n2 − 1) ∪ ( n2 + 1, n]. Moreover, M has only one end.
2. Preliminaries Let M n be an n-dimensional complete submanifold in Hn+m , and let △ be the Hodge Laplace–Beltrami operator of M n acting on the space of differential p-forms. Given two p-forms ω and θ, we define a pointwise inner product ⟨ω, θ⟩ =
n
ω(ei1 , . . . , eip )θ(ei1 , . . . , eip ).
i1 ,...,ip =1
Here we omit the normalizing factor 1/p!. The Weitzenb¨ock formula [19] gives △ = ∇2 − W p ,
(2.1)
where ∇2 is the Bochner Laplacian and Wp is an endomorphism depending upon the curvature tensor of M n . Let {θ1 , . . . , θn } be an orthonormal basis dual to {e1 , . . . , en }, then n k ⟨Wp (ω), ω⟩ = θ ∧ iej R(ek , ej )ω, ω j,k=1
for any p-form ω. In particular, if ω is a 1-form, then ⟨W1 (ω), ω⟩ = Ric(ω ♯ , ω ♯ ),
H. Lin / Nonlinear Analysis 148 (2017) 126–137
where ω ♯ is the vector field dual to ω. For any ω ∈ H p (L2 (M )), by (2.1) we obtain n 1 2 2 k △|ω| = |∇ω| + θ ∧ iej R(ek , ej )ω, ω 2 j,k=1 p−1 Rijkl ω iji3 ···ip ωikl3 ···ip . = |∇ω|2 + p Rij ω ii2 ···ip ωij2 ···ip − 2
129
(2.2)
Since ω is closed and co-closed [20], we have the Kato’s inequality [2] |∇ω|2 ≥ (1 + Kp )|∇|ω| |2 , where Kp =
1 n−p
if 1 ≤ p ≤ n/2, and Kp =
1 p
(2.3)
if n/2 ≤ p ≤ n − 1. On the other hand, we have
1 △|ω|2 = |ω|△|ω| + |∇|ω| |2 . 2 Substituting (2.3) and (2.4) into (2.2), it follows that p−1 iji3 ···ip kl 2 ii2 ···ip j Rijkl ω ωi3 ···ip . |ω|△|ω| ≥ Kp |∇|ω| | + p Rij ω ωi2 ···ip − 2
(2.4)
(2.5)
When p = 1, using the Ricci curvature estimate in [10], we get 1 |∇|ω| |2 + Ric(ω ♯ , ω ♯ ) n−1 1 n−1 2 n−1 2 2 ≥ |∇|ω| | + −(n − 1) − |φ| + (n − 1)|H| − (n − 2) |H| |φ| |ω|2 . (2.6) n−1 n n When 2 ≤ p ≤ n − 2, assume further that M has flat normal bundle. By the Gauss equation, we have |ω|△|ω| ≥
α α α Rijkl = −(δik δjl − δil δjk ) + hα ik hjl − hil hjk ,
and α α Rij = −(n − 1)δij + nH α hα ij − hik hjk .
Thus, Rij ω ii2 ···ip ωij2 ···ip −
p−1 Rijkl ω iji3 ···ip ωikl3 ···ip = F1 (ω) + F2 (ω), 2
where F1 (ω) = −(n − 1)
δij ω ii2 ···ip ωij2 ···ip +
i,j,i2 ,...,ip
p−1 (δik δjl − δil δjk )ω iji3 ···ip ωikl3 ···ip 2
= −(n − p)|ω|2 ,
(2.7)
and p−1 α α α iji3 ···ip kl (hik hjl − hα ωi3 ···ip il hjk )ω 2 α ii2 ···ip j α iji3 ···ip kl − hα ωi2 ···ip − (p − 1)hα ωi3 ···ip . ik hjk ω ik hjl ω
α α ii2 ···ip j F2 (ω) = (nH α hα ωi2 ···ip − ij − hik hjk )ω ii2 ···ip j = nH α hα ωi2 ···ip ij ω
From the computation in [14,13], it follows that 1 2 F2 (ω) ≥ n |H|2 − max{p, n − p}|A|2 . 2 Substituting (2.7), (2.8) into (2.5), we conclude that 1 2 |ω|△|ω| ≥ Kp |∇|ω| |2 − p(n − p)|ω|2 + n |H|2 − max{p, n − p}|A|2 |ω|2 2 1 n 2 2 ≥ Kp |∇|ω| | − p(n − p)|ω| − (n − 1)|φ|2 |ω|2 + |H|2 |ω|2 . 2 2
(2.8)
(2.9)
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Recall that an end E of a complete manifold M is non-parabolic if E admits a positive Green’s function with Neumann boundary condition. To characterize the non-parabolicity of ends, we recall the following result due to Li and Wang: Lemma 2.1 ([12]). Let E be an end of a complete Riemannian manifold. Suppose for some µ ≥ 1, E satisfies a Sobolev type inequality of the form µ1 2µ ≤C f dv |∇f |2 dv (2.10) E
E
for all compactly supported function f ∈ W be non-parabolic.
1,2
(E) defined on E, then E must either have finite volume or
If the ambient manifold has nonpositive sectional curvature, then the L1 -Sobolev inequality due to Hoffman and Spruck [9]
n
n−1 n
g n−1 dv
≤ D(n)
(|∇g| + n|H|g) dv
(2.11)
M
M
−1/n
with σn = volume of the unit ball holds for any g ∈ C01 (M ), where D(n) = 2n (1 + n)(n+1)/n (n − 1)−1 σn n in R . In order to estimate the number of ends of complete manifolds, we also need the following result due to Li and Tam: 0 (M ) be the space of bounded Lemma 2.2 ([11]). Let M be a complete Riemannian manifold and let HD harmonic functions with finite energy. Then the number of non-parabolic ends of M is bounded from above 0 (M ) ≤ dim H 1 (L2 (M )) + 1. by dim HD
3. Estimate of the first eigenvalue Let M be a complete noncompact Riemannian manifold and let Ω be a compact domain in M . Denote by λ1 (Ω ) > 0 the first eigenvalue of the Dirichlet boundary value problem △f + λf = 0 in Ω f =0 on ∂Ω where △ is the Laplace operator on M . If Ω1 ⊂ Ω2 are bounded domains, then λ1 (Ω1 ) ≥ λ1 (Ω2 ). Thus, the first eigenvalue λ1 (M ) of a complete noncompact manifold M can be defined by λ1 (M ) = inf λ1 (Ω ), Ω
where the infimum is taken over all compact domains in M . In this section, we attempt to estimate the first eigenvalue of a complete noncompact submanifold in terms of its mean curvature. Let us recall the previous results in this direction. In [5], Cheung and Leung showed Theorem A ([5]). Let M n be a complete non-compact submanifold in Hn+m . If n|H| ≤ α for some constant 0 ≤ α < n − 1, then λ1 (M ) ≥
(n − 1 − α)2 . 4
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Later, Bessa and Montenegro [1] extended this estimate to complete noncompact submanifolds in a complete simply connected Riemannian manifold with sectional curvature bounded above by a negative constant. More precisely, they proved Theorem B ([1]). Let N n+m be a complete simply connected Riemannian manifold with sectional curvature KN satisfying KN ≤ −a2 for a positive constant a > 0. Let M n be a complete noncompact submanifold in N with n|H| ≤ b < (n − 1)a. Then λ1 (M ) ≥
[(n − 1)a − b]2 . 4
When the ambient space N n+m is of nonpositive curvature, using the L1 -Sobolev inequality (2.11), we establish an estimate of the first eigenvalue in terms of M |H|n dv and prove Theorem 1.1. Proof of Theorem 1.1. Let ρ(x) be the geodesic distance on N from a fixed point x0 ∈ N \ M to x. From the proof of [1], we have △ρ ◦ i ≥ (n − 1)a − n|H|. Denote r = ρ ◦ i. For any f ∈ C0∞ (M ), we have div(f 2 ∇r) = ⟨∇f 2 , ∇r⟩ + f 2 △r ≥ −|∇f 2 | + (n − 1)af 2 − n|H|f 2 . Integrating over M , we have 0≥−
|∇f 2 |dv + (n − 1)a
M
f 2 dv − n
M
|H|f 2 dv.
(3.1)
M
On the other hand, using the Sobolev inequality (2.11), we infer n1 n−1 2n n f n−1 dv |H|n dv n |H|f 2 dv ≤ n M M M ≤ nD(n)∥H∥n |∇f 2 | + n|H|f 2 dv. M
Hence, n
|H|f 2 dv ≤
M
nD(n)∥H∥n 1 − nD(n)∥H∥n
|∇f 2 |dv.
M
Substituting into (3.1) yields 1 |∇f 2 |dv + (n − 1)a f 2 dv 1 − nD(n)∥H∥n M M 2 =− |f | |∇f |dv + (n − 1)a f 2 dv. 1 − nD(n)∥H∥n M M
0≥−
(3.2)
Using the Cauchy–Schwarz inequality, we have 1 −2|f | |∇f | ≥ − |∇f |2 − ϵf 2 ϵ for any ϵ > 0. Hence it follows from (3.2) that 1 ϵ 0≥− |∇f |2 dv + (n − 1)a − f 2 dv, 1 − nD(n)∥H∥n M ϵ 1 − nD(n)∥H∥n M which implies that M
|∇f |2 dv ≥ ϵ 1 − nD(n)∥H∥n (n − 1)a −
ϵ 1 − nD(n)∥H∥n
M
f 2 dv.
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132
Let Q(ϵ) = ϵ 1 − nD(n)∥H∥n (n − 1)a −
ϵ . 1 − nD(n)∥H∥n
Then it is easy to compute that max Q(ϵ) = ϵ>0
2 (n − 1)2 a2 1 − nD(n)∥H∥n . 4
Therefore,
2 |∇f |2 dv (n − 1)2 a2 ≥ 1 − nD(n)∥H∥ , n 4 f 2 dv M
M
λ1 (M ) = inf∞ f ∈C0
which completes the proof of Theorem 1.1.
Remark 3.1. In particular, if M is a complete non-compact minimal submanifold in N n+m , we have 2 2 a . λ1 (M ) ≥ (n−1) 4 4. Vanishing theorems under global conditions Let f ∈ C01 (M ). Substituting g = f |f |
2n n−2
n−1 n dv
M
2(n−1) n−2
, n ≥ 3 in (2.11), and using the H¨older inequality, we have
2(n − 1)D(n) ≤ n−2 ≤
|f |
n n−2
|∇f |dv + nD(n)
M
|H|f
2(n−1) n−2
dv
M
12 12 2n 2(n − 1)D(n) |f | n−2 dv |∇f |2 dv n−2 M M 12 21 2n + nD(n) |f | n−2 dv , |H|2 f 2 dv M
M
2
which implies the following L -Sobolev inequality
2n
n−2 n
|f | n−2 dv
≤ c(n)
M
(|∇f |2 + |H|2 f 2 )dv,
∀f ∈ C01 (M ),
(4.1)
M
where c(n) = 2n2 D(n)2 . Using the above L2 -Sobolev inequality, we deduce the following general global vanishing result, from which we immediately have Theorem 1.2. n Proposition 4.1. , n ≥ 5,be a complete non-compact submanifold in Hn+m with ∥H∥n ≤ α, where Let M p(n−p) 1 2 0 < α < nD(n) 1 − n−1 . Assume that 1+Kp
M
|φ|n dv
n2
<
2 4p(n − p) 1 + Kp − . (n − 1)c(n) (n − 1)2 (1 − nαD(n))2
(i) When 2 ≤ p ≤ n − 2, assume further that M has flat normal bundle. Then H p (L2 (M )) = {0}. (ii) When p = 1, then H 1 (L2 (M )) = {0} and M has only one end.
(4.2)
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Proof. (i) Given ω ∈ H p (L2 (M )) for 2 ≤ p ≤ n − 2. Let η ∈ C0∞ (M ). Multiplying (2.9) by η 2 and integrating over M , we obtain n−1 |φ|2 η 2 |ω|2 dv η 2 |ω|△|ω|dv ≥ Kp η 2 |∇|ω| |2 dv − p(n − p) η 2 |ω|2 dv − 2 M M M M n + |H|2 η 2 |ω|2 dv. (4.3) 2 M For the term at the left hand side, integrating by parts over M , we have η 2 |ω|△|ω|dv = −2 η|ω|⟨∇η, ∇|ω|⟩dv − η 2 |∇|ω| |2 dv. M
M
(4.4)
M
Using the H¨ older inequality together with (4.1), it follows that
2 2
2
n
|φ| η |ω| dv ≤
n2
|φ| dv
M
η|ω|
2n n−2
n−2 n dv
M
supp(η)
n
≤ c(n)
n2
|φ| dv 2
|∇(η|ω|)|2 + |H|2 η 2 |ω|2 dv
M
supp(η)
2
2
2
2 2
2
(η |∇|ω| | + |ω| |∇η| + |H| η |ω| )dv + 2S(η)
= S(η) M
η|ω|⟨∇η, ∇|ω|⟩dv, M
(4.5) where S(η) = c(n)( that
2
supp(η)
|φ|n dv) n . By the variational characterization of λ1 (M ), we have from Theorem 1.1
4 η |ω| dv ≤ |∇(η|ω|)|2 dv (n − 1)2 (1 − nD(n)∥H∥n )2 M M 4 (η 2 |∇|ω| |2 + |ω|2 |∇η|2 )dv = (n − 1)2 (1 − nD(n)∥H∥n )2 M 8 + η|ω|⟨∇η, ∇|ω|⟩dv. (n − 1)2 (1 − nD(n)∥H∥n )2 M 2
2
Combining with (4.3)–(4.5), and using the Cauchy–Schwarz inequality, we infer (1 + Kp − E) η 2 |∇|ω| |2 dv M ≤ 2(E − 1) η|ω|⟨∇η, ∇|ω|⟩dv + E |ω|2 |∇η|2 dv − F |H|2 η 2 |ω|2 dv M M M |E − 1| 2 2 2 2 |ω| |∇η| dv − F |H|2 η 2 |ω|2 dv ≤ |E − 1|ϵ η |∇|ω| | dv + E + ϵ M M M for all ϵ > 0, where n−1 4p(n − p) S(η) + , 2 (n − 1)2 (1 − nD(n)∥H∥n )2 n n−1 F = − S(η). 2 2
E=
Hence, |E − 1| |ω|2 |∇η|2 dv − F |H|2 η 2 |ω|2 dv. (4.6) η 2 |∇|ω| |2 dv ≤ E + ϵ M M M
(1 + Kp − E − |E − 1|ϵ)
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Fix a point x0 ∈ M and let ρ(x) be the geodesic distance on M from x0 to x. Let us choose η ∈ C0∞ (M ) satisfying 1 if ρ(x) ≤ r, η(x) = 0 if 2r ≤ ρ(x) and |∇η|(x) ≤
2 r
if r ≤ ρ(x) ≤ 2r
for r > 0. The hypothesis (4.2) implies that 1 + Kp − E > 0 and F ≥ 0. Choose ϵ small enough such that 1 + Kp − E − |E − 1|ϵ > 0. Then it follows from (4.6) that (1 + Kp − E − |E − 1|ϵ)
2
|∇|ω| | dv ≤ (1 + Kp − E − |E − 1|ϵ)
Bx0 (r)
≤ Since
M
η 2 |∇|ω| |2 dv
M
|E − 1| 4 E + r2 ϵ
|ω|2 dv − F
M
|H|2 |ω|2 dv.
Bx0 (r)
2
|ω| dv < ∞, letting r → ∞ gives ∇|ω| = 0
and |H| |ω| = 0.
Hence, |ω| = constant. If |ω| is not identically zero, then H = 0, which implies that the volume of M is infinite. Thus we have a contradiction, since M |ω|2 dv < ∞. Therefore, H p (L2 (M )) = {0}. (ii) Using the Cauchy–Schwarz inequality, we deduce from (2.6) that (n − 1)(n − 2)2 2 2 n − 1 1 |∇|ω| |2 − (n − 1)|ω|2 + (n − 1) − H |ω| − + ϵ |φ|2 |ω|2 |ω|△|ω| ≥ n−1 4nϵ n (4.7) for any ϵ > 0. Let ϵ =
(n−1)(n−2) , 2n
|ω|△|ω| ≥
then
1 1 n |∇|ω| |2 − (n − 1)|ω|2 − (n − 1)|φ|2 |ω|2 + |H|2 |ω|2 . n−1 2 2
Using the same arguments as in the above case, we conclude that H 1 (L2 (M )) = {0}. From the discussion of Fu–Xu in [8], the assumption M |H|n dv ≤ α < ∞ implies that each end of M is non-parabolic. Hence according to Lemma 2.2, M must have only one end. p(n−p) 2 ± 1, we have 1 − n−1 1+Kp < 0. p(n−p) 2 is odd, then for p = n±1 2 , we also have 1 − n−1 1+Kp < 0. Hence for these
Remark 4.1. When the dimension of M n is even, then for p = When the dimension of M n
n 2
or
n 2
integers p, the above vanishing result for L2 harmonic p-forms does not hold. We are now in the position to prove Theorem 1.3. Proof of Theorem 1.3. Let ω ∈ H p (L2 (M )). When 1 ≤ p < p(n−p) 2 that 1 − n−1 1+Kp > 0. Set C0 =
n 2
− 1 and
n 2
+ 1 < p ≤ n − 1, it is easy to check
2 4p(n − p) min 1 + Kp − 2 2 (n − 1)c(n) p (n − 1) (1 − nαD(n))
H. Lin / Nonlinear Analysis 148 (2017) 126–137
with 0 < α <
1 nD(n)
1−
135
, then the condition M |A|n dv < C0 implies that 2 1 p(n − p) 1− , ∥H∥n ≤ n−n/2 |A|n dv < nD(n) n−1 1 + Kp M 2 n−1
p(n−p) 1+Kp
and
|φ|n dv ≤
M
|A|n dv <
M
4p(n − p) 2 1 + Kp − . (n − 1)c(n) (n − 1)2 (1 − nαD(n))2
Hence, it follows from Proposition 4.1 that ω = 0. When p = 0 or n, for any f ∈ H 0 (L2 (M )), applying a theorem of Yau in [20], we conclude that f is constant. On the other hand, by the discussion in [15], the fact ∥H∥n < ∞ implies that the volume of M is infinite. Since M f 2 dv < +∞ and f is constant, we conclude that f = 0, which means H 0 (L2 (M )) = {0}. By Poincar´e duality, we have H n (L2 (M )) = {0}. The proof of the theorem is completed.
5. Vanishing theorems under pointwise conditions In this section, we obtain some vanishing results under pointwise curvature pinching assumptions. n n+m Proposition 5.1. Let with n|H| ≤ α, where M , n ≥ 5, be a complete non-compact submanifold in H p(n−p) 0 < α < n − 1 − 2 1+Kp for 1 ≤ p ≤ n − 1.
(i) When 2 ≤ p ≤ n − 2, assume that M has flat normal bundle, and that sup |φ|2 < x∈M
(n − 1 − α)2 (1 + Kp ) + 2n inf |H|2 − 4p(n − p) . 2(n − 1)
(5.1)
Then H p (L2 (M )) = {0}. (ii) When p = 1, assume that sup |φ|2 < x∈M
n(n − 1 − α)2 − 4(n − 1)2 . n(n − 1)
Then H 1 (L2 (M )) = {0} and M has only one end. Proof. (i) Given ω ∈ H p (L2 (M )) for 2 ≤ p ≤ n − 2. Combining the relations (4.3) and (4.4), and using Theorem A, we have 2 2 2 2 Kp η |∇|ω| | dv ≤ −2 η|ω|⟨∇η, ∇|ω|⟩dv − η |∇|ω| | dv + E |∇(η|ω|)|2 dv, M
M
M
M
where 4p(n − p) − 2n inf |H|2 + 2(n − 1) sup |φ|2 x∈M
E=
x∈M
(n − 1 − α)2
Using the Cauchy–Schwarz inequality, a direct computation gives A(ϵ) η 2 |∇|ω| |2 dv ≤ B(ϵ) |ω|2 |∇η|2 dv, M
M
.
(5.2)
H. Lin / Nonlinear Analysis 148 (2017) 126–137
136
where A(ϵ) = Kp + 1 − E − ϵ|1 − E|, 1 B(ϵ) = E + |1 − E| ϵ for all ϵ > 0. The assumption (5.1) is equivalent to Kp + 1 − E > 0, which implies that there exists a sufficiently small ϵ > 0 such that A(ϵ) = Kp + 1 − E − ϵ|1 − E| > 0. Hence it follows from (5.2) that 4B(ϵ) |ω|2 dv. A(ϵ) |∇|ω| |2 dv ≤ A(ϵ) η 2 |∇|ω| |2 dv ≤ 2 r M Bx0 (r) M 2 Letting r → ∞, and noting that M |ω| dv < ∞, we conclude that |ω| = constant. Since the mean curvature of M is bounded from above, according to Corollary 2.1 of [4], the volume of M is infinite. Therefore, we have ω = 0. (ii) Let ϵ =
(n−2)2 4n
in (4.7), then we have |ω|△|ω| ≥
1 n |∇|ω| |2 − (n − 1)|ω|2 − |φ|2 |ω|2 . n−1 4
Arguing as in the proof of case (i), we conclude that H 1 (L2 (M )) = {0}. Since λ1 (M ) > 0 by Theorem A, the Sobolev type inequality (2.10) with µ = 1 holds on M . On the other hand, by Corollary 2.1 of [4], the bounded mean curvature implies that each end of M has infinite volume. According to Lemma 2.1, each end of M is non-parabolic. Therefore, using Lemma 2.2, M has only one end. We now prove Theorem 1.4. Proof of Theorem 1.4. Since the condition n|H| < c1 < ∞ implies that the volume of M is infinite, the cases of p = 0 or n follow the line of the proof of Theorem 1.3. The other cases of p are already proved in Proposition 5.1. Remark 5.1. In particular, if the dimension of M n is even and the second fundamental form of M n identically vanishes, that is, M n is the totally geodesic submanifold Hn in Hn+m , by the main theorem in [6] there are no nontrivial L2 harmonic p-forms of degree p ̸= n2 on Hn . Acknowledgment The author would like to thank the referee for the helpful comments and detailed corrections. References [1] G.P. Bessa, J.F. Montenegro, Eigenvalue estimates for submanifolds with locally bounded mean curvature, Ann. Global Anal. Geom. 24 (2003) 279–290. [2] D.M.J. Calderbank, P. Gauduchon, M. Herzlich, Refined Kato inequalities and conformal weights in Riemannian geometry, J. Funct. Anal. 173 (1) (2000) 214–255. [3] M.P. Cavalcante, H. Mirandola, F. Vit´ orio, L2 -harmonic 1-forms on submanifolds with finite total curvature, J. Geom. Anal. 24 (2014) 205–222. [4] X. Cheng, L.F. Cheung, D.T. Zhou, The structure of weakly stable constant mean curvature hypersurfaces, Tohoku Math. J. 60 (1) (2008) 101–121. [5] L.F. Cheung, P.F. Leung, Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space, Math. Z. 236 (2001) 525–530. [6] J. Dodziuk, L2 harmonic forms on rotationally symmetric Riemannian manifolds, Proc. Amer. Math. Soc. 77 (3) (1979) 395–400.
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