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Estimates for the first eigenvalue of the drifting Laplace and the p-Laplace operators on submanifolds with bounded mean curvature in the hyperbolic space
Accepted Manuscript Estimates for the first eigenvalue of the drifting Laplace and the p-Laplace operators on submanifolds with bounded mean curvature in the hyperbolic space
Feng Du, Jing Mao
PII: DOI: Reference:
S0022-247X(17)30707-2 http://dx.doi.org/10.1016/j.jmaa.2017.07.044 YJMAA 21575
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
13 July 2016
Please cite this article in press as: F. Du, J. Mao, Estimates for the first eigenvalue of the drifting Laplace and the p-Laplace operators on submanifolds with bounded mean curvature in the hyperbolic space, J. Math. Anal. Appl. (2017), http://dx.doi.org/10.1016/j.jmaa.2017.07.044
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Estimates for the first eigenvalue of the drifting Laplace and the p-Laplace operators on submanifolds with bounded mean curvature in the hyperbolic space Feng Dub and Jing Maoa,∗ a Faculty
of Mathematics and Statistics, Key Laboratory of Applied Mathematics of Hubei Province, Hubei University, Wuhan, 430062, China Email: [email protected], [email protected] b School of Mathematics and Physics Science, Jingchu University of Technology, Jingmen, 448000, China
Abstract In this paper, we successfully give two interesting lower bounds for the first eigenvalue of submanifolds (with bounded mean curvature) in a hyperbolic space. More precisely, let M be an n-dimensional complete noncompact submanifold in a hyperbolic space and the norm of its mean curvature vector H satisfies H α < n − 1, then we prove that the first
eigenvalue λ1,p (M) of the p-Laplacian Δ p on M satisfies λ1,p (M)
p n−1−α , 1
∞,
nwith equality achieved when M is totally geodesic and p = 2; let dimensional complete noncompact smooth metric measure space with M being a submanifold in a hyperbolic space, and H α < n − 1, ∇ϕ C with ∇ the gradient operator on M, then we show that the first eigenvalue λ1,ϕ (M) of the weighted Laplacian Δϕ on M satisfies
λ1,ϕ (M)
(n−1−α −C)2 , 4
with equality attained when M is totally geodesic and ϕ = constant.
1 Introduction Denote by M be an n-dimensional complete noncompact Riemannian manifold with the Laplace operator Δ. For an open bounded connected domain Ω ⊆ M , the classical Dirichlet eigenvalue problem is to find all possible real numbers λ such that there exists a nontrivial solution u to the boundary value problem Δu + λ u = 0 in Ω, (1.1) u=0 on ∂ Ω, ∗ Corresponding 0
author MSC 2010: 35P15, 53C20. Key Words: Eigenvalues, Laplacian, drifting Laplacian, p-Laplacian, smooth metric measure spaces.
1
F. Du and J. Mao
2
where Δ = div ◦ ∇ is the Laplace operator on M . The desired real numbers are called eigenvalues of Δ, and for a given λ , the space of solutions of (1.1) is a vector space since the first equation of (1.1) is linear in u. This space is called the eigenspace of λ , and the non-zero elements of each eigenspace are called eigenfunctions. It is well-known that for the eigenvalue problem (1.1), Δ only has discrete spectrum whose elements (i.e., eigenvalues) can be listed increasingly as follows 0 < λ1 < λ2 · · · ↑ ∞, and each associated eigenspace has finite dimension. λi (i ≥ 1) is called the ith Dirichlet eigenvalue of Δ. By Domain monotonicity of eigenvalues with vanishing Dirichlet data (cf. [7, pp. 17-18]), we know that λ1 (Ω1 ) λ1 (Ω2 ) if Ω1 ⊇ Ω2 . Let B(q, ) be a geodesic ball, with center q and radius , on M . Then the first Dirichlet eigenvalue λ1 (B(q, )) of the Laplacian Δ on B(q, ) decreases as increases, and it has a limit independent of the choice of the center q. Therefore, one can define this limit λ1 (M ) by λ1 (M ) := lim λ1 (B(q, )). Clearly, λ1 (M ) 0. Schoen and Yau [14, p. 106] →∞
suggested that it is an important question to find conditions which will imply λ1 (M ) > 0. Speaking in other words, manifolds with λ1 (M ) > 0 might have some special geometric properties. There are many interesting results supporting this. For instance, Cheung and Leung [2] proved that if M is an n-dimensional complete minimal submanifold in the hyperbolic m-space Hm (−1), then 2 λ1 (M ) (n−1) > 0, and moreover, M is non-parabolic, i.e., there exists a non-constant bounded 4 subharmonic function on M . They also showed that if furthermore M has at least two ends, then there exists on M a non-constant bounded harmonic function with finite Dirichlet energy. The purpose of this paper is trying to improve Cheung-Leung’s estimate for the first eigenvalue of the Laplacian in [2] to the case of the p-Laplacian and the weighted Laplacian. Let M be an n-dimensional complete noncompact Riemannian manifold and, as before, let B(q, ) be a geodesic ball, with center q and radius , on M. Consider the following nonlinear Dirichlet eigenvalue problem Δ p u + λ |u| p−2 u = 0 in B(q, ), (1.2) u=0 on ∂ B(q, ), where Δ p u = div(∇u p−2 ∇u) is the p-Laplacian with 1 < p < ∞, ∇u is the gradient of u, and ∇u is the norm of ∇u with respect to the metric on M. Without specification, in the sequel, · denotes the norm of some prescribed vector field on M. It is true that (1.2) has a positive weak solution, which is unique modulo the scaling, in W01,p (B(q, )), the completion of the set C0∞ (B(q, )) of smooth functions compactly supported on B(q, ) under the Sobolev norm u1,p =
1
{ B(q,) (|u| p + ∇u p )dv} p , where dv is the Riemannian volume element. One can get a simple proof of this fact in [1] for bounded simply connected domains with sufficiently smooth boundary in Euclidean space. Besides, the first Dirichlet eigenvalue λ1,p (B(q, )) of the p-Laplacian on B(q, ) can be characterized by p B(q,) ∇ f dv 1,p λ1,p (B(q, )) = inf (1.3) f ∈ W0 (B(q, )), f = 0 . p B(q,) | f | dv The (closed or Dirichlet) eigenvalue problem of the p-Laplacian has been studied by the third author and some interesting conclusions have been obtained (see, e.g., [3, 10, 11]). Similar to the case of the Laplacian, by applying the Max-min principle, we can obtain the following result.
F. Du and J. Mao
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Lemma 1.1. Domain monotonicity of eigenvalues with vanishing Dirichlet data also holds for the Dirichlet eigenvalues of the p-Laplacian. Proof. If replace L2 (M) by L p (M) in the proof of Domain monotonicity of eigenvalues with vanishing Dirichlet data (cf. [7, pp. 17-18]), and naturally in this setting the corresponding inner product and norm should be defined by ( f , g) p :=
M
p 2
p 2
| f | |g| dv,
f :=
p
M
| f | p dv
for f , g ∈ L p (M), then similar conclusion (i.e., Lemma 1.1) can also be obtained since for the nonlinear p-Laplacian, the max-min method also works. By Lemma 1.1, we know that λ1,p (B(q, )) decreases as increases and then it has a limit independent of the choice of the center q. So, one can make the following definition. Definition 1.2. The first eigenvalue of the p-Laplacian Δ p on M is defined by
λ1,p (M) := lim λ1,p (B(q, )) . →∞
For λ1,p (M), we can prove the following conclusion. Theorem 1.3. Let M be an n-dimensional complete noncompact submanifold in the hyperbolic m-space Hm (−1). Denote H by the mean curvature vector of M in Hm (−1). If H α for some constant α < n − 1, then
n−1−α p λ1,p (M) > 0. (1.4) p Remark 1.4. (1) Clearly, when p = 2, the nonlinear p-Laplacian Δ p degenerates into the linear Laplacian Δ and correspondingly λ1,p (M) = λ1 (M), which leads to the fact that the estimate (1.4) is exactly the one in [2, Theorem 2] for λ1 (M). So, Theorem 1.3 here covers [2, Theorem 2]. (2) If M is furthermore a complete minimal submanifold in the hyperbolic space Hm (−1), then by Theorem 1.3 we have
n−1 p λ1,p (M) . (1.5) p By McKean’s results [6], one has λ1 (Hn (−1)) = 14 (n − 1)2 , which shows that the estimate (1.5) is sharp for the totally geodesic submanifold Hn (−1) in Hm (−1) for p = 2 and might be also sharp for p = 2. A smooth metric measure space (also known as the weighted measure space) is actually a Riemannian manifold equipped with some measure which is conformal to the usual Riemannian measure. More precisely, for a given complete Riemannian manifold (M, g) with the metric g, the triple (M, g, e−ϕ dv) is called a smooth metric measure space, where ϕ is a smooth real-valued function on M and, as before, dv is the Riemannian volume element associated with g (sometimes, we also call dv the volume density). On a smooth metric measure space (M, g, e−ϕ dv), we can define the so-called drifting Laplacian (also called weighted Laplacian) Δϕ as follows Δϕ := Δ − g(∇ϕ , ∇·),
F. Du and J. Mao
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where, as before, ∇ and Δ are the gradient and the Laplace operators on M, respectively. When the weighted Ricci curvature is bounded from below, the Myers’s theorem, Bishop-Gromov’s volume comparison, Cheeger-Gromoll’s splitting theorem and Abresch-Gromoll excess estimate cannot hold as the Riemannian case. Therefore, it is interesting to know whether a classical result in the Riemannian geometry could be extended to the weighted case or not. The third author here has been walking on this way and some interesting results has also been obtained (see, e.g., [4, 5, 12, 13]). Now, let (M, g, e−ϕ dv) be a complete noncompact smooth metric measure space and, as before, let B(q, ) be a geodesic ball, with center q and radius , on M. Consider the following Dirichlet eigenvalue problem Δϕ u + λ u = 0 in B(q, ), (1.6) u=0 on ∂ B(q, ), where Δϕ is the weighted Laplacian on M defined as above. Similar to the case of the Laplacian, it is known that the weighted Laplacian Δϕ (with the Dirichlet boundary data) only has discrete spectrum on bounded open connected domains and all the eigenvalues can be also listed increasingly. By Rayleigh Theorem and the Max-min principle, the first Dirichlet eigenvalue λ1,ϕ (B(q, )) of the weighted Laplacian on B(q, ) can be characterized by 2 −ϕ dv B(q,) ∇ f e 1,2 λ1,ϕ (B(q, )) = inf (1.7) f ∈ W0 (B(q, )), f = 0 . 2 −ϕ dv B(q,) f e Similar to the case of the Laplacian, by applying the Max-min principle, we can get the following result. Lemma 1.5. Domain monotonicity of eigenvalues with vanishing Dirichlet data also holds for the Dirichlet eigenvalues of the weighted Laplacian. Proof. The proof is almost the same with that of Domain monotonicity of eigenvalues with vanishing Dirichlet data (cf. [7, pp. 17-18]) and the only difference is that one needs to replace the normal Riemannian volume density dv by the weighted one e−ϕ dv, which leads to the fact that all the equalities and inequalities in the proof of Domain monotonicity of eigenvalues with vanishing Dirichlet data still hold. By Lemma 1.5, we know that λ1,p (B(q, )) decreases as increases and then it has a limit independent of the choice of the center q. So, one can make the following definition. Definition 1.6. The first eigenvalue of the weighted Laplacian Δϕ on M is defined by
λ1,ϕ (M) := lim λ1,ϕ (B(q, )) . →∞
For λ1,ϕ (M), we can prove the following conclusion. Theorem 1.7. Let (M, g, e−ϕ dv) be an n-dimensional complete noncompact smooth metric measure space with M being a submanifold in the hyperbolic m-space Hm (−1). Denote H by the mean curvature vector of M in Hm (−1). If H α for some constant α < n − 1 and ∇ϕ C for some constant C, where ∇ is the gradient operator on M, then
λ1,ϕ (M)
(n − 1 − α −C)2 > 0. 4
(1.8)
F. Du and J. Mao
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Remark 1.8. If ϕ = constant, then the constant C can be chosen to be C = 0 and correspondingly the estimate (1.8) degenerates into the one in [2, Theorem 2] for λ1 (M). Therefore, [2, Theorem 2] is included by Theorem 1.8 as a special case.
2 Some useful facts In this section, we would like to recall several facts and also prove some new ones which will be used in the last section for proving estimates (1.4) and (1.8). Lemma 2.1. ([8, Lemma 2]) Let f ∈ C∞ (N m ) where N m is an m-dimensional Riemannian manifold and suppose Qn is an n-dimensional submanifold of N m . Denote by ∇, Δ the connection and Laplacian on N m respectively and Δ the Laplacian on Qn , then the following formula relating Δ to Δ holds: m 2 Δ ( f |Q ) = Δ f |Q + H, ∇ f |Q − ∑ ∇ f (ek , ek )|Q , k=n+1
where {ek : k = n + 1, · · · , m} is a local orthonormal frame field of the normal space of Qn in N m . Using Lemma 2.1 and together with the definition of the weighted Laplacian Δϕ mentioned in Section 1, we can easily obtain the following fact. Lemma 2.2. Under the assumptions of Lemma 2.1, we have Δϕ ( f |Q ) = Δ f |Q + H, ∇ f |Q −
m
∑
2
∇ f (ek , ek )|Q − g (∇ f |Q , ∇ϕ ) ,
k=n+1
Δϕ is the weighted Laplacian related to the weighted where g is the Riemannian metric of − ϕ n volume density e dv of Q , and ∇ is the gradient operator on Qn related to the metric g. Qn ,
Lemma 2.3. ([8, Lemma 3], [2, Corollary 4]) Let ∇ and Δ be the connection and Laplacian of Hm (−1) respectively, and let , be its metric tensor, then we have 2
∇ cosh r = cosh r · , and Δ cosh r = m cosh r, where r is the distance function on Hm (−1) measured from a fixed point in Hm (−1). By applying Lemmas 2.1 and 2.3, the following fact can be proven directly. Lemma 2.4. ([2, Lemmas 5 and 6]) Let M be an n-dimensional submanifold in the hyperbolic m-space Hm (−1), then we have Δ cosh r = n cosh r + H, ∇r |M · sinh r and
Δr = n − ∇r2 coth r + H, ∇r |M ,
where r is measured from a fixed point in Hm (−1) \ M.
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Combining Lemma 2.2 and Lemma 2.4, we can prove the following fact. Lemma 2.5. Assume that (M, g, e−ϕ dv) is an n-dimensional smooth metric measure space and M is a submanifold in the hyperbolic m-space Hm (−1), then we have Δϕ cosh r = n cosh r + H, ∇r |M · sinh r − sinh r · g (∇r, ∇ϕ ) and
Δϕ r = n − ∇r2 coth r + H, ∇r |M − g (∇r, ∇ϕ ) where r is measured from a fixed point in Hm (−1) \ M. Proof. Applying Lemmas 2.2, 2.3 and 2.4 to the function cosh r, we can obtain Δϕ (cosh r|M ) =
Δ cosh r |M + H, ∇ cosh r |M −
m
∑
2
∇ cosh r(ek , ek )|M − g (∇ cosh r, ∇ϕ )
k=n+1
= m cosh r + H, ∇r |M · sinh r − (m − n) cosh r − sinh r · g(∇r, ∇ϕ ) = n cosh r + H, ∇r |M · sinh r − sinh r · g(∇r, ∇ϕ ), which is the first result in Lemma 2.5. Combing the fact that Δϕ cosh r = Δ cosh r − g (∇ϕ , ∇ cosh r) = div(∇ cosh r) − g (∇ϕ , ∇ cosh r) = cosh r∇r2 + Δr · sinh r − g (∇ϕ , ∇ cosh r) , it is easy to get that n cosh r + H, ∇r |M · sinh r − sinh r · g(∇r, ∇ϕ ) = cosh r · ∇r2 + (sinh r)Δr − g (∇r, ∇ cosh r) , which is equivalent to sinh r · Δϕ r = n cosh r − cosh r · ∇r2 + H, ∇r |M · sinh r − sinh r · g (∇r, ∇ϕ ) . Then the second result in Lemma 2.5 follows.
3
Proofs of main results
By using the facts in Section 2, we can prove Theorems 1.3 and 1.7 as follows. Proof of Theorem 1.3. By (1.3) and Definition 1.2, we know that if one wants to get (1.4), it is sufficient to show that for any f ∈ C0∞ (M), the inequality
n−1−α p p | f | dv ∇ f p dv (3.1) p M M holds.
F. Du and J. Mao
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Let r be the distance function measured from a fixed point in Hm (−1) \ M. We know that ∇r = 1 and hence ∇r 1. By Lemma 2.4, one has
Δr = n − ∇r2 coth r + H, ∇r |M n − 1 − H n − 1 − α. Denote by g the Riemannian metric of M. Taking f ∈ C0∞ (M) and using the above inequality, we have div(| f | p · ∇r) = g (∇(| f | p ), ∇r) + | f | p Δr −p| f | p−1 ∇ f + | f | p · (n − 1 − α ) ⎡ ∇ f p
p ⎤ p−1 p−1 ε | f | ε ⎦ + | f | p · (n − 1 − α ) + −p ⎣ p p p−1 = −
p ∇ f p p−1 | f | p + | f | p · (n − 1 − α ), − (p − 1) ε εp
(3.2)
where ε > 0 is a parameter determined later. Here the following Young’s inequality
p ∇ f p ε | f | p−1 p−1 ε p−1 p−1 ∇ f | f | ∇ f = ε | f | · + p ε p p−1 has been used in the second inequality of (3.2). Integrating (3.2) over M and using the divergence theorem, we can obtain p ∇ f p p p−1 + (p − 1)ε | f | dv | f | p · (n − 1 − α )dv, εp M M that is, M
p ∇ f p dv ε p n − 1 − α − (p − 1)ε p−1 | f | p dv.
(3.3)
M
p Consider the function F(ε ) := ε p n − 1 − α − (p − 1)ε p−1 , ε > 0. It is easy to check that F(ε ) gets its maximum at the value p−1
n−1−α p ε= . p Combing this fact with (3.3) yields p−1 p p−1 p
n−1−α p n − 1 − α p · p−1 n − 1 − α − (p − 1) ∇ f dv | f | p dv p p M M p
n−1−α | f | p dv, p M
p
F. Du and J. Mao
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which is exactly (3.1). Then (1.4) follows. This completes the proof of Theorem 1.3.
Proof of Theorem 1.7. By (1.7) and Definition 1.6, we know that if one wants to get (1.8), it is sufficient to show that for any f ∈ C0∞ (M), the inequality (n − 1 − α −C)2 4
M
f 2 e−ϕ dv
M
∇ f 2 e−ϕ dv
(3.4)
holds. As in the proof of Theorem 1.3, for the metric measure space (M, g, e−ϕ dv), we choose r to be the distance function measured from a fixed point in Hm (−1) \ M. By Lemma 2.5, one has
Δϕ r = n − ∇r2 coth r + H, ∇r |M − g (∇r, ∇ϕ ) n − 1 − H · ∇r − ∇ϕ · ∇r n − 1 − α −C. Taking f ∈ C0∞ (M) and using the above inequality results in
div( f 2 ∇r · e−ϕ ) = e−ϕ g ∇ f 2 , ∇r + f 2 e−ϕ Δr − f 2 e−ϕ g(∇r, ∇ϕ )
= e−ϕ g ∇ f 2 , ∇r + f 2 e−ϕ Δϕ r −e−ϕ ∇ f 2 + f 2 e−ϕ · (n − 1 − α −C) e−ϕ −2| f | · ∇ f + f 2 · (n − 1 − α −C) ∇ f 2 −ϕ 2 2 + f · (n − 1 − α −C) , −ε f − e ε
(3.5)
where ε > 0 is a parameter determined later. Here the following inequality ∇ f 2 ε has been used. Integrating (3.5) over M and using the divergence theorem, we can obtain
∇ f 2 −ϕ 2 e dv εf + f 2 · (n − 1 − α −C)e−ϕ dv, ε M M 2| f | · ∇ f ε | f |2 +
that is,
∇ f 2 e−ϕ dv ε (n − 1 − α −C − ε )
M
f 2 e−ϕ dv.
(3.6)
M
On the other hand,
ε (n − 1 − α −C − ε ) with equality holds if and only if ε = M
2 −ϕ
∇ f e
ε + n − 1 − α −C − ε 2 n−1−α −C . 2
2 =
(n − 1 − α −C)2 4
Combining this fact with (3.6), we have
(n − 1 − α −C)2 dv 4
f 2 e−ϕ dv,
M
which is exactly (3.4). Then (1.8) follows. This completes the proof of Theorem 1.7.
F. Du and J. Mao
9
Acknowledgments This work was partially supported by the project sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, the NSF of China (Grant No. 11401131), and Hubei Key Laboratory of Applied Mathematics (Hubei University). The authors would like to thank the anonymous referee for his or her careful reading and valuable comments such that the article appears as its present version.
References [1] M. Belloniand and B. Kawohl, A direct uniqueness proof for equations involving the pLaplace operator, Manuscripta Math. 109 (2002) 229–231. [2] L.-F. Cheung and P.-F. Leung, Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space, Math. Z. 236 (2001) 525–530. [3] F. Du and J. Mao, Reilly-type inequalities for the p-Laplacian on compact Riemannian manifolds, Frontiers of Mathematics in China 10(3) (2015) 583–594. [4] F. Du and J. Mao, Hardy and Rellich type inequalities on metric measure spaces, J. Math. Anal. Appl. 429 (1) (2015) 354–365. [5] F. Du, J. Mao, Q.-L. Wang and C.-X. Wu, The Gagliardo-Nirenberg inequality on metric measure spaces, submitted. [6] H.-P. McKean, An upper bound to the spectrum of Δ on a manifold of negative curvature, J. Differential Geom. 4 (1970) 359–366. [7] I. Chavel, Eigenvalues in Riemannian geometry, Academic Press, New York, 1984. [8] J. Choe and R. Gulliver, Isoperimetric inequalities on minimal submanifolds of space forms, Manuscripta Math. 77 (1992) 169–189. [9] H.-I. Choi and A.-N. Wang, A first eigenvalue estimate for minimal hypersurfaces, J. Differential Geom. 18 (1983) 559–562. [10] J. Mao, Eigenvalue estimation and some results on finite topological type, Ph.D. thesis, ISTUTL, 2013. [11] J. Mao, Eigenvalue inequalities for the p-Laplacian on a Riemannian manifold and estimates for the heat kernel, J. Math. Pures Appl. 101(3) (2014) 372–393. [12] J. Mao, The Caffarelli-Kohn-Nirenberg inequalities and manifolds with non-negative weighted Ricci curvature, J. Math. Anal. Appl. 428(2) (2015) 866–881. [13] J. Mao, The Gagliardo-Nirenberg inequalities and manifolds with non-negative weighted Ricci curvature, Kyushu J. Math. 70(1) (2016) 29–46. [14] R. Schoen and S.-T. Yau, Lectures on differential geometry, International Press, Boston, 1994.
Feng Du, Jing Mao
PII: DOI: Reference:
S0022-247X(17)30707-2 http://dx.doi.org/10.1016/j.jmaa.2017.07.044 YJMAA 21575
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
13 July 2016
Please cite this article in press as: F. Du, J. Mao, Estimates for the first eigenvalue of the drifting Laplace and the p-Laplace operators on submanifolds with bounded mean curvature in the hyperbolic space, J. Math. Anal. Appl. (2017), http://dx.doi.org/10.1016/j.jmaa.2017.07.044
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Estimates for the first eigenvalue of the drifting Laplace and the p-Laplace operators on submanifolds with bounded mean curvature in the hyperbolic space Feng Dub and Jing Maoa,∗ a Faculty
of Mathematics and Statistics, Key Laboratory of Applied Mathematics of Hubei Province, Hubei University, Wuhan, 430062, China Email: [email protected], [email protected] b School of Mathematics and Physics Science, Jingchu University of Technology, Jingmen, 448000, China
Abstract In this paper, we successfully give two interesting lower bounds for the first eigenvalue of submanifolds (with bounded mean curvature) in a hyperbolic space. More precisely, let M be an n-dimensional complete noncompact submanifold in a hyperbolic space and the norm of its mean curvature vector H satisfies H α < n − 1, then we prove that the first
eigenvalue λ1,p (M) of the p-Laplacian Δ p on M satisfies λ1,p (M)
p n−1−α , 1
∞,
nwith equality achieved when M is totally geodesic and p = 2; let dimensional complete noncompact smooth metric measure space with M being a submanifold in a hyperbolic space, and H α < n − 1, ∇ϕ C with ∇ the gradient operator on M, then we show that the first eigenvalue λ1,ϕ (M) of the weighted Laplacian Δϕ on M satisfies
λ1,ϕ (M)
(n−1−α −C)2 , 4
with equality attained when M is totally geodesic and ϕ = constant.
1 Introduction Denote by M be an n-dimensional complete noncompact Riemannian manifold with the Laplace operator Δ. For an open bounded connected domain Ω ⊆ M , the classical Dirichlet eigenvalue problem is to find all possible real numbers λ such that there exists a nontrivial solution u to the boundary value problem Δu + λ u = 0 in Ω, (1.1) u=0 on ∂ Ω, ∗ Corresponding 0
author MSC 2010: 35P15, 53C20. Key Words: Eigenvalues, Laplacian, drifting Laplacian, p-Laplacian, smooth metric measure spaces.
1
F. Du and J. Mao
2
where Δ = div ◦ ∇ is the Laplace operator on M . The desired real numbers are called eigenvalues of Δ, and for a given λ , the space of solutions of (1.1) is a vector space since the first equation of (1.1) is linear in u. This space is called the eigenspace of λ , and the non-zero elements of each eigenspace are called eigenfunctions. It is well-known that for the eigenvalue problem (1.1), Δ only has discrete spectrum whose elements (i.e., eigenvalues) can be listed increasingly as follows 0 < λ1 < λ2 · · · ↑ ∞, and each associated eigenspace has finite dimension. λi (i ≥ 1) is called the ith Dirichlet eigenvalue of Δ. By Domain monotonicity of eigenvalues with vanishing Dirichlet data (cf. [7, pp. 17-18]), we know that λ1 (Ω1 ) λ1 (Ω2 ) if Ω1 ⊇ Ω2 . Let B(q, ) be a geodesic ball, with center q and radius , on M . Then the first Dirichlet eigenvalue λ1 (B(q, )) of the Laplacian Δ on B(q, ) decreases as increases, and it has a limit independent of the choice of the center q. Therefore, one can define this limit λ1 (M ) by λ1 (M ) := lim λ1 (B(q, )). Clearly, λ1 (M ) 0. Schoen and Yau [14, p. 106] →∞
suggested that it is an important question to find conditions which will imply λ1 (M ) > 0. Speaking in other words, manifolds with λ1 (M ) > 0 might have some special geometric properties. There are many interesting results supporting this. For instance, Cheung and Leung [2] proved that if M is an n-dimensional complete minimal submanifold in the hyperbolic m-space Hm (−1), then 2 λ1 (M ) (n−1) > 0, and moreover, M is non-parabolic, i.e., there exists a non-constant bounded 4 subharmonic function on M . They also showed that if furthermore M has at least two ends, then there exists on M a non-constant bounded harmonic function with finite Dirichlet energy. The purpose of this paper is trying to improve Cheung-Leung’s estimate for the first eigenvalue of the Laplacian in [2] to the case of the p-Laplacian and the weighted Laplacian. Let M be an n-dimensional complete noncompact Riemannian manifold and, as before, let B(q, ) be a geodesic ball, with center q and radius , on M. Consider the following nonlinear Dirichlet eigenvalue problem Δ p u + λ |u| p−2 u = 0 in B(q, ), (1.2) u=0 on ∂ B(q, ), where Δ p u = div(∇u p−2 ∇u) is the p-Laplacian with 1 < p < ∞, ∇u is the gradient of u, and ∇u is the norm of ∇u with respect to the metric on M. Without specification, in the sequel, · denotes the norm of some prescribed vector field on M. It is true that (1.2) has a positive weak solution, which is unique modulo the scaling, in W01,p (B(q, )), the completion of the set C0∞ (B(q, )) of smooth functions compactly supported on B(q, ) under the Sobolev norm u1,p =
1
{ B(q,) (|u| p + ∇u p )dv} p , where dv is the Riemannian volume element. One can get a simple proof of this fact in [1] for bounded simply connected domains with sufficiently smooth boundary in Euclidean space. Besides, the first Dirichlet eigenvalue λ1,p (B(q, )) of the p-Laplacian on B(q, ) can be characterized by p B(q,) ∇ f dv 1,p λ1,p (B(q, )) = inf (1.3) f ∈ W0 (B(q, )), f = 0 . p B(q,) | f | dv The (closed or Dirichlet) eigenvalue problem of the p-Laplacian has been studied by the third author and some interesting conclusions have been obtained (see, e.g., [3, 10, 11]). Similar to the case of the Laplacian, by applying the Max-min principle, we can obtain the following result.
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Lemma 1.1. Domain monotonicity of eigenvalues with vanishing Dirichlet data also holds for the Dirichlet eigenvalues of the p-Laplacian. Proof. If replace L2 (M) by L p (M) in the proof of Domain monotonicity of eigenvalues with vanishing Dirichlet data (cf. [7, pp. 17-18]), and naturally in this setting the corresponding inner product and norm should be defined by ( f , g) p :=
M
p 2
p 2
| f | |g| dv,
f :=
p
M
| f | p dv
for f , g ∈ L p (M), then similar conclusion (i.e., Lemma 1.1) can also be obtained since for the nonlinear p-Laplacian, the max-min method also works. By Lemma 1.1, we know that λ1,p (B(q, )) decreases as increases and then it has a limit independent of the choice of the center q. So, one can make the following definition. Definition 1.2. The first eigenvalue of the p-Laplacian Δ p on M is defined by
λ1,p (M) := lim λ1,p (B(q, )) . →∞
For λ1,p (M), we can prove the following conclusion. Theorem 1.3. Let M be an n-dimensional complete noncompact submanifold in the hyperbolic m-space Hm (−1). Denote H by the mean curvature vector of M in Hm (−1). If H α for some constant α < n − 1, then
n−1−α p λ1,p (M) > 0. (1.4) p Remark 1.4. (1) Clearly, when p = 2, the nonlinear p-Laplacian Δ p degenerates into the linear Laplacian Δ and correspondingly λ1,p (M) = λ1 (M), which leads to the fact that the estimate (1.4) is exactly the one in [2, Theorem 2] for λ1 (M). So, Theorem 1.3 here covers [2, Theorem 2]. (2) If M is furthermore a complete minimal submanifold in the hyperbolic space Hm (−1), then by Theorem 1.3 we have
n−1 p λ1,p (M) . (1.5) p By McKean’s results [6], one has λ1 (Hn (−1)) = 14 (n − 1)2 , which shows that the estimate (1.5) is sharp for the totally geodesic submanifold Hn (−1) in Hm (−1) for p = 2 and might be also sharp for p = 2. A smooth metric measure space (also known as the weighted measure space) is actually a Riemannian manifold equipped with some measure which is conformal to the usual Riemannian measure. More precisely, for a given complete Riemannian manifold (M, g) with the metric g, the triple (M, g, e−ϕ dv) is called a smooth metric measure space, where ϕ is a smooth real-valued function on M and, as before, dv is the Riemannian volume element associated with g (sometimes, we also call dv the volume density). On a smooth metric measure space (M, g, e−ϕ dv), we can define the so-called drifting Laplacian (also called weighted Laplacian) Δϕ as follows Δϕ := Δ − g(∇ϕ , ∇·),
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where, as before, ∇ and Δ are the gradient and the Laplace operators on M, respectively. When the weighted Ricci curvature is bounded from below, the Myers’s theorem, Bishop-Gromov’s volume comparison, Cheeger-Gromoll’s splitting theorem and Abresch-Gromoll excess estimate cannot hold as the Riemannian case. Therefore, it is interesting to know whether a classical result in the Riemannian geometry could be extended to the weighted case or not. The third author here has been walking on this way and some interesting results has also been obtained (see, e.g., [4, 5, 12, 13]). Now, let (M, g, e−ϕ dv) be a complete noncompact smooth metric measure space and, as before, let B(q, ) be a geodesic ball, with center q and radius , on M. Consider the following Dirichlet eigenvalue problem Δϕ u + λ u = 0 in B(q, ), (1.6) u=0 on ∂ B(q, ), where Δϕ is the weighted Laplacian on M defined as above. Similar to the case of the Laplacian, it is known that the weighted Laplacian Δϕ (with the Dirichlet boundary data) only has discrete spectrum on bounded open connected domains and all the eigenvalues can be also listed increasingly. By Rayleigh Theorem and the Max-min principle, the first Dirichlet eigenvalue λ1,ϕ (B(q, )) of the weighted Laplacian on B(q, ) can be characterized by 2 −ϕ dv B(q,) ∇ f e 1,2 λ1,ϕ (B(q, )) = inf (1.7) f ∈ W0 (B(q, )), f = 0 . 2 −ϕ dv B(q,) f e Similar to the case of the Laplacian, by applying the Max-min principle, we can get the following result. Lemma 1.5. Domain monotonicity of eigenvalues with vanishing Dirichlet data also holds for the Dirichlet eigenvalues of the weighted Laplacian. Proof. The proof is almost the same with that of Domain monotonicity of eigenvalues with vanishing Dirichlet data (cf. [7, pp. 17-18]) and the only difference is that one needs to replace the normal Riemannian volume density dv by the weighted one e−ϕ dv, which leads to the fact that all the equalities and inequalities in the proof of Domain monotonicity of eigenvalues with vanishing Dirichlet data still hold. By Lemma 1.5, we know that λ1,p (B(q, )) decreases as increases and then it has a limit independent of the choice of the center q. So, one can make the following definition. Definition 1.6. The first eigenvalue of the weighted Laplacian Δϕ on M is defined by
λ1,ϕ (M) := lim λ1,ϕ (B(q, )) . →∞
For λ1,ϕ (M), we can prove the following conclusion. Theorem 1.7. Let (M, g, e−ϕ dv) be an n-dimensional complete noncompact smooth metric measure space with M being a submanifold in the hyperbolic m-space Hm (−1). Denote H by the mean curvature vector of M in Hm (−1). If H α for some constant α < n − 1 and ∇ϕ C for some constant C, where ∇ is the gradient operator on M, then
λ1,ϕ (M)
(n − 1 − α −C)2 > 0. 4
(1.8)
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Remark 1.8. If ϕ = constant, then the constant C can be chosen to be C = 0 and correspondingly the estimate (1.8) degenerates into the one in [2, Theorem 2] for λ1 (M). Therefore, [2, Theorem 2] is included by Theorem 1.8 as a special case.
2 Some useful facts In this section, we would like to recall several facts and also prove some new ones which will be used in the last section for proving estimates (1.4) and (1.8). Lemma 2.1. ([8, Lemma 2]) Let f ∈ C∞ (N m ) where N m is an m-dimensional Riemannian manifold and suppose Qn is an n-dimensional submanifold of N m . Denote by ∇, Δ the connection and Laplacian on N m respectively and Δ the Laplacian on Qn , then the following formula relating Δ to Δ holds: m 2 Δ ( f |Q ) = Δ f |Q + H, ∇ f |Q − ∑ ∇ f (ek , ek )|Q , k=n+1
where {ek : k = n + 1, · · · , m} is a local orthonormal frame field of the normal space of Qn in N m . Using Lemma 2.1 and together with the definition of the weighted Laplacian Δϕ mentioned in Section 1, we can easily obtain the following fact. Lemma 2.2. Under the assumptions of Lemma 2.1, we have Δϕ ( f |Q ) = Δ f |Q + H, ∇ f |Q −
m
∑
2
∇ f (ek , ek )|Q − g (∇ f |Q , ∇ϕ ) ,
k=n+1
Δϕ is the weighted Laplacian related to the weighted where g is the Riemannian metric of − ϕ n volume density e dv of Q , and ∇ is the gradient operator on Qn related to the metric g. Qn ,
Lemma 2.3. ([8, Lemma 3], [2, Corollary 4]) Let ∇ and Δ be the connection and Laplacian of Hm (−1) respectively, and let , be its metric tensor, then we have 2
∇ cosh r = cosh r · , and Δ cosh r = m cosh r, where r is the distance function on Hm (−1) measured from a fixed point in Hm (−1). By applying Lemmas 2.1 and 2.3, the following fact can be proven directly. Lemma 2.4. ([2, Lemmas 5 and 6]) Let M be an n-dimensional submanifold in the hyperbolic m-space Hm (−1), then we have Δ cosh r = n cosh r + H, ∇r |M · sinh r and
Δr = n − ∇r2 coth r + H, ∇r |M ,
where r is measured from a fixed point in Hm (−1) \ M.
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Combining Lemma 2.2 and Lemma 2.4, we can prove the following fact. Lemma 2.5. Assume that (M, g, e−ϕ dv) is an n-dimensional smooth metric measure space and M is a submanifold in the hyperbolic m-space Hm (−1), then we have Δϕ cosh r = n cosh r + H, ∇r |M · sinh r − sinh r · g (∇r, ∇ϕ ) and
Δϕ r = n − ∇r2 coth r + H, ∇r |M − g (∇r, ∇ϕ ) where r is measured from a fixed point in Hm (−1) \ M. Proof. Applying Lemmas 2.2, 2.3 and 2.4 to the function cosh r, we can obtain Δϕ (cosh r|M ) =
Δ cosh r |M + H, ∇ cosh r |M −
m
∑
2
∇ cosh r(ek , ek )|M − g (∇ cosh r, ∇ϕ )
k=n+1
= m cosh r + H, ∇r |M · sinh r − (m − n) cosh r − sinh r · g(∇r, ∇ϕ ) = n cosh r + H, ∇r |M · sinh r − sinh r · g(∇r, ∇ϕ ), which is the first result in Lemma 2.5. Combing the fact that Δϕ cosh r = Δ cosh r − g (∇ϕ , ∇ cosh r) = div(∇ cosh r) − g (∇ϕ , ∇ cosh r) = cosh r∇r2 + Δr · sinh r − g (∇ϕ , ∇ cosh r) , it is easy to get that n cosh r + H, ∇r |M · sinh r − sinh r · g(∇r, ∇ϕ ) = cosh r · ∇r2 + (sinh r)Δr − g (∇r, ∇ cosh r) , which is equivalent to sinh r · Δϕ r = n cosh r − cosh r · ∇r2 + H, ∇r |M · sinh r − sinh r · g (∇r, ∇ϕ ) . Then the second result in Lemma 2.5 follows.
3
Proofs of main results
By using the facts in Section 2, we can prove Theorems 1.3 and 1.7 as follows. Proof of Theorem 1.3. By (1.3) and Definition 1.2, we know that if one wants to get (1.4), it is sufficient to show that for any f ∈ C0∞ (M), the inequality
n−1−α p p | f | dv ∇ f p dv (3.1) p M M holds.
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Let r be the distance function measured from a fixed point in Hm (−1) \ M. We know that ∇r = 1 and hence ∇r 1. By Lemma 2.4, one has
Δr = n − ∇r2 coth r + H, ∇r |M n − 1 − H n − 1 − α. Denote by g the Riemannian metric of M. Taking f ∈ C0∞ (M) and using the above inequality, we have div(| f | p · ∇r) = g (∇(| f | p ), ∇r) + | f | p Δr −p| f | p−1 ∇ f + | f | p · (n − 1 − α ) ⎡ ∇ f p
p ⎤ p−1 p−1 ε | f | ε ⎦ + | f | p · (n − 1 − α ) + −p ⎣ p p p−1 = −
p ∇ f p p−1 | f | p + | f | p · (n − 1 − α ), − (p − 1) ε εp
(3.2)
where ε > 0 is a parameter determined later. Here the following Young’s inequality
p ∇ f p ε | f | p−1 p−1 ε p−1 p−1 ∇ f | f | ∇ f = ε | f | · + p ε p p−1 has been used in the second inequality of (3.2). Integrating (3.2) over M and using the divergence theorem, we can obtain p ∇ f p p p−1 + (p − 1)ε | f | dv | f | p · (n − 1 − α )dv, εp M M that is, M
p ∇ f p dv ε p n − 1 − α − (p − 1)ε p−1 | f | p dv.
(3.3)
M
p Consider the function F(ε ) := ε p n − 1 − α − (p − 1)ε p−1 , ε > 0. It is easy to check that F(ε ) gets its maximum at the value p−1
n−1−α p ε= . p Combing this fact with (3.3) yields p−1 p p−1 p
n−1−α p n − 1 − α p · p−1 n − 1 − α − (p − 1) ∇ f dv | f | p dv p p M M p
n−1−α | f | p dv, p M
p
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which is exactly (3.1). Then (1.4) follows. This completes the proof of Theorem 1.3.
Proof of Theorem 1.7. By (1.7) and Definition 1.6, we know that if one wants to get (1.8), it is sufficient to show that for any f ∈ C0∞ (M), the inequality (n − 1 − α −C)2 4
M
f 2 e−ϕ dv
M
∇ f 2 e−ϕ dv
(3.4)
holds. As in the proof of Theorem 1.3, for the metric measure space (M, g, e−ϕ dv), we choose r to be the distance function measured from a fixed point in Hm (−1) \ M. By Lemma 2.5, one has
Δϕ r = n − ∇r2 coth r + H, ∇r |M − g (∇r, ∇ϕ ) n − 1 − H · ∇r − ∇ϕ · ∇r n − 1 − α −C. Taking f ∈ C0∞ (M) and using the above inequality results in
div( f 2 ∇r · e−ϕ ) = e−ϕ g ∇ f 2 , ∇r + f 2 e−ϕ Δr − f 2 e−ϕ g(∇r, ∇ϕ )
= e−ϕ g ∇ f 2 , ∇r + f 2 e−ϕ Δϕ r −e−ϕ ∇ f 2 + f 2 e−ϕ · (n − 1 − α −C) e−ϕ −2| f | · ∇ f + f 2 · (n − 1 − α −C) ∇ f 2 −ϕ 2 2 + f · (n − 1 − α −C) , −ε f − e ε
(3.5)
where ε > 0 is a parameter determined later. Here the following inequality ∇ f 2 ε has been used. Integrating (3.5) over M and using the divergence theorem, we can obtain
∇ f 2 −ϕ 2 e dv εf + f 2 · (n − 1 − α −C)e−ϕ dv, ε M M 2| f | · ∇ f ε | f |2 +
that is,
∇ f 2 e−ϕ dv ε (n − 1 − α −C − ε )
M
f 2 e−ϕ dv.
(3.6)
M
On the other hand,
ε (n − 1 − α −C − ε ) with equality holds if and only if ε = M
2 −ϕ
∇ f e
ε + n − 1 − α −C − ε 2 n−1−α −C . 2
2 =
(n − 1 − α −C)2 4
Combining this fact with (3.6), we have
(n − 1 − α −C)2 dv 4
f 2 e−ϕ dv,
M
which is exactly (3.4). Then (1.8) follows. This completes the proof of Theorem 1.7.
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Acknowledgments This work was partially supported by the project sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, the NSF of China (Grant No. 11401131), and Hubei Key Laboratory of Applied Mathematics (Hubei University). The authors would like to thank the anonymous referee for his or her careful reading and valuable comments such that the article appears as its present version.
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