Eigenvalue estimates for submanifolds of warped product spaces

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(1)AperTO - Archivio Istituzionale Open Access dell'Università di Torino. Eigenvalue estimates for submanifolds of warped product spaces This is a pre print version of the following article: Original Citation:. Availability: This version is available. http://hdl.handle.net/2318/1693187. since. 2019-02-18T17:53:45Z. Published version: DOI:10.1017/S0305004113000443 Terms of use: Open Access Anyone can freely access the full text of works made available as "Open Access". Works made available under a Creative Commons license can be used according to the terms and conditions of said license. Use of all other works requires consent of the right holder (author or publisher) if not exempted from copyright protection by the applicable law.. (Article begins on next page). 30 July 2021.

(2) Under consideration for publication in Math. Proc. Camb. Phil. Soc.. 1. Eigenvalue estimates for submanifolds of warped product spaces By G.P. Bessa, S.C. Garc´ıa-Mart´ınez, L. Mari and H.F. Ramirez-Ospina Departamento de Matem´ atica, Universidade Federal do Cear´ a, 60455-760 Fortaleza-CE, Brazil, e-mail : bessa@mat.ufc.br, lucio.mari@libero.it Departamento de Matem´ aticas, Universidad de Murcia, Campus de Espinardo, 30100 Espinardo, Murcia, Spain, e-mail : scarolinagarciam@gmail.com, hectorfabian.ramirez@um.es (Received month revised month ) Abstract In this paper, we give lower bounds for the fundamental tone of open sets in minimal submanifolds immersed into warped product spaces of type N n ×f Qq , where f ∈ C ∞ (N ). This setting allows to deal, among others, with minimal submanifolds of pieces of cylinders, cones, spheres and pseudo-hyperbolic spaces, most of these examples being not covered by the previous literature. Applications also include the study of the essential spectrum of hyperbolic graphs over compact regions of the boundary at infinity.. 1. Introduction Let M be a connected Riemannian manifold, possibly incomplete, and let ∆ = div ◦ ∇ be the Laplace-Beltrami operator acting on Cc∞ (M ), the space of smooth functions with compact support. To study the spectrum of ∆, we shall fix a self-adjoint extension: hereafter, will always consider the Friedrichs extension of ∆, that is, the unique extension of (∆, Cc∞ (M )) whose domain lies in that of the closure of the associated quadratic form Z Q : ϕ ∈ Cc∞ (M ) 7−→ |∇ϕ|2 . M. We remark that, when M is geodesically complete, by a result in [17, 19, 30] ∆ is essentially self-adjoint, that is, the Friedrichs extension is indeed the unique self-adjoint extension of (∆, Cc∞ (M )). Denote with σ(−∆) and σess (−∆), respectively, the spectrum and the essential spectrum of −∆. Given an open subset Ω ⊂ M , the fundamental tone of Ω, λ∗ (Ω), is defined by ∫Ω |∇f |2 1 ; f ∈ H (Ω)\{0} . λ∗ (Ω) = inf σ(−∆) = inf 0 ∫Ω f 2 When Ω has compact closure and Lipschitz boundary, λ∗ (Ω) coincides with the first eigenvalue λ1 (Ω) of Ω, with Dirichlet boundary data on ∂Ω. Its associated eigenspace is 1-dimensional and spanned by any nontrivial solution u of ( ∆u + λ1 (Ω)u = 0 on Ω, u=0. on ∂Ω..

(3) 2. Bessa, Garc´ıa-Mart´ınez, Mari and Ramirez-Ospina. The relations between the fundamental tone of open sets of M and their geometric invariants has been the subject to an intensive research in the past 50 years. Among a huge literature, we limit ourselves to quote the classics [5, 6, 15] and references therein for a detailed picture. In particular, a great effort has been done to estimate the fundamental tone of minimal submanifolds of well-behaved ambient spaces (for instance, in [8, 9, 14, 16, 18]). In this paper, we move a step further by giving lower bounds for the fundamental tone of manifolds which are minimally immersed in ambient spaces N n ×f Qq carrying a warped product structure. As we shall see in the last section, the generality of our setting allows applications to submanifolds in pieces of cylinders, cones, tubes, spheres, also improving certain recent results in the literature ([7, 8, 9]). We remark that there have been an increasing interest in the study of minimal and constant mean curvature submanifolds in product spaces N × R, after the discovery of many beautiful examples such as those in [24, 25], and this motivates a thorough investigation of the spectrum of such submanifolds. To introduce our main result, Theorem 8 below, we shall need some preliminary material, and we therefore prefer to postpone the statement of Theorem 8 after some definitions and a brief overview of basic results. Section 2 also contains a key technical lemma, Lemma 3 below. The statement and proof of Theorem 8 will appear in Section 3, and the final Section 4 is devoted to applications. 2. Preliminaries Isometric immersions Let M and W be smooth Riemannian manifolds of dimension m and n + q respectively and ϕ : M ,→ W be an isometric immersion. Consider a smooth function F : W → R and the composition F ◦ ϕ : M → R. Identifying X with dϕ(X), the Hessian of F ◦ ϕ at x ∈ M is given by Hess M (F ◦ ϕ)(x) (X, Y ) = Hess W F (ϕ(x)) (X, Y ) + h∇F, σ(X, Y )iϕ(x) ,. (2·1). where σ(X, Y ) is the second fundamental form of ϕ. Tracing (2·1) with respect to an orthonormal basis {e1 , . . . em }, ( ) m m X X ∆M (F ◦ ϕ)(x) = Hess W F (ϕ(x)) (ei , ei ) + h∇F, σ(ei , ei )i i=1. =. m X. i=1. Hess W F (ϕ(x)) (ei , ei ) + mh∇F, Hi,. (2·2). i=1. where H = m−1 tr(σ) is the normalized mean curvature vector. Formulae (2·1) and (2·2) are well known in the literature, see [22]. Models and Hessian comparisons + 2 Hereafter, we denote with R+ 0 = [0, +∞). Let g ∈ C (R0 ) be positive in (0, R0 ), for some 0 < R0 ≤ ∞, and satisfying g(0) = 0,. g 0 (0) = 1.. The κ-dimensional model manifold Qκg constructed from the function g is just the ball BR (o) ⊆ Rκ with metric given, in polar geodesic coordinates centered at o, by ds2g = dr2 + g(r)2 h , iSκ−1 ,.

(4) Eigenvalue estimates for submanifolds. 3. where h , iSκ−1 is the standard metric on the unit (κ − 1)-sphere. The radial sectional curvature and the Hessian of the distance function r on Qκg are given by the expressions g 0 (r) 2 g 00 (r) ds − dr ⊗ dr . , Hess r = K rad = − g(r) g(r) From the first relation, we see that a model can, equivalently, be specified by prescribing its radial sectional curvature G ∈ C ∞ (R+ 0 ) and recovering g as the solution of ( g 00 − Gg = 0, (2·3) g(0) = 0, g 0 (0) = 1, on the maximal interval (0, R0 ) where g > 0. For future use, we will denote with G− the negative part of G, i.e. G− = max{0, −G}. For the proof of our main results we will make use of the following version of the Hessian Comparison Theorem, see [21] and [27, Chapter 2]. Theorem 1. Let Qq be a complete Riemannian q-manifold. Fix a point o ∈ Q, denote by ρQ (x) the Riemannian distance function from o and let Do = Q\cut(o) be the domain of the normal geodesic coordinates centered at o. Given G ∈ C ∞ (R+ 0 ), let g be the solution of the Cauchy problem (2·3), and let (0, R0 ) ⊆ [0, +∞) be the maximal interval where g is positive. If the radial sectional curvature of Q satisfies rad KQ ≤ −G(ρQ ). rad (respectively, KQ ≥ −G(ρQ )),. (2·4). on B(o, R0 ), then Hess Q ρQ ≥. g 0 (ρQ ) h , iQ− dρQ ⊗ dρQ g(ρQ ). (respectively, ≤). on Do ∩ B(o, R0 )\{o}, in the sense of quadratic forms. Eigenvalues and Eigenfunctions The generalized version of Barta’s Eigenvalue Theorem [4], proved in [9] will be important in the sequel. Theorem 2. Let Ω be an open set in a Riemannian manifold M and let f ∈ C 2 (Ω), f > 0 on Ω. Then ∆f λ∗ (Ω) ≥ inf − . (2·5) Ω f We recall that, given a model Qκg with g > 0 on (0, R0 ), and given R ∈ (0, R0 ), the first eigenfunction v of the geodesic ball Bg (R) centered at o is radial. This can be easily seen by proving that its spherical mean Z 1 v¯(r) = v g(r)κ−1 ∂Bg (R) is still an eigenfunction associated to λ1 (Bg (R)) and using the fact that the space of first eigenfunctions has dimension 1. With a slight abuse of notation, we can thus identify the first eigenfunction v ∈ C ∞ (Bg (R)) of Bg (R) with the solution v : [0, R] → R of  0  v 00 + (κ − 1) g v 0 + λ1 (Bg (R))v = 0 on (0, R), g (2·6)  v(0) = 1, v 0 (0) = 0, v(R) = 0, v > 0 on [0, R)..

(5) 4. Bessa, Garc´ıa-Mart´ınez, Mari and Ramirez-Ospina. Note that, multiplying the ODE by g κ−1 , integrating and using the initial condition, one can easily argue that v 0 < 0 on (0, R]. We will need the following technical lemma, which extends a result due to Bessa-Costa, see [7, Lemma 2.4]. Lemma 3. Let Qκg be a model manifold with radial sectional curvature −G(r), and suppose that g 0 > 0 on [0, R). Let v ∈ C 2 (Bg (R)) be a first positive eigenfunction of Bg (R) ⊂ Qκg . If λ1 (Bg (R)) ≥ κkG− kL∞ ([0,R]) .. (2·7). Then the following inequality holds: g 0 (t) 0 v (t) + λ1 (Bg (R))v(t) ≤ 0, t ∈ (0, R]. (2·8) g(t) Proof. For simplicity of notation, we denote by λ = λ1 (Bg (R)). Multiplying (2·6) by g κ−1 we deduce that v(t) satisfies the following differential equation: ( κ−1 0 0 (g v ) + λg κ−1 v = 0 on (0, R), (2·9) v(0) = 1, v 0 (0) = 0, v(R) = 0, v > 0 on [0, R). κ. Our aim is to deduce (2·8) via some modified Sturm-type arguments. In order to do so, we search for a positive function µ solving g 0 (t) + λµ(t) = 0 on (0, R). g(t) Z λ t g(s) ds is a solution, giving Integrating, we get that log µ(t) = − κ 0 g 0 (s)   Z λ t g(s) −  ds κ 0 g 0 (s) µ(t) = e . κµ0 (t). (2·10). The above expression is well defined since g 0 > 0 on [0, R). Since µ0 (t) = −. λ g(t) µ(t) κ g 0 (t). we deduce .  −. =− From (2·11) we see that κ. 1 g(t) e κ g 0 (t). λ κ. t. Z. λ − λ g(t) κ µ0 (t)v(t)− v 0 (t)µ(t) = − 0 e κ g (t). 0. Z 0. t. . . λ g(s) − ds 0 g (s) κ v(t)−v 0 (t)e. Z 0. t. . g(s) ds g 0 (s). . g(s) ds 0 g (t) 0 g 0 (s) κ v (t) + λv(t) . g(t). (2·11). g 0 (t) 0 v (t) + λv(t) ≤ 0 on (0, R) if and only if g(t). µ0 (t)v(t) − v 0 (t)µ(t) ≥ 0. on (0, R),. and we are going to prove this last inequality. Differentiating (2·10) and multiplying by (1/κ) both sides of the equality, we have " # 0 2 g 0 (t) g (t) λ 0 00 µ (t) + µ (t) G(t) − + = 0, g(t) g(t) κ.

(6) Eigenvalue estimates for submanifolds. 5. that is, " # 0 2 g (t) λ g(t) G(t) − + . µ (t) = −µ (t) 0 g (t) g(t) κ 00. g(t) λ Since µ (t) 0 = − µ(t) g (t) κ 0. 0. . g(t) g 0 (t). 2. we can rewrite µ00 (t) in the following way:. " 2 2 # λ g(t) λ g(t) µ (t) = µ(t) G(t) −1+ . κ g 0 (t) κ g 0 (t) 00. Multiplying the above equation by g κ−1 (t), and then adding and subtracting the term (κ − 1)g κ−2 (t)g 0 (t)µ0 (t), we obtain # " 2 2 g(t) λ G(t) g(t) κ−1 0 0 κ−1 − 2 +1 . (2·12) (g µ ) (t) = −λg (t)µ(t) − κ g 0 (t) κ g 0 (t) Next, we multiply (2·12) by v(t) and (2·9) by −µ(t), and we add them to get 2 g(t) λ λ G(t) + . (g κ−1 µ0 )0 (t)v(t) − (g κ−1 v 0 )0 (t)µ(t) = g κ−1 (t)µ(t)v(t) κ g 0 (t) κ Integrating from 0 to t gives g κ−1 (µ0 v − v 0 µ) (t) =. Z 0. t. λ κ−1 g (s) κ. . g(s) g 0 (s). 2 λ G(s) + µ(s)v(s)ds. κ. (2·13). Now, from (2·7) we deduce that 2 g(t) λ λ κ−1 g (t) G(t) + µ(t)v(t) ≥ 0, κ g 0 (t) κ whence µ0 (t)v(t) − v 0 (t)µ(t) ≥ 0 for t ∈ (0, R), as claimed. Remark 4. It is important to find conditions to ensure the lower bound (2·7). For instance, if −G(r) = B 2 , where B is a positive constant, then the solution gB of (2·3) is gB (r) = B −1 sin(Br),. gB0 > 0. thus Qκg B. κ. on [0, π/(2B)).. (2·14). 2. The function gB yields the model manifold = S (B ), the κ-dimensional sphere 2 of constant sectional curvature B and diameter diamSκ (B 2 ) = π/B. Note that the first eigenvalue of the geodesic ball of Sκ (B 2 ) of radius R = π/2B is λ1 (BSκ (B 2 ) (π/2B)) = κB 2 and v(r) = cos(Br) is its first eigenfunction. When −G(r) ≤ B 2 and R ≤ π/(2B), by Sturm’s argument a solution g of (2·3) satisfies h π g0 g0 ≥ B > 0 on 0, . g gB 2B By Cheng’s Comparison Theorem (version proved by Bessa-Montenegro in [10]), λ1 (Bg (R)) ≥ λ1 (BgB (R)),. h π R ∈ 0, . 2B. In order to get λ1 (Bg (R)) ≥ κkG− kL∞ ([0,R)) it is sufficient to have λ1 (BgB (R)) = λ1 (BSκ (B 2 ) (R)) ≥ κkG− kL∞ ([0,R)) .. (2·15).

(7) 6. Bessa, Garc´ıa-Mart´ınez, Mari and Ramirez-Ospina. e On the other hand, we can see κkG− kL∞ ([0,R)) as a first eigenvalue of a ball of radius R 2 e in a κ-dimensional sphere of sectional curvature B , i.e. e κkG− kL∞ ([0,R)) = λ1 (BSκ (Be2 ) (R)), π e= p e 2 = kG− kL∞ ([0,R)) . where R and B 2 kG− kL∞ ([0,R)) We then conclude that the inequality (2·15) holds whenever π · R≤ p 2 kG− kL∞ ([0,R)) Remark 5. We remark that if Z ∞ 1 t G− (s)ds ≤ 4 t. for every t ∈ R+ ,. then both g and g 0 are strictly positive on R+ . This criterion has been proved in [13, Prop. 1.21]. A preliminary computation. From now on, we will consider the case when the ambient space is a warped product W n+q = N ×f Q of two Riemannian manifolds (N n , h, iN ) and (Qq , h, iQ ), with the Riemannian metric on W given by hh , ii = h , iN + f 2 h , iQ for some smooth positive function f : N → R+ . We fix the index convention 1 ≤ j, k ≤ n,. n + 1 ≤ α, β ≤ n + q.. For (p, q) ∈ W , we choose a chart (U, ψ) on N around p, with coordinate tangent basis {∂j } = {∂/∂ψj }, and a chart (V, φ) on Q around q, with basis {∂α } = {∂/∂φα }. Then, with respect to the product chart (U × V, ψ × φ) around (p, q), the Hessian of F at (p, q) has components  Hess W F (∂j , ∂κ ) = Hess N F (∂j , ∂κ ),      1 Hess W F (∂j , ∂α ) = ∂j ∂α F − ∂j f ∂α F, f  E   1D N   Hess W F (∂α , ∂β ) = Hess Q F (∂α , ∂β ) + ∇ f, ∇F hh∂α , ∂β ii , f N. (2·16). where Hess NF and Hess QF mean respectively Hess (F ◦ iN ) and Hess (F ◦ iQ ) and the inclusions are given by iN : (N, h, iN ) → N ×f {q} ⊆ N ×f Q,. x 7→ (x, q),. iQ : (Q, h, iQ ) → {p} ×f Q ⊆ N ×f Q,. y 7→ (p, y).. From (2·16) we observe that if F (p, q) = f (p) · h(q), where f is the warping function and h : Q → R is a smooth function on Q, then Hess W F has a block structure, that is Hess W F (X, Z) = 0 ∀X ∈ T(p,q) N ×f {q} , Z ∈ T(p,q) {p} ×f Q . More precisely, we have the following result. Lemma 6. Let F ∈ C ∞ (N ×f Q) be given by F (p, q) = f (p) · h(q), where h ∈ C ∞ (Q)..

(8) Eigenvalue estimates for submanifolds. 7. Then  Hess W F (X, Y ) = hHess N f (X, Y ),     Hess W F (X, Z) = 0,

(9) N

(10) 2 

(11) ∇ f

(12)   N  Hess W F (Z, W ) = f Hess Q h(Z, W ) + h hhZ, W ii , f for every X, Y ∈ T(p,q) N ×f {q} and Z, W ∈ T(p,q) {p} ×f Q .. (2·17). 3. Main results m. n. q. Let ϕ : M → N ×f Q , m > n, be a minimal immersion. Hereafter, we shall require the following Assumption 7. Suppose that the radial sectional curvature of Q satisfies rad KQ ≤ −G(ρQ ), where G ∈ C ∞ (R+ 0 ),. where ρQ (x) = distQ (o, x). We assume that the solution g of (2·3) is positive and g 0 > 0 on [0, R), and that BQ (o, R) ⊆ Q\cut(o). Let v : Bg (R) → R be the first eigenfunction of the ball Bg (R) ⊂ Qm−n . As remarked, g up to normalizing and possibly changing sign v > 0 on Bg (R), v is radial and solves  0  v 00 (t) + (m − n − 1) g (t) v 0 (t) + λ1 (Bg (R))v(t) = 0, t ∈ (0, R) g(t) (3·1)  v(0) = 1, v(R) = 0, v > 0 on [0, R), v 0 < 0 on (0, R]. Observe that, when m = n + 1, the equation simply becomes v 00 (t) + λ1 (Bg (R))v(t) = 0. Theorem 8. Let ϕ : M m → N n ×f Qq be an m-dimensional submanifold minimally immersed into the warped product space (N n ×f Qq , hh , ii), where hh , ii = h , iN + f 2 h , iQ , 0 < f ∈ C ∞ (N ), Q satisfies Assumption 7 and m > n. Suppose that the warping function f satisfies

(13) N

(14) 2

(15) ∇ f

(16) N Hess N f (·, ·) − h, iN ≤ 0. (3·2) f Let U ⊆ N be an open subset, and let Ω ⊂ ϕ−1 (U ×f BQ (o, R)) be a connected component. Then, if R is such that π R≤ p (3·3) 2 kG− kL∞ ([0,R)) the following estimate holds: . ∗. λ (Ω) ≥ inf. p∈U. λ1 (Bg (R)) − m |∇N f |2N (p) |f (p)|2. ,. (3·4). where Bg (R) is the geodesic ball of radius R in the model manifold Qgm−n or the interval [−R, R] if m = n + 1..

(17) 8. Bessa, Garc´ıa-Mart´ınez, Mari and Ramirez-Ospina. Proof. We start defining F : U ×f BQ (o, R) → R by F (p, q) = f (p) · h(q), where h ∈ C ∞ (BQ (o, R)) is given by h(q) = (v ◦ ρQ )(q) and v ∈ C ∞ ([0, R]) is the solution of (3·1). By Theorem 2, we have that ∆(F ◦ ϕ) λ∗ (Ω) ≥ inf − . Ω F ◦ϕ. (3·5). We are going to give a lower bound for −∆(F ◦ϕ)/(F ◦ϕ). Let x ∈ Ω and let {e1 , . . . , em } be an orthonormal basis for Tx Ω. Let ϕ(x) = (p(x), q(x)), t(x) = ρQ (q(x)) and denote by PN : T(p,q) N ×f Q → T(p,q) N ×f {q} and PQ : T(p,q) N ×f Q → T(p,q) {p} ×f Q the orthogonal projections onto the tangent spaces of the two fibers. Then, by (2·2) and the minimality of M , the Laplacian of F ◦ ϕ at x has the expression ∆ (F ◦ ϕ)(x) = =. m X. Hess W F (ϕ(x))(ei , ei ). i=1 m h X. Hess W F (ϕ(x))(PNei , PNei ) + Hess W F (ϕ(x))(PQei , PQei ). i. i=1. where W = N ×f Q. Using Lemma 6, and writing t = t(x) for the ease of notation, we deduce ∆ (F ◦ ϕ)(x) = v(t). m X. Hess N f (PNei , PNei )(p) + f (p). Hess Q v(t)(PQei , PQei ). i=1. i=1. + v(t). m X.

(18) N

(19) 2

(20) ∇ f

(21). N. f. (p). m X. PQei , PQei .. (3·6). i=1. n } be an Let {E1 , . . . , E orthonormal basis for Tp N , and consider the tangent basis n+q n+q ∂/∂ρQ , {∂/∂θγ }γ=n+2 , associated to normal coordinates at Q. Then the set {ξl }l=1 given by. ξj = Ej ∀j = 1, . . . , n,. ξn+1 =. 1 ∂ , f ∂ρQ. ξγ =. 1 ∂ ∀γ = n + 2, . . . , n + q f ∂θγ. is an orthonormal basis of T(p,q) N ×f Q . So, we can write ei as a linear combination of vectors of this basis in the following way:. ei =. n X. aji · ξj + bi · ξn+1 +. n+q X. cγi · ξγ ,. γ=n+2. j=1. for constants aji , bi , cγi satisfying n X. (aji )2 + b2i +. n+q X. (cγi )2 = 1, ∀i = 1, . . . , m.. γ=n+2. j=1. From ∇Q v(t) = v 0 (t). ∂ , ∂ρQ. Hess Q v(t) = v 0 (t)Hess Q ρQ + v 00 (t)dρQ ⊗ dρQ ,. (3·7).

(22) Eigenvalue estimates for submanifolds. 9. we can rewrite (3·6) in the following way: m X. m h X ∂ , PQei PQei (v 0 (t)) ∂ρQ Q i=1 i=1

(23) N

(24) 2 m i

(25) ∇ f

(26) X. N + v 0 (t)Hess Q ρQ (PQei , PQei ) + v(t) (p) PQei , PQei f i=1 !

(27) N

(28) 2 m

(29) ∇ f

(30) X N 1 − hhPNei , PNei ii (p) =v(t) Hess N f (PNei , PNei ) + f i=1 ! n+q m m X X X γ 2 1 ∂ ∂ 00 2 0 + v (t) , . bi + v (t) (ci ) Hess Q ρQ f (p) ∂θγ ∂θγ i=1 i=1 γ=n+2. ∆(F ◦ ϕ)(x) =v(t). Hess N f (PNei , PNei )(p) + f (p). Using (3·2) and the fact that v is positive we have m X 1 h mv(t)|∇N f |2N (p) + v 00 (t) b2i f (p) i=1 i n+q m X X γ ∂ ∂ 2 0 , . (ci ) Hess Q ρQ + v (t) ∂θγ ∂θγ i=1 γ=n+2. −∆(F ◦ ϕ)(x) ≥ −. Since v 0 (t) ≤ 0, we can apply the Hessian Comparison Theorem, to obtain −∆(F ◦ ϕ)(x) ≥ −. m

(31)

(32) 2 X 1 h

(33)

(34) b2i mv(t)

(35) ∇N f

(36) (p) + v 00 (t) f (p) N i=1. +v 0 (t). n+q m g 0 (t) X X γ 2 i (c ) g(t) i=1 γ=n+2 i. m m X n m X X g 0 (t) 1 h 00 X 2 v (t) bi + v 0 (t) m− (aji )2 − b2i f (p) g(t) i=1 i=1 j=1 i=1

(37)

(38) 2 i

(39) N

(40) + mv(t)

(41) ∇ f

(42) (p). =−. N. where the last equality follows by an algebraic manipulation that uses (3·7) summed for i = 1, . . . , m. Now, by a simple rearranging, ! m X g 0 (t) 1 h 00 0 00 2 −∆(F ◦ ϕ)(x) ≥ − v (t) + (m − n − 1)v (t) − v (t) 1− bi f (p) g(t) i=1 ! m X n m

(43)

(44) 2 i X X g 0 (t)

(45)

(46) 0 j 2 2 n− (ai ) +1 − bi +mv(t)

(47) ∇N f

(48) (p) . +v (t) g(t) N i=1 j=1 i=1. From (3·1) we get

(49) 2

(50)

(51)

(52) λ1 (Bg (R)) −m

(53) ∇N f

(54) (p) N " ! !# m m X n m X X X g 0 (t) 1 + v 00 (t) 1− b2i −v 0 (t) n− (aji )2 +1 − b2i . f (p) g(t) i=1 i=1 j=1 i=1. −∆(F ◦ ϕ)(x) ≥. v(t) f (p). . We claim that the last line of (3·8) is nonnegative, that is,   ! m m X n m 0 X X X g (t)  v 00 (t) 1 − b2i − v 0 (t) n− (aji )2 + 1 − b2i  ≥ 0. g(t) i=1 i=1 j=1 i=1. (3·8). (3·9).

(55) 10. Bessa, Garc´ıa-Mart´ınez, Mari and Ramirez-Ospina. To prove this, we substitute v 00 (t) = −(m − n − 1)v 0 (t). g 0 (t) − λ1 (Bg (R))v(t) in (3·9) to g(t). get v 00 (t) 1 −. m X. ! b2i. i=1.   m X n m 0 X X g (t) n − − v 0 (t) (aji )2 + 1 − b2i  = g(t) i=1 j=1 i=1. ! m X g 0 (t) 2 + λ1 (Bg (R))v(t) 1− − (m − n)v (t) bi g(t) i=1   m X n 0 X g (t)  n− −v 0 (t) (aji )2  , g(t) i=1 j=1 . 0. (3·10). so that (3·9) is equivalent to show that ! m X g 0 (t) 2 + λ1 (Bg (R))v(t) 1− bi − (m − n)v (t) g(t) i=1   m X n 0 X g (t)  −v 0 (t) n− (aji )2  ≥ 0. g(t) i=1 j=1 . 0. (3·11). Now, in our assumption (3·3), by Remark 4 it holds λ1 (Bg (R)) ≥ (m − n)kG− kL∞ ([0,R]) . Hence, applying Lemma 3 we infer that g 0 (t) + λ1 (Bg (R))v(t) ≤ 0. g(t) Pm Moreover, it is clear that 1 − i=1 b2i ≥ 0, and finally we observe the inequality ! m X n n m n n n X X X X X X 2 2 j 2 (ai ) = hhei , ξj ii = |PM ξj | ≤ |ξj | = 1 = n, (m − n)v 0 (t). i=1 j=1. j=1. i=1. j=1. j=1. j=1. where PM is the projection on M . Keeping in mind that v 0 ≤ 0, this concludes the proof of the claimed (3·11). From (3·8) we have

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(57) 2 ∆(F ◦ ϕ) 1 (x) ≥ 2 λ1 (Bg (R)) − m

(58) ∇N f

(59) N (p) . − (3·12) F ◦ϕ f (p) Therefore, by (3·5) we conclude the desired (3·4). Remark 9. In the case N = R, we observe that the mean curvature function of the fibers {p} ×f Q is given by H(y) = f 0 (y)/f (y). Therefore, condition (3·2) is equivalent to f H0 ≤ 0, that is, H0 ≤ 0. There exists a large class of functions for which H0 ≤ 0. For instance, f (y) = constant, f (y) = y and f (y) = ecy , where c ∈ R. 4. Applications To show the generality of Theorem 8, we conclude this paper with a number of different examples, and we discuss the sharpness of the estimates produced..

(60) Eigenvalue estimates for submanifolds. 11. 4·1. Balls In the limiting case when N is a point, computations hold with the obvious simplifications and we get the following result, which is basically known (see QUOTATION!!) Corollary 10. Let ϕ : M m → Qq be an m-dimensional submanifold minimally immersed into Qq . Suppose that Q satisfies Assumption 7. Let Ω ⊂ ϕ−1 (BQ (o, R)) be a connected component with π · R≤ p 2 kG− kL∞ ([0,R)) Then λ∗ (Ω) ≥ λ1 (Bgm (R)). Here Bgm (R) is a geodesic ball of radius R in an m-dimensional model manifold Qm g . For instance, a minimal submanifold ϕ : M m → Rq in a ball of radius R has fundamental tone greater or equal than that of the flat m-ball of radius R: c 2 m m λ∗ (M ) ≥ λ1 (BR )= , R where cm is the first zero of the Jm/2−1 -Bessel function. Similarly, the first eigenvalue of a minimal submanifold ϕ : M m → Sq+ contained in half of a unit sphere is greater or equal than that of the upper half m-sphere: λ∗ (M ) ≥ λ1 BSm (π/2) = m. We underline that, although the sphere is well studied, the first eigenvalue λ1 (BSm (r)) for r 6∈ {π/2, π} is still pretty much unknown. Estimates for spherical cups have been developed in [1, 28, 29] for dimension two, [20] for dimension three and [2, 3, 12] for all dimensions. 4·2. Cylinders A very similar situation occurs when the ambient space is a cylinder R×Qq . Considering f = 1 and N = R in Theorem 8 we obtain a generalized version of Theorem 1.1 of [7]. Corollary 11. Let ϕ : M m → R × Qq be an m-dimensional submanifold minimally immersed into R × Qq with Q satisfying the Assumption 7. Let Ω ⊂ ϕ−1 (R × BQ (o, R)) be a connected component with π · R≤ p 2 kG− kL∞ ([0,R)) Then λ∗ (Ω) ≥ λ1 (Bgm−1 (R)).. (4·1). Here, Bgm−1 (R) is a geodesic ball of radius R in an (m − 1)-dimensional model manifold Qm−1 . g Note that, differently from Corollary 10, here the comparison ball in estimate (4·1) has dimension m − 1 and not m. In particular, when Qq = Rq in the last corollary we get the following result in the Euclidean space proved by Bessa and Costa in [7]. Corollary 12. Let ϕ : M m → Rq+1 be an m-dimensional submanifold minimally.

(61) 12. Bessa, Garc´ıa-Mart´ınez, Mari and Ramirez-Ospina. immersed into Rq+1 . Let Ω ⊂ ϕ−1 (R × BRq (o, R)) be a connected component. Then 2 c m−1 . (4·2) λ∗ (Ω) ≥ λ1 (BRm−1 (o, R)) = R Here cm−1 is the first zero of the J(m−1)/2−1 -Bessel function. 4·3. Pseudo-hyperbolic and hyperbolic spaces The pseudo-hyperbolic spaces, introduced by Tashiro in [31], are warped products R ×f Qq with (i) f (y) = aeby ,. or. (ii) f (y) = a cosh(by),. for some constants a, b > 0. In the case (i), we observe that condition (3·2) is satisfied, as it shows ( 0 in case (i), (f 0 )2 00 f − = 2 f ab / cosh(by) > 0 in case (ii). We state the following corollary in the case f (y) = eby . Corollary 13. Let ϕ : M m → R×eby Qq be an m-dimensional submanifold minimally immersed into R ×eby Qq , with Q satisfying Assumption 7. Let Ω ⊂ ϕ−1 (α, β) ×eby BQ (o, R) be a connected component with π R≤ p · 2 kG− kL∞ ([0,R)) Then, λ∗ (Ω) ≥. λ1 (Bgm−1 (R)) − mb2 . e2bβ. (4·3). . Here Bgm−1 (R) is the geodesic ball of (m − 1)-dimensional model space Qm−1 g Foliating through horospheres, we can represent the hyperbolic space Hq+1 as the warped product R ×ey Rq . By Corollary 13 we have the following eigenvalue estimate. Corollary 14. Let ϕ : M m → Hq+1 be an m-dimensional submanifold minimally immersed into Hq+1 . Let Ω ⊂ ϕ−1 ((−∞, β) ×ey BRq (o, R)) be a connected component. Then 2 λ1 (BRm−1 (o, R)) −2β cm−1 λ∗ (Ω) ≥ − m = e − m, (4·4) e2β R where cm−1 is the first zero of the J(m−1)/2−1 -Bessel function. 4·4. Cones A (q + 1)-dimensional cone C q+1 (Q) ⊆ Rm over an open subset Q ⊂ Sq can be seen as the warped product C q+1 (Q) = (0, +∞) ×f Q where f (y) = y. In order to match with Assumption 7 we shall suppose that Q ⊂ BSq (o, R) for some R ≤ π/2. More generally, we can consider cones C q+1 (Q) over open subsets Q ⊂ W of Riemannian manifolds W , with Q satisfying Assumption 7. We have the following result. Corollary 15. Let ϕ : M m → C q+1 (Q) be a m-dimensional submanifold minimally immersed into C q+1 (Q) with Q satisfying the Assumption 7. Let Ω ⊂ ϕ−1 (0, a) ×y.

(62) Eigenvalue estimates for submanifolds. 13. BQ (o, R) be a connected component with π R≤ p · 2 kG− kL∞ ([0,R)) Then, λ∗ (Ω) ≥. 1 λ1 (Bgm−1 (R)) − m , 2 a. (4·5). where Bgm−1 (R) is the geodesic ball of radius R in the model manifold Qm−1 . g Exploiting the warped product structure of the sphere Sq+1 = (0, π)×sin y Sq , our estimate can be applied for minimal submanifolds in sectors of Sq+1 which are different from spherical cups, leading to the next Corollary 16. Let ϕ : M m → Sq+1 = (o, π) ×sin y Sq be an m-dimensional submanifold minimally immersed into Sq+1 . Let Ω ⊂ ϕ−1 ((o, r) ×sin y BSq (θ)), θ < π/2 be a connected component. Then  λ1 (BSm−1 (θ)) − m   if r ≤ π/2,  (sin r)2 ∗ (4·6) λ (Ω) ≥    if r ≥ π/2. λ1 (BSm−1 (θ)) − m These estimates are effective when θ is small, that is, when the minimal submanifold is contained in a small slice of the sphere. In particular, in this case they improve on the estimates in Corollary 10 when both are applicable. 4·5. Essential spectrum The ideas developed above can be applied to study the essential spectrum of −∆ of submanifolds properly immersed into the hyperbolic spaces with fairly weak bounds on the mean curvature vector. Via Persson formula ([26] and [11, Prop. 3.2]), one can express the bottom of the essential spectrum of −∆ as follows: for every exhaustion of M by relatively compact open sets {Kj } with Lipschitz boundary, inf σess (−∆) = lim λ∗ (M \Kj ). j→+∞. (4·7). It therefore follows that −∆ has purely discrete spectrum if and only if lim λ∗ (M \Kj ) = ∞.. j→+∞. Our next application regards the essential spectrum of graph hypersurfaces of Hq+1 whose boundary lies in a relatively compact region of Hq∞ , the boundary at infinity of Hq+1 . Corollary 17. Consider the upper half-space model of the hyperbolic space Hq+1 , q ≥ 2, with coordinates (x0 , x1 , . . . , xq ) = (x0 , x ¯) and metric 1 h , i = 2 dx20 + dx21 + . . . + dx2q , x0 and let Hq∞ be its boundary at infinity, with chart x ¯. Consider a hypersurface without q q+1 boundary ϕ : M → H that can be written as the graph of a function u over a relatively compact, open set W ⊆ Hq∞ , and denote with H(¯ x) its mean curvature. For z > 0, define

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(64). Hz = sup

(65) H(¯ x)

(66) : x ¯ ∈ W, u(¯ x) = z.

(67) 14. Bessa, Garc´ıa-Mart´ınez, Mari and Ramirez-Ospina. If lim z 2 Hz = 0,. z→0. (4·8). then M has purely discrete spectrum. Proof. Setting y = log x0 , we can rewrite the metric on Hq+1 as the one of the warped product R ×ey Rq . In our assumptions, since M has no boundary and is a graph over W it holds y(ϕ(x)) → −∞ as x diverges in M q . We identify the factor Rq in the warped product structure with Hq∞ endowed with the Euclidean metric, we fix an origin o ∈ Hq∞ and we let R be large enough that W ⊂ BRq (o, R). Let {zj } ↓ 0+ be a chosen sequence, set βj = log zj ↓ −∞ and define Kj = ϕ−1 (βj , +∞) × W , Ωj = M \Kj . In our assumptions, Kj is relatively compact for every j and {Kj } is a smooth exhaustion of M . Consider a positive first eigenfunction v of the geodesic ball BRq−1 (o, 2R), with the normalization kvkL∞ = 1. Define F : (−∞, βj ) ×ey BRq (o, 2R) → R as F (y, p) = ey · h(p), where h(p) = v(ρRq (p)). By Theorem 2 and formula (2·2), −∆(F ◦ ϕ) F ◦ϕ M \Kj " q # X 1 = inf − Hess Hq+1 F (ϕ(x)) (ei , ei ) + qh∇F, Hi . F ◦ ϕ i=1 M \Kj. λ∗ (M \ Kj ) ≥ inf. The proof of Theorem 8, in particular inequality (3·12), shows that, for x ∈ M \Kj , q. −. 1 X λ1 (BRq−1 (o, 2R)) Hess Hq+1 F (ϕ(x)) (ei , ei ) ≥ − q, F ◦ ϕ i=1 e2y(x). therefore, on M \ Kj , −. ∆(F ◦ ϕ) λ1 (BRq−1 (o, 2R)) |∇F | (x) ≥ − q − q |H| (ϕ(x)). 2y(x) F ◦ϕ F e. On the other hand, ∇F = F ∇y + ey ∇h and thus |∇F |/F ≤ 1 + |∇h|/h. Since 1 ≥ h > 0 on BRq−1 (o, R), we infer that sup BRq−1 (o,R). |∇F | ≤ C(R), F. where C(R) = 1 +. |∇h| > 0. BRq−1 (o,R) h sup. From the above, we have . ∗. λ (M \ Kj ) ≥ inf. M \Kj. λ1 (BRq−1 (o, 2R)) − qC(R)|H(x)|e2y(x) − qe2y(x) . e2y(x). In our assumptions, on M \Kj , 2 |H(x)|e2y(x) ≤ Hx0 (x) e2y(x) = Hx0 (x) x0 (x) .. (4·9).

(68) Eigenvalue estimates for submanifolds. 15. By (4·8), this latter goes to zero uniformly for x ∈ M \Kj and divergent j. In particular, for each fixed ε > 0, there exists jε large such that, for j ≥ jε , |H(x)|e2y(x) ≤ ε on M \Kj . It therefore follows that, for j large enough, λ1 (BRq−1 (o, 2R)) − qC(R)ε − qx0 (x)2 . λ∗ (M \ Kj ) ≥ inf x0 (x)2 M \Kj Choosing ε sufficiently small, letting j → +∞ and using that x0 (x)2 ≤ e2βj → 0+ for x ∈ M \Kj and divergent j, we deduce that λ∗ (M \Kj ) → +∞, and the claim follows by Persson formula. To conclude, we consider the essential spectrum of submanifolds satisfying some strong non-properness assumption. This includes submanifolds with bounded image immersed in a complete manifold. We begin with recalling the following Definition 18. Let M , W be Riemannian manifolds and let ϕ : M → W be an isometric immersion. The limit set of ϕ, denoted by lim ϕ, is a closed set defined as follows . lim ϕ = p ∈ W ; ∃ {pk } ⊂ M, distM (o, pk ) → ∞ and distW (p, ϕ(pk )) → 0 . Observe that: • An isometric immersion ϕ : M → W is proper if and only if lim ϕ = ∅. • The closure of the set ϕ−1 [W \ T (lim ϕ)] may not be a compact subset of M . Here T (lim ϕ) = {y ∈ W : distW (y, lim ϕ) < } is the -tubular neighborhood of lim ϕ. Definition 19. An isometric immersion ϕ : M → W is strongly non-proper if for all > 0 the closed subset ϕ−1 (W \ T lim ϕ) is compact in M . Remark 20. A strongly non-proper immersions is not necessarily bounded: for example, the graph immersion ϕ : B1 (0)\{0} ⊂ Rm → Rm × R given by 1 − r(x) ϕ(x) = (w, z) = x, sin r(x)(1 − r(x) r(x) is strongly non-proper, and lim ϕ = {w = 0} ∪ {r(w) = 1, z = 0}. Corollary 21. Let ϕ : M m → N n ×f Qq be a strongly non-proper minimal submanifold. Suppose that Q satisfies Assumption 7. Assume in addition that the warping function f satisfies inf N f > c1 > 0, supN |∇f | ≤ c2 < ∞ and

(69) N

(70) 2

(71) ∇ f

(72) N h, iN ≤ 0. Hess N f (·, ·) − f Then, if lim ϕ ⊂ N ×f {o}, the spectrum of M is discrete. Proof. Let Tj (N ) = N ×f BQ (o, 1/j), for j large enough that BQ (o, 2/j) b M is a regular, convex ball. Let Kj = ϕ−1 [(N ×f Q) \ Tj (N )] be an exhaustion of M by relatively compact, open sets. Note that ϕ(M \ Kj ) ⊂ Tj (N ). We now proceed as in the proof of Corollary 17. Define F = f (p)vj (ρQ (q)), where vj is the first eigenfunction of Bg (2/j) ⊂ Qm−n , normalized according to kvj kL∞ = 1, and note that g k∇ log F kL∞ (Tj (N )) ≤ k∇ log f k + k∇ log vj k ≤. c2 + k∇ log vj k. c1.

(73) 16. Bessa, Garc´ıa-Mart´ınez, Mari and Ramirez-Ospina. By gradient estimates (see for instance, [23, Thm. 6.1].) 0 vj . ≤ C · j, k∇ log vj kL∞ (Tj ) = vj ∞ L ([0,j]) for some absolute constant C > 0, and so k∇ log F kL∞ (Tj ) ≤ Cj. Using formula (4·9) and proceeding as in the proof of Corollary 17, we have that λ1 (Bg (2/j)) − m |∇N f |2N (p) ∗ λ (M \ Kj ) ≥ inf p∈N |f (p)|2 − mkHkL∞ (M ) k∇ log F kL∞ (Tj ) . Since λ1 (Bg (2/j)) − m |∇N f |2N (p) λ1 (Bg (2/j)) − mc22 ≥ , 2 |f (p)| c21 we deduce λ∗ (M \ Kj ) ≥. λ1 (Bg (2/j)) − mc22 − mkHkL∞ (M ) Cj. c21. Taking into account the standard asymptotic λ1 (Bg (2/j)) ∼ Cj 2 , for some C > 0, we conclude that lim λ∗ (M \ Kj ) = +∞,. j→+∞. and the thesis follows by Persson formula.. Acknowledgements: The first author was partially supported by CNPq, grant # 301041/2009-1. The second author was supported by MINECO project MTM2012-34037 and Fundaci´ on Séneca project 04540/GERM/06, Spain, by the Interuniversity Cooperation Programme Spanish-Brazilian project PHB2010-0137. The fourth author was supported by FPI Grant BES-2010-036829, MINECO project MTM201234037 and Fundaci´ on Séneca project 04540/GERM/06, Spain. This research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Regi´ on de Murcia, Spain, by Fundación Séneca, Regional Agency for Science and Technology (Regional Plan for Science and Technology 2007-2010). This work was developed during the third author’s visiting period at the Universidade Federal do Cear´ a-UFC, Fortaleza-Brazil. He wishes to thank the Mathematics Department for the warm hospitality and for the delightful research environment.. [1] [2] [3] [4] [5]. REFERENCES J.L. Barbosa, M.P. Do Carmo, Stability of minimal surfaces and eigenvalues of the Laplacian. Math. Z. 173, (1980), 13–28. C.S. Barroso and G.P. Bessa, A note on the first eigenvalue of spherically symmetric manifolds. Mat. Contemp. 30, (2006), 63–69. C.S. Barroso and G.P. Bessa, Lower bounds for the first Laplacian eigenvalue of geodesic balls of spherically symmetric manifolds. Int. J. Appl. Math. Stat. 6 (2006), 82-86. J. Barta, Sur la vibration fundamentale d’une membrane. C. R. Acad. Sci. 204, (1937), 472–473. P.H. Bérard, Spectral geometry: direct and inverse problems. Lect. Notes in Math. 1207, Springer-Verlag, 1986..

(74) Eigenvalue estimates for submanifolds. 17. [6] M. Berger, P. Gauduchon and E. Mazet, Le Spectre d’une Variété Riemannienes. Lect. Notes in Math. 194, Springer-Verlag, 1974. [7] G.P. Bessa and M.S. Costa, Eigenvalue estimates for submanifolds with locally bounded mean curvature in N × R. Proc. Amer. Math. Soc. 137, (2009), 1093–1102. [8] G.P. Bessa and J.F. Montenegro, Eigenvalue estimates for submanifolds with locally bounded mean curvature. Ann. Global Anal. and Geom. 24, (2003), 279–290. [9] G.P. Bessa and J.F. Montenegro, An Extension of Barta’s Theorem and Geometric Applications. Ann. Global Anal. Geom. 31, (2007), 345–362. [10] G.P. Bessa andJ.F. Montenegro, On Cheng’s eigenvalue comparison theorem. Math. Proc. Cambridge Philos. Soc. 144, (2008), 673-682. [11] G.P. Bessa J.F. Montenegro and Paolo Piccione, Riemannian Submersions with Discrete Spectrum. J. Geom. Anal. 22, (2012), 603-620. [12] C. Betz, G.A. Camera and H. Gzyl, Bounds for first eigenvalue of a spherical cap. Appl. Math. Optm. 10, (1983), 193–202. [13] B. Bianchini L. Mari and M. Rigoli, On some aspects of Oscillation Theory and Geometry, to appear on Mem. Amer. Math. Soc. [14] A. Candel, Eigenvalue estimates for minimal surfaces in Hyperbolic space. Trans. Amer. Math. Soc. 359, (2007), 3567–3575. [15] I. Chavel, Eigenvalues in Riemannian Geometry. Pure and Applied Mathematics, 1984, Academic Press, INC. [16] S.Y. Cheng, P. Li and S.T. Yau, Heat equations on minimal submanifolds and their applications. Amer. J. Math. 106, (1984), 1033–1065. [17] P. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Funct. An. 12, (1973), 401–414. [18] 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. [19] E.B. Davies, Spectral theory and differential operators, Cambrigde University Press, 1995. [20] S. Friedland and W.K. Hayman, Eigenvalue inequalities for the dirichlet problem on spheres and the growth of subharmonic functions. Comment. Math. Helvetici 51, (1976), 133–161. [21] R.E. Greene and H. Wu, Function theory on manifolds which possess a pole. Lect. Notes in Math. 699. Springer, Berlin, 1979. [22] L. Jorge and D. Koutrofiotis, An estimate for the curvature of bounded submanifolds. Amer. J. Math. 103, (1980), 711–725. [23] P. Li, lecture notes on Geometric analysis. Lecture Notes Series, 6. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1993. iv+90 pp. [24] W. Meeks and H. Rosenberg, The theory of minimal surfaces in M 2 × R. Comment. Math. Helv. 80, (2005), 811–858. [25] W. Meeks and H. Rosenberg, Stable minimal surfaces in M 2 × R. J. Differential Geom. 68, (2004), 515–534. [26] A. Persson, Bounds for the discrete part of the spectrum of a semi-bounded Schrdinger operator. Math. Scand. 8, (1960) 143-153. [27] S. Pigola, M. Rigoli and A.G. Setti, Vanishing and finiteness results in geometric analysis. A generalization of the Bochner technique. Progress in Mathematics, 266. Birkhauser Verlag, Basel, 2008. [28] M.A. Pinsky, The first eigenvalue of a sphercial cap. Appl. Math. Opt. 7, (1981), 137–139. [29] S. Sato, Barta’s inequalities and the first eigenvalue of a cap domain of a 2-sphere. Math. Z. 181, (1982), 313–318. [30] R.S. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. An. 52, (1983), 48–79. [31] Y. Tashiro, Complete Riemannian manifolds and some vector fields. Trans. Amer. Math. Soc. 117, (1965), 251–275..

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