Compact manifolds with fixed boundary and large Steklov eigenvalues
aa r X i v : . [ m a t h . SP ] O c t COMPACT MANIFOLDS WITH FIXED BOUNDARY ANDLARGE STEKLOV EIGENVALUES
BRUNO COLBOIS, AHMAD EL SOUFI, AND ALEXANDRE GIROUARD
Abstract.
Let (
M, g ) be a compact Riemannian manifold with boundary.Let b > ≥
3, we prove that for j = b + 1 it is possible to obtain anarbitrarily large Steklov eigenvalue σ j ( M, e δ g ) using a conformal perturbation δ ∈ C ∞ ( M ) which is supported in a thin neighbourhood of the boundary, with δ = 0 on the boundary. For j ≤ b , it is also possible to obtain arbitrarily largeeigenvalues, but the conformal factor must spread throughout the interior of M . In fact, when working in a fixed conformal class and for δ = 0 on theboundary, it is known that the volume of ( M, e δ g ) has to tend to infinity inorder for some σ j to become arbitrarily large. This is in stark contrast withthe situation for the eigenvalues of the Laplace operator on a closed mani-fold, where a conformal factor that is large enough for the volume to becomeunbounded results in the spectrum collapsing to 0. We also prove that itis possible to obtain large Steklov eigenvalues while keeping different bound-ary components arbitrarily close to each other, by constructing a convenientRiemannian submersion. Introduction
The Steklov eigenvalues of a smooth compact connected Riemannian manifold(
M, g ) of dimension n + 1 ≥ σ for whichthere exists a non-zero harmonic function f : M → R which satisfies ∂ ν f = σf on the boundary Σ. Here and further ∂ ν is the outward normal derivative onΣ. It is well known that the Steklov eigenvalues form a discrete spectrum 0 = σ < σ ≤ σ ≤ · · · ր ∞ , where each eigenvalue is repeated according to itsmultiplicity. The interplay between the geometry of M and the Steklov spectrumhas recently attracted substantial attention. See [9] and the references therein forrecent development and open problems.Many developments linking the Steklov eigenvalues of a compact manifold M with the eigenvalues λ j of the Laplace operator on its boundary have appeared.See for instance [14, 3] and more recently [13], where it was proved that for anyEuclidean domain Ω ⊂ R n +1 with smooth boundary, there exists a constant c Ω > λ j ≤ σ j + 2 c Ω σ j , and σ j ≤ c Ω + q c + λ j . These results indicate a strong link between the Steklov eigenvalues of a manifoldand the geometry of its boundary. See also [4, 15] for recent similar results on
During the first week of 2017, AG and BC were supposed to travel to Tours and work withAhmad El Soufi to complete this paper. We learned just a few days before our visit of his untimelydeath. Ahmad was a colleague and a friend. He will be dearly missed.
Riemannian manifolds. In fact, on smooth surfaces the spectral asymptotics iscompletely determined by the geometry of the boundary [8].In the present paper, we investigate the following question:
For a given closed Riemannian manifold Σ , how large can σ ( M ) be amongcompact Riemannian manifolds M with boundary isometric to Σ ? For a manifold (
M, g ) of dimension n +1 ≥ σ arbitrarily large by using conformal perturbations g ′ = h g such that h = 1 on Σ. Of course, this imply that any eigenvalue σ j becomesarbitrarily large under such a conformal perturbation, but the situation is moreinteresting than that. Indeed, let b > σ b +1 arbitrarily large by using a conformalperturbation h g where h is a smooth function which is different from 1 only in athin strip located arbitrarily close to the boundary (with h = 1 identically on Σ).It is also possible to make lower eigenvalues σ j arbitrarily large, but this requiresconformal perturbations which penetrates deeply into the manifold M .One could also ask how small an eigenvalue σ j ( M ) can be. This question is easier,as it is relatively easy to construct small eigenvalues while keeping the boundaryfixed. On surfaces, it is sufficient to create thin passages (see Figure 3 of [9, Section4]) while for manifolds of dimension ≥
3, one can use a conformal perturbationsupported inside the manifold M . See Proposition 2.1. Large eigenvalues on surfaces.
It was proved in [12] that any compact surface M with boundary of length L > σ ( M ) ≤ πL (1 + genus( M )) . (1)In [5], a sequence of surfaces ( M l ) l ∈ N with one boundary component of fixed length L > l →∞ σ ( M l ) = + ∞ . These two results give a complete answer to our initial question for surfaces: it ispossible to obtain arbitrarily large σ , but it is necessary to increase the genus of M in order to do so. Manifolds of higher dimensions.
For any compact Riemannian manifold (
M, g ) ofdimension ≥ σ . More can be said: let b > σ b +1 arbitrarily large using conformal perturbations g ε which aresupported in an arbitrary neighbourhood of the boundary Σ, and which coincidewith g on the boundary. It is also possible to make σ large, but this requiresconformal perturbations away from the boundary (See Proposition 3.2). The nexttheorem is the main result of this paper. Theorem 1.1.
Let ( M, g ) be a compact connected Riemannian manifold of dimen-sion ≥ with b ∈ N boundary components Σ , · · · , Σ b .(i) For every neighborhood V of Σ , there exists a one-parameter family of Rie-mannian metrics g ε conformal to g which coincide with g on Σ and in M \ V , suchthat σ b +1 ( g ε ) → ∞ as ε → . OMPACT MANIFOLDS WITH LARGE STEKLOV EIGENVALUES 3 (ii) There exists a one-parameter family of Riemannian metrics g ε conformal to g which coincide with g on Σ such that σ ( g ε ) → ∞ as ε → . The proof of Theorem 1.1 will be presented in Section 3. It is important to notethat in order to obtain large eigenvalues, it is necessary to perturb the metric neareach points of the boundary.
Proposition 1.2.
Let ( M, g ) be a compact Riemannian manifold with boundary Σ .Let p ∈ Σ and let ε > . Then any Riemannian metric g ′ on M which coincideswith g on B ( p, ε ) ⊂ M satisfies σ k ( M, g ′ ) ≤ λ Dk ( B ( p, ε ) , g ) , where λ Dk ( B ( p, ε ) , g ) is the k -th eigenvalue of a mixed Steklov-Dirichlet eigenvalueproblem. The proof of this observation is an exercise in the use of min-max characterisa-tions of eigenvalues. It will be presented in Section 2.The conformal perturbations which are used in the proof of Theorem 1.1 are suchthat the volume | M | g ε tends to infinity as ε →
0. This is a necessary conditionwhen working in a fixed conformal class [ g ]. Indeed, the following inequality for g ′ ∈ [ g ] was proved in [10]: σ k ( M, g ′ ) | Σ | /n < A + Bk n +1 I ( M ) ( n − /n , where A is a constant which depends on g , B depends on the dimension and I ( M )is the isoperimetric ratio I ( M ) = | Σ | g ′ | M | n/ ( n +1) g ′ . In each of the constructions, the diameter also becomes unbounded. In Theorem 3.5,we construct a sequence g m of Riemannian metrics on M such that g m (cid:12)(cid:12) Σ = g Σ andsuch that ( M, g m ) has uniformly bounded diameter and σ becomes arbitrarilylarge.To conclude this introduction, note that it is difficult to obtain lower boundsfor Steklov eigenvalues. Under relatively strong convexity assumptions this wasalready investigated by Escobar in [7]. More recently Jammes [11] proposed aninteresting inequality in the spirit of Cheeger: σ ( M ) ≥ h ( M ) j ( M )4 . Here h ( M ) is the classical Cheeger constant and j ( M ) was introduced in [11]. It ischallenging to obtain effective lower bounds on σ ( M ) using this inequality, mainlybecause it is difficult to estimate h ( M ) and j ( M ). Moreover, it it is interesting tonote that the metrics which we construct in Theorem 1.1 and 3.5 have small Cheegerand Cheeger-Jammes constants, despite the eigenvalue σ being arbitrarily large. BRUNO COLBOIS, AHMAD EL SOUFI, AND ALEXANDRE GIROUARD
Plan of the paper.
In the next section, we present the variational character-ization of the Steklov and mixed Steklov-Dirichlet eigenvalue problems and deducesome simple consequences. In Section 3 we prove the main results of the paper byfirst working in cylinders and then using quasi-isometric control of eigenvalues toobtain Theorem 1.1. We also prove Theorem 3.5 which provides an example wheretwo boundary components are arbitrarily close to each other.2.
Variational characterisation and quasi-isometric control ofeigenvalues
Let M be a smooth compact Riemannian manifold with boundary Σ. Let H k ( M )be the set of k -dimensional linear subspaces of C ∞ ( M ). It is well known that theSteklov eigenvalue σ k is given by σ k ( M, g ) = min E ∈H k max = f ∈ E ´ M | df | g dv g ´ Σ | f | dv g , (2)where dv g is the volume form. It the following we will use conformal metrics of theform g ′ = h g , where h is a smooth function on M such that h = 1 identically onthe boundary Σ. In the min-max characterization of σ k ( M, h g ), the denominatoris the same as above, while the numerator is ˆ M | df | g ′ dv g ′ = ˆ M | df | g h n − dv g . In the following, we will often write | df | for | df | g .We seize this opportunity to prove one of the simple statement from the intro-duction. Proposition 2.1.
Let M be a compact smooth Riemannian manifold of dimension n + 1 at least 3, with boundary Σ . For each p ∈ Σ and each ε > , there existsa sequence of conformal deformations h m g such that h m > is a smooth functionwhich is identically equal to 1 on Σ and on the complement of the ball B ( p, ε ) ⊂ M ,and such that lim m →∞ σ k ( M, h m g ) = 0 for each k ∈ N .Proof. Given ε >
0, let p ∈ Σ and consider a smooth function f ∈ C ∞ ( M ) which issupported in B ( p, ε ) ⊂ M and which does not vanish at p . Let h m be a sequence ofpositive smooth functions on M such that h m = 1 on Σ and on the complement of B ( p, ε ), which satisfies lim m →∞ h m = 0 uniformly on compact subsets of B ( p, ε ) ∩ interior( M ). It follows that the conformal deformations ˜ g m = h m g satisfylim m →∞ ´ M | df | g dv ˜ g m ´ Σ f dv ˜ g m = ´ M | df | g h n − m dv g ´ Σ f dv g = 0 . Using k functions f j ( j = 1 , · · · , k ) with disjoint support in B ( p, ε ) instead ofa single function f , the result now follows from the min-max characterization of σ k . (cid:3) We will also use the following mixed Steklov-Dirichlet problem on a domainΩ ⊂ M : ∆ f = 0 in Ω , f = 0 on ∂ Ω \ Σ , ∂ ν f = λf on ∂ Ω ∩ Σ . OMPACT MANIFOLDS WITH LARGE STEKLOV EIGENVALUES 5
It has discrete spectrum 0 < λ D ≤ λ D ≤ · · · ր ∞ . The k -th eigenvalue is given by λ Dk = min E ∈H k, max = f ∈ E ´ Ω |∇ f | dv g ´ ∂ Ω ∩ Σ f dv g , (3)where H k, = { E ∈ H k : f = 0 on ∂ Ω \ Σ × { } ∀ f ∈ E } . For more informationon mixed Steklov problems see for instance [2] and [1].We are now ready for the proof of Proposition 1.2. Proof of Proposition 1.2.
Let ( φ k ) be a sequence of eigenfunctions correspondingto λ Dk ( B ( p, ε )), which are extended by 0 elsewhere in M . Using the subspace E k = span( φ , · · · , φ k ) in the min-max characterization of σ k ( M, g ′ ) completes theproof. (cid:3) The following proposition is borrowed from [5]. It is classical and follows directlyfrom the min-max characterization of the eigenvalues. We believe that this principlewas used for the first time in [6], in the context of the Laplace-Beltrami operatoracting on differential forms.
Proposition 2.2.
Let M be a compact manifold of dimension n , with smoothboundary Σ and let g , g be two Riemannian metrics on M which are quasi-isometric with ratio A ≥ , which means that for each x ∈ M and = v ∈ T x M we have A ≤ g ( x )( v, v ) g ( x )( v, v ) ≤ A. Then the Steklov eigenvalues with respect to g and g satisfy the following inequal-ity: A n +1 ≤ σ k ( M, g ) σ k ( M, g ) ≤ A n +1 . Note also that if the metrics g and g are quasi-isometric with ratio A ≥
1, thengiven a smooth function h on M , the two conformal metrics h g and h g are alsoquasi-isometric with the same ratio A . This will be useful in the proof of Theorem1.1 when going from cylindrical boundaries to arbitrary manifolds.3. Large Steklov eigenvalues on manifolds with fixed boundary
Let M be a compact manifold of dimension ( n + 1) with b ≥ ∪ ... ∪ Σ b . We will prove Theorem 1.1 by first working under theextra hypothesis that the boundary Σ of M has a neighbourhood which is isometricto the product Σ × [0 , L ) for some L >
0. This is not a strong hypothesis since itis always satisfied up to a quasi-isometry (See the proof of Theorem 1.1 below).In the present context, we denote g the restriction of the Riemannian metric g to the boundary Σ, and d the corresponding exterior derivative on C ∞ (Σ). Thespectrum of the Laplace operator on Σ is denoted0 = λ = · · · = λ b < λ b +1 ≤ · · · → + ∞ . Theorem 1.1 will follow from Proposition 3.1 and Proposition 3.3 below.
Proposition 3.1.
Let ( M, g ) be a Riemannian manifold of dimension n + 1 ≥ ,with boundary Σ and assume that there exists a neighborhood V of Σ which isisometric to the product Σ × [0 , L ) for some L > . For every ε > sufficiently BRUNO COLBOIS, AHMAD EL SOUFI, AND ALEXANDRE GIROUARD small, there exists a Riemannian metric g ε = h ε g conformal to g which coincideswith g in the complement of Σ × ( ε, ε ) and such that σ b +1 ( g ε ) ≥ Aε , where A = min { λ b +1 (Σ) , } > .Proof. For every positive ε < min { L , L } , define a Riemannian metric g ε = h ε g on M where h ε ≥ × ( ε, ε ) and, for ( x, t ) ∈ Σ × [2 ε, ε ], h ε ( x, t ) = ε − . Let { φ k } k ∈ N be an orthonormal basis of eigenfunctions of the Laplacian on Σ, with∆ φ k = λ k φ k . Denote by Σ , . . . , Σ b the connected components of Σ. One has λ = · · · = λ b = 0 and, for every j ≤ b , one chooses φ j = | Σ j | − on Σ j and φ j = 0elsewhere.Let f be a smooth function on M with ´ Σ f dv g = 0 and ´ Σ f dv g = 1. Therestriction of f to Σ × [0 , L ) is developed in Fourier series: f ( x, t ) = X j ≥ a j ( t ) φ j ( x )with a j (0) = | Σ j | − ˆ Σ j f dv g , for j = 1 , . . . , b and, since ´ Σ f dv g = 0 and ´ Σ f dv g = 1 a (0) | Σ | + · · · a b (0) | Σ b | = 0 (4) X j ≥ a j (0) = 1 . (5)Observe that P bj =1 a j (0) is the square of the L -norm of the orthogonal projectionof f (cid:12)(cid:12) Σ on ker(∆) = span { φ , . . . , φ b } , in L (Σ , g ). From df ( x, t ) = X j ≥ (cid:0) a ′ j ( t ) φ j ( x ) dt + a j ( t ) d φ j ( x ) (cid:1) and ´ Σ | d φ j | dv g = λ j , it follows that the Dirichlet energy of f on ( M, h ε g ) is R ε ( f ) := ˆ M | df | h n − ε dv g ≥ ˆ Σ × (0 ,L ) | df | h n − ε dv g = X j ≥ ˆ L (cid:0) a ′ j ( t ) + λ j a j ( t ) (cid:1) h n − ε ( t ) dt. (6)At this point, observe that either the function a j decreases quickly when movingaway from the boundary (which costs energy from the first term in (6)) or it remainsbig enough, and the second term contributes a large amount to the energy R ε ( f ).This is now explained more precisely. OMPACT MANIFOLDS WITH LARGE STEKLOV EIGENVALUES 7
Fix an integer j ≥
1. If | a j ( t ) | ≥ | a j (0) | for all t ∈ [2 ε, ε ), then ˆ L λ j a j ( t ) h n − ε ( t ) dt ≥ λ j ˆ ε ε a j ( t ) ε − n − dt ≥ λ j ε − n +3 a j (0) ≥ λ j ε a j (0) . (7)Otherwise, there exists t ∈ [2 ε, ε ) with | a j ( t ) | ≤ | a j (0) | , which implies | a j (0) − a j ( t ) | ≥ | a j (0) | − | a j ( t ) | ≥ | a j (0) | and using the Cauchy-Schwarz inequality thisleads to ˆ L a ′ j ( t ) h n − ε ( t ) dt ≥ ˆ ε a ′ j ( t ) dt + ˆ t ε a ′ j ( t ) ε − n − dt (8) ≥ ε (cid:18) ˆ ε a ′ j ( t ) dt (cid:19) + ε − n − t − ε (cid:18) ˆ t ε a ′ j ( t ) dt (cid:19) where ε − n − t − ε ≥ ε − n − ε ≥ ε ≥ ε . Hence (since x + y ≥ ( x + y ) ) ˆ L a ′ j ( t ) h n − ε ( t ) dt ≥ ε "(cid:18) ˆ ε a ′ j ( t ) dt (cid:19) + (cid:18) ˆ t ε a ′ j ( t ) dt (cid:19) ≥ ε (cid:18) ˆ ε a ′ j ( t ) dt + ˆ t ε a ′ j ( t ) dt (cid:19) = 14 ε ( a j (0) − a j ( t )) ≥ ε a j (0) . (9)For j ≥ b + 1 one has λ j ≥ λ b +1 and, combining (7) and (9) leads to ˆ L (cid:0) a ′ j ( t ) + λ j a j ( t ) (cid:1) h n − ε ( t ) dt ≥ min { λ b +1 , } ε a j (0) = Aε a j (0) . Therefore, thanks to (6) and (5), R ε ( f ) ≥ Aε X j ≥ b +1 a j (0) = Aε − X j ≤ b a j (0) . (10)Now, every normalized function f which is orthogonal in L (Σ) to ker(∆) =span { φ , . . . , φ b } , satisfies P j ≤ b a j (0) = 0 and, then R ε ( f ) ≥ Aε . Using the max-min principle we deduce that σ b +1 ( M, g ε ) ≥ Aε . This completes the proof of (i). (cid:3) In Proposition 3.3 below, we will prove that it is also possible to make σ ar-bitrarily large using a conformal perturbation h g . This is more difficult than for σ b +1 . One of the difficulties comes from the fact that the conformal perturbationwill need to be supported everywhere inside the manifold M . This follows from thefollowing easy proposition. Proposition 3.2.
For every Riemannian metric g ′ which coincides with g on Σ and in the complement of Σ × [0 , L ) , one has σ b ( g ′ ) ≤ L .
BRUNO COLBOIS, AHMAD EL SOUFI, AND ALEXANDRE GIROUARD
Proof of Proposition 3.2.
For every j ≤ b , let ψ j be the function on M such that ψ j is constant equal to zero in the complement of Σ j × [0 , L ) and, for ( x, t ) ∈ Σ j × [0 , L ), ψ j ( x, t ) = (cid:26) | Σ j | − in Σ j × [0 , L ] , − tL ) | Σ j | − in Σ × [ L , L ] . For any Riemannian metric g ′ which coincides with g on Σ and on the complementof Σ × [0 , L ], one has ´ M | dψ j | g ′ dv g ′ = L and ´ Σ ψ j dv g ′ = 1. Moreover, ψ , · · · , ψ b are mutually orthogonal on the boundary. Therefore, using the min-max principlewe deduce that σ b ( g ′ ) ≤ L . (cid:3) Proposition 3.3.
Let ( M, g ) be a Riemannian manifold of dimension n +1 ≥ withboundary Σ and assume that there exists a neighbourhood of Σ which is isometricto the product Σ × [0 , L ) for some L > . For every ε > sufficiently small, thereexists a Riemannian metric g ε = h ε g conformal to g which coincides with g in theneighbourhood Σ × [0 , ε ) of Σ and such that σ ( g ε ) ≥ Cε where C is an explicit constant which only depends on g . The following Poincar´e type result will be useful.
Lemma 3.4.
Let ( M, g ) be a compact manifold and denote by µ the first positiveeigenvalue of the Laplacian of ( M, g ) with Neumann boundary condition if ∂M is nonempty. Let V and V be two disjoint measurable subsets of M of positivevolume. Every function f ∈ C ∞ ( M ) satisfies ˆ M | df | dv g ≥ µ | V | g , | V | g ) (cid:18) V f dv g − V f dv g (cid:19) . where ffl M f dv g := | M | g ´ M f dv g .Proof of Lemma 3.4. Denote by m = ffl M f dv g the mean value of f on M . Thefunction f − m is orthogonal to constant functions on M which implies ˆ M | df | dv g ≥ µ ˆ M ( f − m ) dv g ≥ µ ˆ V ( f − m ) dv g + µ ˆ V ( f − m ) dv g . Using the Cauchy-Schwarz inequality, we get for j = 1 , ˆ V j ( f − m ) dv g ≥ | V j | ˆ V j ( f − m ) dv g ! = | V j | V j f dv g − m ! and then (since x + y ≥ ( x − y ) ) ˆ V ( f − m ) dv g + ˆ V ( f − m ) dv g ≥
12 min( | V | , | V | ) (cid:18) V f dv g − V f dv g (cid:19) which ends the proof. (cid:3) The proof of Proposition 3.3 is more subtle than that of Proposition 3.1. Thebehaviour of a smooth function f away from the boundary has to be taken intoaccount in this case. Indeed, the Steklov eigenfunctions corresponding to index j ≤ b can be almost constant on connected components of the boundary. Theydo not spend a lot of energy ”laterally”. This is expressed by the situation where P bj =1 a j (0) is large in the proof below. OMPACT MANIFOLDS WITH LARGE STEKLOV EIGENVALUES 9
Proof of Proposition 3.3.
The situation where the boundary is connected is alreadytreated by Proposition 3.1. Therefore we assume that b ≥ ε < min { L , L } , aRiemannian metric g ε = h ε g on M where h ε ≥ ε − in M \ (Σ × [0 , L/ × [0 , ε ) . Note that unlike the conformal deformation used in the proof of Proposition 3.1,here, the metric g ε tends to infinity everywhere in the interior of M as ε →
0. Let f be a smooth function on M with ´ Σ f dv g = 0 and such that ´ Σ f dv g = 1. Ourgoal is to get a lower bound for the Dirichlet energy R ε ( f ) = ´ M | df | g ε dv g ε of theform C/ε .Let Σ , . . . , Σ b be the connected components of the boundary Σ. As in the proofof Proposition 3.1, we consider an orthonormal basis { φ j } j ∈ N of eigenfunctions ofthe Laplacian on Σ, with ∆ φ j = λ j φ j , λ = · · · = λ b = 0 and, for every j ≤ b , φ j = | Σ j | − on Σ j and φ j = 0 elsewhere. The restriction of f to Σ × [0 , L ) is f ( x, t ) = X j ≥ a j ( t ) φ j ( x ) . In what follows we treat separately the case where P bj =1 a j (0) is small and thecase where it is large. Indeed, following step by step the proof of Proposition 3.1,we get R ε ( f ) ≥ Aε − X j ≤ b a j (0) with A = min { λ b +1 , } >
0. Hence, if P j ≤ b a j (0) ≤ , then R ε ( f ) ≥ A ε . (11)Assume now that P j ≤ b a j (0) ≥ and recall that the number of boundarycomponents b ≥
2. We will prove that two of the boundary components, w.l.o.g.Σ and Σ , are such that a (0) > a (0) < | a ( t ) | and | a ( t ) | decreases quickly away from the boundary. Otherwise,we will appeal to Lemma 3.4. More precisely, let Ω be the complement in M ofΣ × [0 , L ). We will prove that R ε ( f ) ≥ B ε (12)with B = min { µ (Ω , g ) bL , b } b − min j ≤ b | Σ j | max j ≤ b | Σ j | , where, µ (Ω , g ) is the first positive Neumann eigenvalue of the Laplacian in Ω.Indeed, from P j ≤ b a j (0) ≥ we deduce the existence of j ≤ b with a j (0) ≥ b . We assume w.l.o.g. that j = 1 and that a (0) ≥ √ b . From (4) onehas P ≤ j ≤ b a j (0) | Σ j | = − a (0) | Σ | ≤ − | Σ | √ b . This implies w.l.o.g. that a (0) | Σ | ≤ − b − a (0) | Σ | ≤ − | Σ | ( b − √ b . Now, as in (8) and (9), if there exists t ∈ [ L , L ) ⊂ [2 ε, L ) with a ( t ) ≤ a (0),we would have (with a (0) − a ( t ) ≥ a (0)) R ε ( f ) ≥ ˆ t a ′ ( t ) h n − ε ( t ) dt = ˆ ε a ′ ( t ) dt + ˆ t ε a ′ ( t ) ε − n − dt ≥ ε a (0) ≥ bε. (13)Similarly, if there exists t ∈ [ L , L ) ⊂ [2 ε, L ) with a ( t ) ≥ a (0), we would have(with a ( t ) − a (0) ≥ − a (0) > R ε ( f ) ≥ ˆ t a ′ ( t ) h n − ε ( t ) dt ≥ ε a (0) ≥ ε b ( b − | Σ || Σ | . (14)Let us assume now that for each t ∈ [ L , L ), a ( t ) ≥ a (0) and a ( t ) ≤ a (0).We then have, taking into account that, for each j ≤ b , ´ Σ j φ i dv g = 0 if i = j and ´ Σ j φ j dv g = | Σ j | , Σ × [ L ,L ) f ( x, t ) dv g = | Σ | | Σ × [ L , L ) | g ˆ L L a ( t ) dt ≥ | Σ | a (0) > Σ × [ L ,L ) f ( x, t ) dv g = | Σ | | Σ × [ L , L ) | g ˆ L L a ( t ) dt ≤ | Σ | a (0) < . We apply Lemma 3.4 to the function f in the complement Ω in M of Σ × [0 , L ).This leads to ˆ Ω | df | dv g ≥ Lµ (Ω , g )4 min j ≤ b | Σ j | (cid:18) | Σ | a (0) − | Σ | a (0) (cid:19) where µ (Ω , g ) is the first positive Neumann eigenvalue of the Laplacian in Ω. Onehas1 | Σ | a (0) − | Σ | a (0) ≥ | Σ | √ b + 1 | Σ | | Σ | ( b − √ b = | Σ | √ b (cid:18) | Σ | + 1( b − | Σ | (cid:19) ≥ b ( b − √ b min j ≤ b | Σ j | max j ≤ b | Σ j | . Thus, ˆ Ω | df | dv g ≥ bL b − µ (Ω , g ) min j ≤ b | Σ j | max j ≤ b | Σ j | . Since h ε = ε − on Ω, we get R ( f ε ) = ˆ M | df | h n − ε dv g ≥ ε − n − ˆ Ω | df | dv g ≥ µ (Ω , g ) bL b − min j ≤ b | Σ j | max j ≤ b | Σ j | × ε (15)Combining (13) , (14) and (15) leads to (12).In conclusion, the inequality of the proposition holds with C = min { A, B } . (cid:3) OMPACT MANIFOLDS WITH LARGE STEKLOV EIGENVALUES 11
We are know ready to prove Theorem 1.1.
Proof of Theorem 1.1.
Let (
M, g ) be a compact Riemannian manifold with b > δ >
0, define N ( δ ) = { p ∈ M : d g ( p, Σ) < δ } . The normal exponential map along the boundary defines Fermi coordinates ona neighbourhood V of the boundary. The distance t to the boundary is one ofthe coordinates. It follows from the Gauss Lemma that the Riemannian metric isexpressed by g = h t + dt , where h t is the restriction of g to the parallel hypersurfaceΣ t at distance t from the boundary Σ. In particular it follows from h = g andcontinuity that there exists δ > h t to Σ t is quasi-isometric to g with ratio 2 for each t ∈ [0 , δ ]. Let χ : [0 , δ ] → R be a smoothnon-decreasing function with value 0 on [0 , δ ] and value 1 on [2 δ, δ ]. The metric g δ is defined, using Fermi coordinates, by g δ = ( χ ( t )( h t + dt ) + (1 − χ ( t ))( g + dt ) in N (3 δ ) ,g elsewhere.The metric g δ is quasi-isometric to g with ratio 2, and it is isometric to the productmetric g + dt on N ( δ ). One can now apply Proposition 3.1 and Proposition 3.3to ( M, g δ ). In both cases, this leads to a family of smooth functions h ε such thatsome eigenvalue σ j satisfies lim ε → σ j ( M, h ε g δ ) = + ∞ . Note that h ε g is quasi-isometric to h ε g δ with ratio 2. Therefore, one can apply Proposition 2.2 to deducethat lim ε → σ j ( M, h ε g ) = + ∞ . (cid:3) The conformal perturbation g ε = h ε g that is used in the proof of Proposition 3.1is so that the diameter of ( M, g ε ) tends to infinity as ε →
0. Moreover, when b ≥ ε →
0. In the following theorem, a family of metrics on a cylinder is constructed,which coincide with the initial one along the boundary and such that the distancebetween the two components of the boundary is independent of ε , while σ becomesarbitrarily large. Theorem 3.5.
Let (Σ , g ) be a closed connected Riemannian manifold of dimension n ≥ and consider the cylinder M = Σ × [ − L, L ] , with L > , endowed with theproduct metric g = g + dt . For every ε > , sufficiently small, there exists aRiemannian metric g ε = h ε ( t ) g + dt which coincides with g in a neighborhood ofthe boundary Σ × {− L, L } in M and such that σ ( g ε ) ≥ Cε , where C = min( λ (Σ) , ) . The distance between the two boundary components isindependent of ε and may be chosen arbitrarily small. The condition n ≥ Proof of Theorem 3.5.
Given 0 < ε < min( L, L ), let h ε : [ − L, L ] → [1 , + ∞ ) be asmooth even function such that h ε ( t ) = (cid:26) − L, − L + ε ] ∪ [ L − ε, L ] ,ε − in [ − L + 2 ε, L − ε ] . On the manifold M = Σ × [ − L, L ], define the metric g ε by g ε ( t, p ) = h ε ( t ) g + dt . Let f be a smooth function on M = Σ × [ − L, L ] with ´ ∂M f dv g = 0 and such that ´ ∂M f dv g = 1. As before, we develop the function f in Fourier series f ( x, t ) = X j ≥ a j ( t ) φ j ( x ) (16)where { φ j } j ∈ N is an orthonormal basis of eigenfunctions of the Laplacian on Σ, with∆ φ j = λ j φ j . The eigenfunction corresponding to λ = 0 is the constant function φ on Σ and ´ Σ φ j dv g = 0 for all j ≥
2. The conditions ´ ∂M f dv g = ´ Σ ( f ( x, L ) + f ( x, − L )) dv g = 0 and ´ ∂M f dv g = ´ Σ ( f ( x, L ) + f ( x, − L ) ) dv g = 1 amount to a ( − L ) + a ( L ) = 0 , ∞ X j =1 ( a j ( − L ) + a j ( L )) = 1 . From df ( x, t ) = P ∞ j =1 ( a ′ j ( t ) φ i ( x ) dt + a j ( t ) d φ i ( x )) we get R ε ( f ) = ˆ M | df | g ε dv g ε where | df | g ε = X j a ′ j ( t ) φ j ( x ) + h − ε ( t ) a j ( t ) | d φ j ( x ) | g , and dv g ε = h nε dv g so that R ε ( f ) = ∞ X j =1 ˆ L − L ( a ′ j ( t ) h nε ( t ) + λ j a j ( t ) h n − ε ( t )) dt. We set R jε ( f ) = ´ L − L ( a ′ j ( t ) h nε ( t )+ λ j a j ( t ) h n − ε ( t )) dt so that R ε ( f ) = P ∞ j =1 R jε ( f ). Step 1 : For all j ≥
2, we will prove that R jε ( f ) ≥ Aε ( a j ( − L ) + a j ( L ))with A = min( λ , ).Indeed, let us fix an integer j . If there exists t ∈ [ − L + 2 ε, − L + 3 ε ] with | a j ( t ) − a j ( − L ) | ≥ | a j ( − L ) | , then, using the Cauchy-Schwarz inequality ˆ − L a ′ j ( t ) h nε ( t ) dt ≥ ˆ t − L a ′ j ( t ) dt ≥ t + L (cid:18) ˆ t − L a ′ j ( t ) dt (cid:19) ≥ ε ( a j ( t ) − a j ( − L )) ≥ εa j ( − L ) . (17)Otherwise, | a j ( t ) − a j ( − L ) | ≤ | a j ( − L ) | for all t ∈ [ − L + 2 ε, − L + 3 ε ], whichimplies | a j ( t ) | ≥ | a j ( − L ) | and then ˆ − L λ j a j ( t ) h n − ε ( t ) dt ≥ ˆ − L +3 ε − L +2 ε λ j a j ( t ) ε − n − dt ≥ λ j ε n − a j ( − L ) . (18)Thus, in all cases, we have for j ≥ ε n − ≤ ε and λ j ≥ λ ) ˆ − L ( a ′ j ( t ) h nε ( t ) + λ j a j ( t ) h n − ε ( t )) dt ≥ Aε a j ( − L ) (19) OMPACT MANIFOLDS WITH LARGE STEKLOV EIGENVALUES 13 with A = min( λ , ). The same arguments lead to ˆ L ( a ′ j ( t ) h nε ( t ) + λ j a j ( t ) h n − ε ( t )) dt ≥ Aε a j ( L ) . (20)Combining (19) and (20), we get for all j ≥ R jε ( f ) ≥ Aε ( a j ( − L ) + a j ( L )) . (21) Step 2
For j = 1, we will also show that R ε ( f ) ≥ ε ( a ( − L ) + a ( L )) . Indeed, recall that we have a ( − L ) = − a ( L ). Using (17), we see that if thereexists t ∈ [ − L + 2 ε, − L + 3 ε ] with | a ( t ) − a ( − L ) | ≥ | a ( − L ) | , then R ε ( f ) ≥ ˆ − L a ′ ( t ) h nε ( t ) dt ≥ ε a ( − L ) = 124 ε ( a ( − L ) + a ( L )) . Similarly, if there exists t ∈ [ L − ε, L − ε ] with | a ( t ) − a ( L ) | ≥ | a ( L ) | ), then R ε ( f ) ≥ ˆ L a ′ ( t ) h nε ( t ) dt ≥ ε a ( L ) = 124 ε ( a ( − L ) + a ( L )) . Now, assume that we have both | a ( t ) − a ( − L ) | ≤ | a ( − L ) | for all t ∈ [ − L +2 ε, − L + 3 ε ] and | a ( L ) − a ( t ) | ≤ | a ( L ) | for all t ∈ [ L − ε, L − ε ]. Using theCauchy-Schwarz inequality we get R ε ( f ) ≥ ˆ L − ε − L +3 ε a ′ ( t ) h nε ( t ) dt = ε − n ˆ L − ε − L +3 ε a ′ ( t ) dt ≥ ε − n L − ε ˆ L − ε − L +3 ε a ′ ( t ) dt ! ≥ ε − n L ( a ( L − ε ) − a ( − L + 3 ε )) Assume w.l.o.g. a ( L ) >
0. Then, under the conditions above, we have a ( L − ε ) ≥ a ( L ) − | a ( L ) − a ( L − ε ) | ≥ a ( L )and a ( − L +3 ε ) ≤ a ( − L )+ | a ( − L +3 ε ) − a ( − L ) | ≤ a ( − L )+ 12 | a ( − L ) | = − a ( L ) . Therefore, a ( L − ε ) − a ( − L + 3 ε ) ≥ a ( L ) and, then R ε ( f ) ≥ ε − n L a ( L ) = ε − n L ( a ( − L ) + a ( L )) ≥ ε ( a ( − L ) + a ( L ))In conclusion, we have R ε ( f ) = X j ≥ R jε ( f ) ≥ Cε X j ≥ ( a j ( − L ) + a j ( L )) = Cε with C = min( A, ) = min( λ , ). (cid:3) Remark 3.6.
For n = 1 , it follows from Kokarev’s bound (1) that σ (Σ × [ − L, L ] , g )) ≤ π length (Σ) . For n = 2 , and any Riemannian metric of the form h g + dt , with h ≡ on Σ × {− L, L } , the following holds: σ ≤ Lλ . Indeed, in this case one could use the function f ( x, t ) = a ( t ) φ ( x ) as a test functionand obtain R ( f ) = ˆ L − L ( a ′ ( t ) h ( t ) + λ a ( t ) dt. Using a ≡ leads to the claimed inequality. Remark 3.7.
This example shows that if we are far from being a product”, then,immediately, large eigenvalues could appear. Also, in our construction, the naturalprojection from ( M, g ε ) → [ − L, L ] is a Riemannian submersion on [ − L, L ] . Thisimplies that L is also the distance between the two boundaries in ( M, g ε ) , for any ε . It is interesting to see that this distance could be very small, without smalleigenvalues. Acknowledgments.
The authors are grateful to the anonymous referee for point-ing out a mistake in the original proof of Theorem 3.5, which lead to an improvementof the result.
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Universit´e de Neuchˆatel, Institut de Math´ematiques, Rue Emile-Argand 11, CH-2000Neuchˆatel, Switzerland
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