The spectral gap of graphs and Steklov eigenvalues on surfaces

Using expander graphs, we construct a sequence of smooth compact surfaces with boundary of perimeter N, and with the first non-zero Steklov eigenvalue uniformly bounded away from zero. This answers a question which was raised in [9]. The genus grows linearly with N, this is the optimal growth rate.


Introduction
Let Ω be a compact, connected, orientable smooth Riemannian surface with boundary Σ = ∂Ω. The Steklov eigenvalue problem on Ω is where ∆ is the Laplace-Beltrami operator on Ω and ∂ ν denotes the outward normal derivative along the boundary Σ. The Steklov spectrum of Ω is denoted by where each eigenvalue is repeated according to its multiplicity. In [9], the second author and I. Polterovich asked the following question: Is there a sequence {Ω N } of surfaces with boundary such that σ 1 (Ω N )L(∂Ω N ) → ∞ as N → ∞?
In this case, the supremum is attained in the limit by a sequence of surfaces with their number of boundary components tending to infinity. It follows from (1) that the growth of σ ⋆ (γ) is sublinear.
In the construction of {Ω N } that we propose, the number of boundary components also tends to infinity. It would be interesting to know if this condition is necessary.
Remark 3. The problem of constructing closed surfaces M with large normalized first non-zero eigenvalue λ 1 (M )Area(M ) has been considered by several authors. See for instance [4,2,3]. Our proofs uses methods which are related to those of [7].

Plan of the paper
In Section 2, we present the construction of a surface Ω Γ which is obtained from a regular graph Γ = (V, E) by sewing copies of a fundamental piece following the pattern of the graph Γ. In Section 3, we introduce the spectrum of the graph Γ and state a comparison result (Theorem 7) between λ 1 (Γ) and σ 1 (Ω Γ ). This is then used, in conjunction with expander graphs, to prove Theorem 1. In Section 4, we present the comparison argument leading to the proof of Theorem 7.
Remark 4. While this paper was in the final stage of its preparation, we learned that Mikhail Karpukhin also has developed a method for construction surfaces with large normalized Steklov eigenvalue σ 1 L. His approach is different, and his work will appear in [11].

Constructing manifolds from graphs
Let Γ be a finite connected regular graph of degree k. The set of vertices of Γ is denoted V = V (Γ), the set of edges is denoted E = E(Γ). The number of vertices of Γ is |V (Γ)|. We will construct a Riemannian surface Ω Γ modelled on the graphs Γ from a fixed orientable Riemannian surface M 0 which we call the fundamental piece (See Figure 1) and which is assumed to satisfy the following hypotheses: 1. The boundary of M 0 has k + 1 components Σ 0 , B 1 , · · · , B k .
2. Each of the boundary component is a geodesic curve of length 1.

The component Σ 0 has a neighbourhood which is isometric to the cylinder
The manifold Ω Γ is obtained by sewing copies of the fundamental piece M 0 following the pattern of the graph Γ: The following lemma shows that the genus of Ω Γ grows linearly with the number of vertices of the graph Γ.
where γ(M 0 ) is the genus of the fundamental piece M 0 , and |V (Γ)| is the number of vertices of Γ.
Remark 6. Because the number of vertices of odd degree is always even, Proof of Lemma 5. The genus γ and the Euler-Poincaré characteristic χ of a smooth compact orientable surface with b boundary components are related by the formula Let K : Ω Γ → R be the Gauss curvature. Since the boundary curves Σ 0 , B 1 , · · · , B k are geodesics, it follows from the Gauss-Bonnet formula hat where we have used that the number of boundary components of M 0 is k + 1. It follows that the genus of Ω Γ is

Comparing eigenvalues on graphs to Steklov eigenvalues
Our main reference for spectral theory on graphs is [5]. The space The discrete Laplacian ∆ Γ acts on ℓ 2 (V (Γ)) and is defined by the quadratic form where the symbol v ∼ w means that the vertices v and w of Γ are adjacent, and the sum appearing in (2) is over all non-oriented edges of Γ. The discrete Laplacian ∆ Γ has a finite non-negative spectrum which we denote by where each eigenvalue is repeated according to its multiplicity. The first non zero eigenvalue admits the following variational characterization: In order to compare λ 1 (Γ) to the fist non-zero Steklov eigenvalue of Ω Γ , the following variational characterization will be used: The main result of this paper will follow from the following estimate.
Theorem 7. There exist constants α, β > 0 depending only on the fundamental piece M 0 such that The proof of Theorem 7 will be presented in Section 4.

Expander graphs and the proof of Theorem 1
To prove Theorem 1, we will use expander graphs, through one of their many characterizations. See [10] for a survey of their properties and applications. Consider a fundamental piece M 0 of genus 0, with 5 boundary components, that is with k = 4. Let {Γ N } be a family of 4-regular expander graphs such that the number of vertices |V (Γ N )| = N . The existence of this family of expander graphs follows from the classical probabilistic method [14]. It follows from Lemma 5 that the genus of Ω ΓN is γ(Ω ΓN ) = 1 + N.
By definition, there is a constant c > 0 such that λ 1 (Γ N ) ≥ c for each N ∈ N. Since the boundary of Ω ΓN has N boundary components of length 1, Theorem 7 leads to This completes the proof of Theorem 1.

Proof of the comparison results
Let f : Ω Γ → R be a smooth function. Given a vertex v ∈ V (Γ), the function f v is defined to be the restriction of f to the cylinder C v . On each cylinder C v , the function f v admits a decomposition is the average of f on the corresponding slice of C v . It follows that for each r ∈ [0, 1], The function f is defined to be f v on each cylinder C v , and similarly the functionf is defined to bef v on each C v .
Let f ∈ C ∞ (Ω Γ ) be a Steklov eigenfunction corresponding to σ 1 (Ω Γ ). The function is defined to be the average of f over the boundary component Σ v . Since |Σ v | = 1 for each vertex v, this is expressed by the function x can be used as a trial function in the variational characterization (3) of λ 1 (Γ). It follows from the orthogonality of f andf on the boundary Σ = ∂Ω Γ that The two terms on the right-hand side of the previous inequality will be bounded above in terms of ∇f 2 L 2 (ΩΓ) . In order to bound ∂ΩΓf 2 dV Σ , it will be sufficient to consider the behaviour off locally on each cylinders C v . More work will be required to bound q Γ (x).

Local estimate of smooth functions on cylindrical neighbourhoods
On the model cylinder C 0 = [0, 1] × Σ 0 , consider the following mixed Neumann-Steklov spectral problem: This problem is related to the sloshing spectral problem. See [1,13] for details. Lemma 9. Let µ be the first non-zero eigenvalue of the sloshing problem (5). For any smooth function f : Proof. The first non-zero eigenvalue of this problem is characterized by Sincef is orthogonal to constants on each boundary component Σ v , it follows from (7) that The proof of Lemma 9 is completed by observing that the Dirichlet energy of

Global estimate and graph structure
The proof of Lemma 10 is based on the following general estimate.

Lemma 11.
Let Ω be a smooth compact connected Riemannian surface with boundary. Let A and B be two of the connected components of the boundary ∂Ω, both of length 1. There exists a constant C > 0 depending only on Ω such that any smooth function f ∈ C ∞ (Ω) satisfies In fact, we will use this estimate only for harmonic functions, in which case it is also possible to prove it using a method similar to that of [7].
Proof of Lemma 11. Let x = A f , y = B f be the average of f on the two boundary components A, B.
be the average of f on the surface Ω.
where µ > 0 is the first non zero Neumann eigenvalue of Ω. It follows that In other words, the Dirichlet energy of f controls how far f is from its average f in H 1 -norm. This is essentially a version of the Poincaré Inequality. The restriction of f to A and B are also close to the average f in L 2 -norm. Indeed, it follows from the fact that the trace operators τ a : H 1 (Ω) → L 2 (A) and τ B : and similarly |y − f | ≤ τ B g H 1 (Ω) , where τ A and τ B are the operator norms. These two inequalities together lead to In combination with (10) this imply One can take C = ( τ A + τ B ) 2 (µ −1 + 1)). The proof is completed.
Proof of Lemma 10. For each adjacent vertices v ∼ w of the graph Γ, we apply Lemma 11 to the surface M v ∪ M w with A = Σ v and B = Σ w to get Since the graph Γ is regular of degree k, it follows that v∼w

The proof of Theorem 7
The upper bound Let f be a Steklov eigenfunction corresponding to the first non-zero Steklov eigenvalue σ 1 (Ω Γ ). Combining the local estimate obtained in Lemma 9 and the global estimate of Lemma 10 with Inequality (4) leads to which of course can be rewritten Now, because we are on a regular graph of degree k, λ 1 ≤ k, so that and taking β = 1 k+1 min µ, 1 C0 , the proof of Theorem 7 is completed.
Using x, a function f x : Ω Γ → R is defined to be x(v) on Σ v and to decay linearly to zero on the cylinder and can therefore be used in the variational characterization of σ 1 (Ω Γ ). The estimates of the Rayleigh quotient are simple and follows [7, p. 290] verbatim. We will not reproduce it here.