Courant-sharp eigenvalues of compact flat surfaces: Klein bottles and cylinders

The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, M\"obius strips,\ldots . A natural toy model for further investigations is the flat Klein bottle, a non-orientable surface with Euler characteristic $0$, and particularly the Klein bottle associated with the square torus, whose eigenvalues have higher multiplicities. In this note, we prove that the only Courant-sharp eigenvalues of the flat Klein bottle associated with the square torus (resp. with square fundamental domain) are the first and second eigenvalues. We also consider the flat cylinders $(0,\pi) \times \mathbb{S}^1_r$ where $r \in \{0.5,1\}$ is the radius of the circle $\mathbb{S}^1_r$, and we show that the only Courant-sharp Dirichlet eigenvalues of these cylinders are the first and second eigenvalues.


Introduction
Given a compact Riemannian surface (M, g), we write the eigenvalues of the Laplace-Beltrami operator −∆ g , Courant's nodal domain theorem (1923) states that any eigenfunction associated with the eigenvalue λ k has at most k nodal domains (connected components of the complement of the zero set of u). The eigenvalue λ k is called Courant-sharp if there exists an associated eigenfunction with precisely k nodal domains. It follows from Courant's theorem that the eigenvalues λ 1 and λ 2 are Courant-sharp, and that λ k−1 < λ k whenever λ k is Courant-sharp. Courant-sharp eigenvalues appear naturally in the context of partitions, [11].
for the other compact flat surfaces, Klein bottles and cylinders. The purpose of the present paper is to prove the following theorems. Theorem 1.2. Let C r denote the flat cylinder (0, π) × S 1 r , with the product metric. Here S 1 r denotes the circle with radius r. Then, for r ∈ 1 2 , 1 , the only Courantsharp Dirichlet eigenvalues of C r , are λ 1 and λ 2 . Courant-sharp eigenvalues have previously been determined for several compact surfaces. We refer to the following papers and their bibliographies.
Most of the papers mentioned above adapt the method introduced by Pleijel in [24] to the example at hand.
The paper is organized as follows. In Section 2, we recall the main lines of Pleijel's method. More precisely, we show how a lower bound on the ratio λ k (M ) k and a lower bound on the Weyl counting function can be used to restrict the search for Courantsharp eigenvalues to a finite set of eigenvalues. To actually determine the Courantsharp eigenvalues one then needs to analyze the nodal patterns of eigenfunctions in a finite set of eigenspaces. In Section 3, we recall some basic facts concerning Klein bottles. In Sections 4 and 5, we adapt Pleijel's method (Section 2) to the flat Klein bottles K 1 and K 2 . In Section 6, we adapt Pleijel's method (Section 2) to the flat cylinders C r , with r ∈ 1 2 , 1 .

Pleijel's method summarized
In order to prove that the number of Courant-sharp (Dirichlet) eigenvalues of a compact Riemannian surface (M, g) is finite, one only needs two ingredients, (2) an inequality à la Faber-Krahn for domains ω ⊂ M with small enough area, for some ε ∈ [0, 1), and for some constant C(M, ε). Here, δ 1 (ω) denotes the least Dirichlet eigenvalue of ω, and j 0,1 is the first positive zero of the Bessel function J 0 (j 0,1 ≈ 2.404825).
Note that the right-hand side of (2.1) is equal to (1 − ε) 2 δ 1 (ω * ), where ω * denotes a disk in R 2 with area equal to |ω|. The existence of such an inequality (for a given ε ∈ (0, 1)) follows from the asymptotic isoperimetric inequality proved in [6, Appendice C], and from the usual symmetrization argument to translate the isoperimetric inequality into a Faber-Krahn type inequality for the first Dirichlet eigenvalue, [6,Lemme 15].
In order to give more quantitative information on the Courant-sharp eigenvalues, one needs a lower bound on the Weyl function, in the form, for some constants A(M ) and B(M ) depending on the geometry of (M, g).
Since j 0,1 > 2, choosing ε < 1 − 2 j 0,1 , the coefficient of the leading term in F M,ε is positive, and the function tends to infinity when λ tends to infinity.
To conclude whether the eigenvalue λ k (M ) is actually Courant-sharp, it remains to determine the maximum number of nodal domains of an eigenfunction in the eigenspace E (λ k (M )).

Remarks 2.3.
(1) In Sections 4 and 6, we will use the fact that we can choose ε = 0 in the isoperimetric inequality (2.1) for the flat Klein bottles and for the flat cylinders. This follows from [17], and was already used in [19,4,5] for the flat torus and for the Möbius band. (2) When ∂M = ∅, and for Neumann or Robin eigenfunctions of (M, g), the inequality (2.1) can only be applied to a nodal domain which does not touch the boundary ∂M . To prove a result à la Pleijel for Neumann or Robin eigenfunctions, as in [25,21,16,10,9], it is necessary to take care of this difficulty.

(3) The proof of Lemma 2.1 also yields the following inequality. Let κ(k) denote the maximal number of nodal domains of an eigenfunction associated with
This inequality, which generalizes Pleijel's inequality [24] valid for plane domains, is actually a particular case of a general result valid for any closed Riemannian manifold, [6]. The interesting feature is that the upper bound γ(n) < 1 only depends on the dimension.

Preliminaries on Klein bottles
3.1. Klein bottles. In this note, we are interested in the flat Klein bottles. More precisely, given a, b > 0, we consider the isometries of R 2 given by . We denote by G 2 (resp. G) the group generated by τ 1 and τ 2 (resp. by τ 1 and τ ). These groups act properly and freely by isometries on R 2 equipped with the usual scalar product. Since τ 2 = τ 2 , the group G 2 is a subgroup of index 2 of the group G. We denote by T a,b (resp. K a,b ) the torus R 2 /G 2 (resp. the Klein bottle R 2 /G). We equip T a,b and K a,b with the induced flat Riemannian metrics.
A fundamental domain for the action of G 2 (resp. G) on R 2 is the rectangle T a,b = (0, a) × (0, b) (resp. the rectangle K a,b = (0, a 2 ) × (0, b), see Figure 3.1 (A)). The horizontal sides of K a,b are identified with the same orientation, the vertical sides are identified with the opposite orientations.
The geodesics of the Klein bottle are the images of the lines in R 2 under the Riemannian covering map R 2 → K a,b (see [8]). They can be looked at in the fundamental domain K a,b , taking into account the identifications (x, 0) ∼ (x, b) and (0, y) ∼ ( a 2 , b − y). Among them, we have some special geodesics, see Figure 3.
which is a periodic geodesic of length a; the two horizontal lines in blue in Figure 3.1 (C) yield a periodic geodesic of the Klein bottle; divides the surface into two Möbius strips whose center lines are the geodesics t → (t, 0) and t → (t, b 2 ), see Figure 3.1 (D). The isometry τ of R 2 induces an isometry on the torus T a,b so that we can identify K a,b with the quotient T a,b / {Id, τ }. It follows that the eigenfunctions of the Klein bottle K a,b are precisely the eigenfunctions of the torus T a,b which are invariant under the map τ . Because τ is orientation reversing, the surface K a,b is non-orientable with orientation double cover T a,b .
with associated eigenvaluesλ(m, n) = 4π 2 m 2 a 2 + n 2 b 2 . Given some eigenvalue λ of T a,b , we introduce the set, or, equivalently, if and only if We can rewrite a τ -invariant eigenfunction φ as, The following lemma follows readily.  Here, N denotes the set of non-negative integers, and N • the set of positive integers.
the multiplicity of λ is even.

Choices for a and b.
In this paper, we restrict our attention to the case a = b = 2π, i.e. to the flat Klein bottle K 1 := K 2π,2π , whose fundamental domain is the rectangle K 1 = (0, π)×(0, 2π), and to the case a = 2π, b = π, i.e., to the flat Klein bottle K 2 := K 2π,π , whose fundamental domain is the square K 2 = (0, π) × (0, π). As in [13] for flat tori, we could consider other values of the pair (a, b), in particular we could look at what happens when a is fixed, b tends to zero, and vice-versa.
As points of the spectrum, the eigenvalues of the flat Klein bottle K c , c ∈ {1, 2} are the numbers of the formλ(p, q) = p 2 + c 2 q 2 , with p, q ∈ N, and the extra condition that p is even when q = 0. As usual, the eigenvalues of K c are listed in nondecreasing order, multiplicities accounted for, starting from the label 1, For λ ≥ 0, we introduce the Weyl counting function, Weyl's asymptotic law tells us that where |K c | denotes the area of the Klein bottle, namely |K c | = 2π 2 c . For later purposes, we also introduce the set (3.11) L c (λ) := (m, n) ∈ N 2 | m 2 + c 2 n 2 < λ and the counting function,

Courant-sharp eigenvalues of the Klein bottles
The purpose of this section is to determine the Courant-sharp eigenvalues of the Klein bottles K c , following Pleijel's method, Section 2.

This proves inequalities (4.1) and (4.3).
To prove the other inequalities, consider the set Then, E c (λ) ⊂ where x denotes the integer part of x ≥ 0. It follows that, for all λ ≥ 0, Inequalities (4.2) and (4.4) follow from the previous inequalities. Table 4.1 displays the eigenvalues of K c less than or equal to 25 (c = 1), resp. 47 (c = 2), the corresponding labeled eigenvalues, and the ratio λ k min (K c ) k min which should be larger than or equal to Since the eigenvalues λ 1 (K c ), λ 2 (K c ) are Courant-sharp, we conclude from    when c = 1, it remains to investigate λ 3 (K 1 ) and λ 5 (K 1 ). This is done in the next section.

The eigenvalue λ 3 (K 1 ) is not Courant-sharp.
A general eigenfunction associated with λ 3 has the form (A cos(x) + B sin(x)) sin(y). It is sufficient to look at eigenfunctions of the form sin(x − α) sin(y). These eigenfunctions have exactly two nodal domains in K 1 . It follows that λ 3 is not Courant-sharp, see Figure 5.1.
It follows that an eigenfunction associated with λ 5 has at most 4 nodal domains in K 1 , and hence that λ 5 (K 1 ) is not Courant-sharp. has multiple solutions within the above range.
As usual, we arrange the Dirichlet eigenvalues of C r in non-decreasing order, starting from the label 1, multiplicities accounted for, The purpose of this section is to prove Theorem 1.2, i.e. to determine the Courantsharp eigenvalues in two specific cases r ∈ 1 2 , 1 whose eigenvalues have higher multiplicities. The corresponding cylinder are the orientation covers of the Möbius band we studied in [5]. As in [12,13], we could also consider other values of r, in particular r close to zero or r very large. Remark 6.1. As pointed out in the introduction, λ 1 (C r ) and λ 2 (C r ) are always Courant-sharp, and any Courant-sharp eigenvalue λ k (C r ) satisfies the inequality λ k (C r ) > λ k−1 (C r ). Following Pleijel's method, Section 2, we introduce Weyl's counting function, According to Weyl's law, where |C r | denotes the area of the cylinder.
2r . The following inequalities hold.
Proof. The arguments are similar to those used in the proof of Lemma 4.1. Inequality (6.6) follows from (6.5). The inequality in (6.7) follows from the description of the Dirichlet spectrum of C r . The other inequalities follow from (6.7) and (6.6).
The following tables give the eigenvalues, the corresponding range of labels, and the ratios λ k min k min for C 1 2 (left) and C 1 (right). For a Courant-sharp eigenvalue, this ratio should be greater than 1.840844 for C 1 2 , and greater than 0.920422 for C 1 . According to Remark 6.1, the eigenvalues λ 1 and λ 2 are always Courant-sharp.
The (left) shows that it only remains to analyze the eigenvalue 5 = ). The associated eigenspace is generated by the eigenfunctions sin(x) cos(2y) and sin(x) sin(2y). Functions in this eigenspace have 2 nodal domains. This finishes the proof of Theorem 1.2 in the case r = 1 2 . The table for C 1 (right) shows that it only remains to analyze two eigenvalues 4 and 5. The eigenvalue 4 = λ 4 (C 1 ) is simple, with associated eigenfunction sin(2x) which has two nodal domains. The eigenvalue 5 = λ 5 (C 1 ) = · · · = λ 8 (C 1 ) has multiplicity 4. The corresponding eigenspace is generated by the eigenfunctions sin(2x) cos(x), sin(2x) sin(x) and sin(x) cos(2x), sin(x) sin(2y) which, according to [5], turn out to span the second eigenspace E(λ 2 (M 1 )) of the square Möbius strip, whose orientation cover is C 1 . Since the eigenfunctions in E(λ 2 (M 1 )) have two nodal domains, it follows that the eigenfunctions in the eigenspace E (λ 5 (C 1 )) have at most four nodal domains. This finishes the proof of Theorem 1.2 in the case r = 1. Note that one can give a precise analysis of nodal patterns in the eigenspace E (λ 5 (C 1 )) by using the same arguments as in [5,Section 4].