Maximization of Laplace-Beltrami eigenvalues on closed Riemannian surfaces
MMaximization of Laplace-Beltrami eigenvalueson closed Riemannian surfaces
Chiu-Yen Kao ∗ , Rongjie Lai † , and Braxton Osting ‡ March 29, 2016
Abstract
Let (
M, g ) be a connected, closed, orientable Riemannian surface and denote by λ k ( M, g )the k -th eigenvalue of the Laplace-Beltrami operator on ( M, g ). In this paper, we consider themapping (
M, g ) (cid:55)→ λ k ( M, g ). We propose a computational method for finding the conformalspectrum Λ ck ( M, [ g ]), which is defined by the eigenvalue optimization problem of maximizing λ k ( M, g ) for k fixed as g varies within a conformal class [ g ] of fixed volume vol( M, g ) = 1.We also propose a computational method for the problem where M is additionally allowed tovary over surfaces with fixed genus, γ . This is known as the topological spectrum for genus γ and denoted by Λ tk ( γ ). Our computations support a conjecture of N. Nadirashvili (2002)that Λ tk (0) = 8 πk , attained by a sequence of surfaces degenerating to a union of k identicalround spheres. Furthermore, based on our computations, we conjecture that Λ tk (1) = π √ +8 π ( k − k − k -th Laplace-Beltrami eigenvalue hasa local maximum with value λ k = 4 π (cid:6) k (cid:7) (cid:16)(cid:6) k (cid:7) − (cid:17) − . Several properties are also studiedcomputationally, including uniqueness, symmetry, and eigenvalue multiplicity. Keywords.
Extremal Laplace-Beltrami eigenvalues, conformal spectrum, topological spectrum,closed Riemannian surface, spectral geometry, isoperimetric inequality
Let (
M, g ) be a connected, closed, orientable Riemannian surface and ∆
M,g : C ∞ ( M ) → C ∞ ( M )denote the Laplace-Beltrami operator. The Laplace-Beltrami eigenproblem is to find eigenvalues λ ( M, g ) and eigenfunctions, ψ ( x ; M, g ) for x ∈ M , satisfying − ∆ M,g ψ ( x ; M, g ) = λ ( M, g ) ψ ( x ; M, g ) x ∈ M. (1) ∗ Department of Mathematical Sciences, Claremont McKenna College, CA 91711 (
[email protected] ).Chiu-Yen Kao is partially supported by NSF DMS-1318364. † Department of Mathematics, Rensselaer Polytechnic Institute, NY 12180 ( [email protected] ). Rongjie Lai is partiallysupported by NSF DMS-1522645. ‡ Corresponding author. Department of Mathematics, University of Utah, Salt Lake City, UT 84112( [email protected] ). Braxton Osting is partially supported by NSF DMS-1103959 and DMS-1461138. a r X i v : . [ m a t h . DG ] M a r enote the spectrum of − ∆ M,g by σ ( M, g ) := { λ ( M, g ) < λ ( M, g ) ≤ . . . } . For a generalintroduction to properties of ∆ M,g and σ ( M, g ), we refer to [Cha84, BB85]. Given a fixed manifold M , consider the mapping g (cid:55)→ σ ( M, g ). Let G ( M ) denote the class of Riemannian metrics g on M . We recall that a metric g is conformal to g if there exists a smooth function ω : M → R + suchthat g = ωg . The conformal class , [ g ], consists of all metrics conformal to g . Following [CS03],for k fixed, we define the conformal k -th eigenvalue of ( M, [ g ]) to beΛ ck ( M, [ g ]) := sup { Λ k ( M, g ) : g ∈ [ g ] } , (2)where Λ k ( M, g ) := λ k ( M, g ) · vol( M, g ). Let M ( γ ) denote the class of orientable, closed sur-faces with genus γ and consider the mapping ( M, g ) (cid:55)→ σ ( M, g ). For k fixed, the k -th topologicaleigenvalue for genus γ is definedΛ tk ( γ ) := sup { Λ k ( M, g ) : M ∈ M ( γ ) , g ∈ G ( M ) } . (3)The conformal and topological eigenvalues are finite; see §
2. We refer to the conformal eigenval-ues and topological eigenvalues collectively as the conformal spectrum and topological spectrum,respectively.For some conformal classes, the first few conformal eigenvalues are known explicitly. However,little is known about the larger conformal eigenvalues of any conformal class, ( M, [ g ]). The topo-logical spectrum is only known for γ = 0 with k = 1 , γ = 1 with k = 1 (a conjecture existsfor γ = 2, k = 1). We discuss these results and provide some references in § ω + > ω − >
0, we define the admissible set, A ( M, g , ω − , ω + ) := { ω ∈ L ∞ ( M ) : ω − ≤ ω ≤ ω + a.e. } . For a fixed Riemannian surface (
M, g ) and a function ω ∈ A ( M, g , ω − , ω + ), we consider thegeneralized eigenvalues, characterized by the Courant-Fischer formulation λ k − ( M, g , ω ) = min E k ⊂ H ( M )subspace of dim k max ψ ∈ E k ,ψ (cid:54) =0 (cid:82) M |∇ ψ | dµ g (cid:82) M ψ ωdµ g , (4)where E k is in general a k -dimensional subspace of H ( M ) and dµ g is the measure induced bythe metric g . Note that for ω ∈ C ∞ ∩ A ( M, g , ω − , ω + ), the identity ∆ M,ωg = ω ∆ M,g impliesthat λ k ( M, g , ω ) = λ k ( M, ωg ). As above, we define a volume-normalized quantity, Λ k ( M, g, ω ) = λ k ( M, g, ω ) · (cid:82) M ωdµ g and consider the optimization problem,Λ (cid:63)k ( M, g , ω − , ω + ) = sup { Λ k ( M, g , ω ) : ω ∈ A ( M, g , ω − , ω + ) } . (5) Proposition 1.1.
Fix k ∈ N . Let ( M, g ) be a smooth, closed Riemannian surface and < ω − <ω + . Then there exists an ω (cid:63) ∈ A ( M, g , ω − , ω + ) which attains Λ (cid:63)k ( M, g , ω − , ω + ) , the supremum in (5) . Furthermore, for any (cid:15) > , there exist constants ω + ( (cid:15) ) and ω − ( (cid:15) ) satisfying ω + ( (cid:15) ) > ω − ( (cid:15) ) > such that Λ ck ( M, [ g ]) − (cid:15) ≤ Λ (cid:63)k ( M, g , ω − ( (cid:15) ) , ω + ( (cid:15) )) ≤ Λ ck ( M, [ g ]) . Note that by the dilation property of eigenvalues, λ k ( M, cg ) = c − λ k ( M, g ), this is equivalent to minimizing λ k ( M, g ) over { g ∈ [ g ] : vol( M, g ) = 1 } . § § ω + and ω − such that ω + ↑ ∞ and ω − ↓
0. The bound in Proposition 1.1 justifiesthis strategy. Similarly, we approximate (3), the topological spectrum for genus γ , bysup { Λ k ( M, g , ω ) : M ∈ M ( γ ) , g ∈ G ( M ) , and ω ∈ A ( M, g , ω − , ω + ) } . (6)For a given closed Riemannian surface ( M, g ) and constants k ≥ ω + > ω − >
0, we developa computational method for seeking the conformal factor ω ∈ A ( M, g , ω − , ω + ) which attains thesupremum in (5). To achieve this aim, we evolve ω within A ( M, g , ω − , ω + ) to increase Λ k ( M, g , ω ).If ω were assumed smooth, this would be equivalent to evolving a metric g within its conformalclass, [ g ] to increase Λ k ( M, g ). We also develop a computational method for approximating thetopological spectrum for genus γ = 0 and γ = 1 via (6). The method depends on an explicitparameterization of moduli space, and in principle could be extended to higher genus [IT92, Bus10].Our computations support a conjecture of N. Nadirashvili [Nad02] that Λ tk (0) = 8 πk , attainedby a sequence of surfaces degenerating to a union of k identical round spheres (see § n = 2, and a genus γ = 0 surface, the inequality, Λ tk (0) ≥ πk , of [CS03, Corollary 1] istight. Based on our computations, we further conjecture that Λ tk (1) = π √ + 8 π ( k − k − § k has a local maximum with value Λ k = 4 π (cid:6) k (cid:7) (cid:16)(cid:6) k (cid:7) − (cid:17) − . Weconjecture that this is the global maximum among flat tori. A detailed study of the first non-trivialconformal eigenvalue of flat tori is also conducted in § Outline. In §
2, we provide some background material and review related work. This includes adiscussion of properties of the Laplace-Beltrami eigenproblem and its solution, a brief discussion ofmoduli spaces, variations of eigenvalues with respect to the conformal structure, and the spectrumof the disconnected union of a surface and a sphere. We also provide a proof of Proposition 1.1.In §
3, we discuss the Laplace-Beltrami eigenproblem on a sphere and flat tori, which are centralto later sections. In §
4, we describe our computational methods. In §
5, we compute the conformalspectrum of several Riemannian surfaces and the topological spectrum for genus γ = 0 and γ = 1surfaces. We conclude in § Let (
M, g ) be a connected, closed, smooth Riemannian manifold of dimension n ≥
2. The first fun-damental form on M can be written (using Einstein notation) in local coordinates as g = g ij dx i dx j ,where g ij = g ( ∂ x i , ∂ x j ). Let dµ g denote the measure on ( M, g ) induced by the Riemannian metric.Let (cid:104)· , ·(cid:105) g denote the L -inner product on ( M, g ) and denote (cid:107) f (cid:107) g = (cid:104) f, f (cid:105) g . In local coordinatesthe divergence and gradient are written ( ∇ f ) i = ∂ i f = g ij ∂ j f and div X = 1 (cid:112) | g | ∂ i (cid:112) | g | X i . Here g ij is the inverse of the metric tensor g = g ij and | · | is the determinant. The Laplace-Beltrami3perator, ∆ M,g : C ∞ ( M ) → C ∞ ( M ) is written in local coordinates∆ M,g f = div ∇ f = 1 (cid:112) | g | ∂ i (cid:112) | g | g ij ∂ j f. (7)Denote the spectrum of − ∆ M,g by σ ( M, g ). For a general introduction to properties of ∆
M,g and σ ( M, g ), we refer to [Cha84, BB85].
Properties of ∆ M,g and σ ( M, g ) .
1. The eigenvalues λ k ( M, g ) are characterized by the Courant-Fischer formulation λ k − ( M, g ) = min E k ⊂ H ( M )subspace of dim k max ψ ∈ E k ,ψ (cid:54) =0 (cid:82) M |∇ ψ | dµ g (cid:82) M ψ dµ g , (8)where E k is in general a k -dimensional subspace of H ( M ) and at the minimizer, E k =span( { ψ j ( · ; M, g ) } kj =1 ).2. For fixed ( M, g ), λ k ( M, g ) ↑ ∞ as k ↑ ∞ and each eigenspace is finite dimensional. We have λ = 0 and the corresponding eigenspace is one dimensional and spanned by the constantfunction. Eigenspaces belonging to distinct eigenvalues are orthogonal in L ( M ) and L ( M )is spanned by the eigenspaces. Every eigenfunction is C ∞ on M .3. (dilation property) For ( M, g ) fixed, the quantity λ k ( M, g ) vol(
M, g ) n , where n is the dimen-sion, is invariant to dilations of the metric g . That is, for any α ∈ R + , λ k ( M, αg ) vol(
M, αg ) n = λ k ( M, g ) vol(
M, g ) n . Since vol(
M, αg ) = α n vol( M, g ), this is equivalent to λ k ( M, αg ) = α − λ k ( M, g ). For surfaces( n = 2), Λ k ( M, g ) = λ k ( M, g ) vol(
M, g ) is invariant to dilations of the metric g .4. (Spectrum of disconnected manifolds) If ( M, g ) is a disconnected manifold, M = M ∪ M ,then σ ( M, g ) = σ ( M , g ) ∪ σ ( M , g ).5. (Weyl’s Law) Let N ( λ ) := { λ k ( M, g ) : λ k ( M, g ) ≤ λ } , counted with multiplicity. Then N ( λ ) ∼ ω n vol( M, g )(2 π ) n λ n/ as λ ↑ ∞ , where ω n = π n Γ( n +1) is the volume of the unit ball in R n . In particular, λ k ∼ (2 π ) ω n n vol( M, g ) n k n as k ↑ ∞ . .1 Related work We briefly summarize some related work. A recent review was given by Penskoi [Pen13a].Although eigenvalue optimization problems were already proposed by Lord Rayleigh in the late1870s [Ray77] (see also the surveys [Hen06, AB07]), eigenvalue optimization problems posed onmore general surfaces were not studied until the 1970s. The first result in this direction is due toJ. Hersch, who showed that Λ t (0) = Λ ( S , g ) = 8 π ≈ . , attained only by the standard metric (up to isometry) on S [Her70] (see also [Cha84, p.94] or[SY94, Chapter III]). P. C. Yang and S.-T. Yau generalized this result in [YY80], provingΛ t ( γ ) ≤ π (1 + γ ) . In [Kor93], N. Korevaar generalized this result to larger eigenvalues, showing there exists a constant C , such that Λ tk ( γ ) ≤ C (1 + γ ) k. This result shows that the topological spectrum is finite and since Λ ck ( M, [ g ]) ≤ Λ tk ( γ ) for any M ∈ M ( γ ) and g ∈ G ( M ), that conformal eigenvalues are finite as well. In [Nad96], N. Nadirashviliproved that Λ t (1) = Λ ( T , g ) = 8 π √ ≈ . , attained only by the flat metric on the equilateral torus (generated by (1 ,
0) and ( , √ ), see § over all flat tori is attainedonly by the equilateral torus [Ber73]. For k = 2, N. Nadirashvili showed thatΛ t (0) = 16 π ≈ . , attained by a sequence of surfaces degenerating to a union of two identical round spheres [Nad02].Nadirashvili also conjectured that Λ tk (0) = 8 πk , attained by a sequence of surfaces degenerating toa union of k identical round spheres. In [JLN + γ = 2 surfaces arestudied both analytically and computationally and it is conjectured thatΛ t (2) = 16 π ≈ . , (9)attained by a Bolza surface, a singular surface which is realized as a double branched covering ofthe sphere.We next state several relevant results of B. Colbois and A. El Soufi [CS03], from whom we havealso adopted notation for the present work. It is shown that for any Riemannian surface ( M, g )and any integer k ≥
0, Λ ck ( M, [ g ]) ≥ Λ tk (0). Furthermore, for all k ,Λ ck +1 ( M, [ g ]) − Λ ck ( M, [ g ]) ≥ Λ t (0) = 8 π (10)which implies that Λ ck ( M, [ g ]) ≥ πk . This implies thatΛ tk ( γ ) ≥ Λ t(cid:96) ( γ ) + 8 π ( k − (cid:96) ) , for k ≥ (cid:96) ≥ . (11) We state the 2-dimensional results here for simplicity, but several of these results are proven for general dimension. k -th topological eigenvalue must be at least as large as the eigenvalueassociated with the surface constructed by gluing k − (cid:96) balls of the appropriate volume to the surfacewhich maximizes the (cid:96) -th eigenvalue; see § (cid:96) = 0, (11) givesΛ tk ( γ ) ≥ πk. Finally, for any fixed integer k ≥
0, the function γ (cid:55)→ Λ tk ( γ ) is non-decreasing.Recently it has been shown (independently by several authors) that the supremum in (2) forthe first conformal eigenvalue, Λ c ( M, [ g ]), is attained by an extremal metric, g (cid:63) ∈ [ g ], and severalresults on the regularity of g (cid:63) have been proven [NS10, Pet13, Kok14]. In particular, g (cid:63) is smoothand positive, up to a finite set of some conical singularities on M . G. Kokarev also studies theexistence and regularity of higher conformal eigenvalues Λ ck ( M, [ g ]) [Kok14].Closely related to conformal and topological spectra is the study of extremal metrics on closedsurfaces, on which there has recently been significant development [JNP06, Lap08, Pen12, Pen13c,Pen13b, Kar13, Kar14]. A Riemannian metric g on a closed surface M is said to be an extremalmetric for Λ k ( M, g ) if for any analytic deformation g t such that g = g the following inequalityholds: ddt Λ k ( M, g t ) (cid:12)(cid:12)(cid:12) t ↓ ≤ ≤ ddt Λ k ( M, g t ) (cid:12)(cid:12)(cid:12) t ↑ . Recently, M. Karpukhin [Kar13] investigated a number of extremal metrics studied in [Pen12,Pen13b, Kar14, Lap08] and showed, by direct comparison with the equilateral torus glued to kissingspheres, that none are maximal. This is precisely the configuration which, based on numericalevidence, is conjectured to be maximal in the present paper.For dimension n ≥
3, the topological spectrum does not exist. Indeed, H. Urakawa [Ura79] founda sequence of Riemannian metrics, { g n } n , of volume one on the sphere S such that λ ( S , g n ) → ∞ .B. Colbois and J. Dodziuk showed that every compact manifold, M , with dimension n ≥ g with arbitrarily large first eigenvalue, λ ( M, g ) [CD94].In [Fri79], S. Friedland studies the problem of finding a metric with L ∞ constraints within itsconformal class to minimize (increasing) functions of the Laplace-Beltrami eigenvalues. For thesphere, S , he shows that the infimum is attained at a metric which is bang-bang, i.e. , activatesthe pointwise constraints almost everywhere. Note that these results do not shed light on the maximization problem , (5); we do not expect a conformal factor achieving the supremum in (5) tobe bang-bang.There are also a number of other types of bounds for eigenvalues on Riemannian manifolds. Inparticular, there are a number of both upper and lower bounds for Laplace-Beltrami eigenvalues ofmanifolds with positive Ricci curvature (see, for example, [Cha84, Ch. III], [Kro92], and [LL10]).[GNP09, Pet14] give upper bounds on the second eigenvalue of n -dimensional spheres for confor-mally round metrics. [PS09, CSG10, CDS10] study isoperimetric problems for Laplace-Beltramieigenvalues of compact submanifolds. Given two oriented, 2-dimensional Riemannian manifolds, ( M , g ) and ( M , g ), a conformal map-ping is an orientation-preserving diffeomorphism h : M → M such that h ∗ ( g ) = ωg where ω isa real-valued positive smooth function on M . We say that ( M , g ) and ( M , g ) are conformally6quivalent (or have the same complex structure if one identifies the induced Riemann surface) ifthere exists a conformal mapping between them. The moduli space of genus γ , M γ , is the set ofall conformal equivalence classes of closed Riemannian surfaces of genus γ . Roughly speaking, themoduli space parameterizes the conformal classes of metrics for a given genus.Here, we introduce some very basic results from moduli theory for genus γ = 0 and γ =1 surfaces. By the Uniformization Theorem, every closed Riemann surface of genus γ = 0 isconformally equivalent to the Riemann sphere, so the moduli space consists of a single point [IT92].Every genus γ = 1 Riemann surface is conformally equivalent to a Riemann surface C / Γ τ where,for given τ ∈ H , Γ τ = { m + nτ : m, n ∈ Z } is a lattice group on C . Here H = { τ ∈ C : (cid:61) τ > } denotes the upper half plane. Theorem 2.1. [IT92, Theorem 1.1] For any two points τ and τ (cid:48) in the upper half-plane, the twotori C / Γ τ and C / Γ τ (cid:48) are conformally equivalent if and only if τ (cid:48) ∈ P SL (2 , Z ) τ := (cid:26) aτ + bcτ + d : a, b, c, d ∈ Z and ad − bc = 1 (cid:27) where P SL (2 , Z ) denotes the projective special linear group of degree two over the ring of integers. Thus, the moduli space for genus γ = 1, can be represented as the quotient space H/P SL (2 , Z )and the fundamental domain is the green shaded area in Figure 2(right). The moduli spaces forsurfaces with genus γ ≥ M γ is non-trivial.To find the topological spectrum (3) in practice, we use the moduli space to parameterize theconformal classes of metrics [ g ]. In the following section we discuss how the conformal factor ω isvaried within each conformal class. In this section, we compute the variation of a simple Laplace-Beltrami eigenvalue within the con-formal class. General variations of a Laplace-Beltrami eigenvalue with respect to the conformalfactor are discussed in [SI08]. In this work, we only require the variation of a simple eigenvalue.Let (
M, g ) be a fixed Riemannian manifold and consider the conformal class, consisting ofmetrics ωg , where ω is a smooth, positive-valued function on M . Using (7), the Laplace-Beltramioperator on ( M, ωg ) is expressed as∆
M,ωg f = 1 ω n/ (cid:112) | g | ∂ i (cid:16) ω n − (cid:112) | g | g ij ∂ j f (cid:17) . (12) Proposition 2.2.
Let ( λ, ψ ) be a simple eigenpair of − ∆ M,ωg . The variation of λ with respect toa perturbation of the conformal function ω is given by (cid:28) δλδω , δω (cid:29) ωg = 1 (cid:104) ψ, ψ (cid:105) ωg (cid:28) − n λω − ψ + n − ω − (cid:107)∇ ωg ψ (cid:107) ωg , δω (cid:29) ωg (13) In particular, for n = 2 , (cid:28) δλδω , δω (cid:29) ωg = − λ (cid:10) ω − ψ , δω (cid:11) ωg (cid:104) ψ, ψ (cid:105) ωg = − λ (cid:10) ψ , δω (cid:11) g (cid:104) ωψ, ψ (cid:105) g (14)7 roof. Taking variations with respect to ω , taking the ( M, ωg )-inner product with ψ , and usingthe eigenvalue equation, − ∆ M,ωg ψ = λ ψ , yields δλ (cid:104) ψ, ψ (cid:105) ωg = (cid:28) ψ, n ω − δω ( − λψ ) − n −
22 div (cid:2) ( ω − δw ) ∇ ωg ψ (cid:3)(cid:29) ωg . Applying Green’s formula yields (13).
It is useful to consider the spectrum of a disconnected union of a surface (
M, g ) and the sphere( S , g ), denoted ( M (cid:48) , g (cid:48) ). Generally, the spectrum of disconnected manifolds consists of a union ofthe spectra of the connected components. Here, we consider the case where the sphere is dilatedsuch that the k -th eigenvalue of ( M, g ) is equal to the first eigenvalue of ( S , g ). Consider thedilation ( S , g ) (cid:55)→ ( S , αg ) . We choose α such that λ ( S , αg ) = λ k ( M, g ) implying α − λ ( S , g ) = λ k ( M, g ) . Since ( S , αg ) contributes an extra zero eigenvalue, the ( k + 1)-th eigenvalue of the disjoint union( M (cid:48) , g (cid:48) ) is then λ k ( M, g ). The ( k + 1)-th volume-normalized eigenvalue of ( M (cid:48) , g (cid:48) ) is thenΛ k +1 ( M (cid:48) , g (cid:48) ) = λ k ( M, g ) · (cid:0) vol( S , αg ) + vol( M, g ) (cid:1) = λ k ( M, g ) · α vol( S , g ) + λ k ( M, g ) · vol( M, g )= Λ ( S , g ) + Λ k ( M, g ) . We remark that ( M (cid:48) , g (cid:48) ) can be viewed as the degenerate limit of a sequence of surfaces [CS03]. Fix k ≥
1. Let (
M, g ) be a smooth, closed Riemannian surface and 0 < ω − < ω + . Write A = A ( M, g , ω − , ω + ). Our proof of existence employs the direct method in the calculus of variationsand follows [CM90, Hen06]. We first show that the supremum of Λ k ( M, g , · ) on A , as defined in(5), is finite and Λ (cid:63)k ( M, g , ω − , ω + ) ≤ λ ck ( M, [ g ]). Let ω ∈ A be arbitrary. By assumption, ( M, g )is compact, so A ⊂ L . Thus, C ∞ is dense in A . Using the weak* continuity of Λ k ( M, g , · ), thereexists an ˜ ω ∈ C ∞ ∩ A with Λ k ( M, g , ω ) ≤ Λ k ( M, g , ˜ ω ) + (cid:15). Taking (cid:15) ↓ (cid:63)k ( M, g , ω − , ω + ) ≤ λ ck ( M, [ g ]) < ∞ .Let { ω (cid:96) } ∞ (cid:96) =1 be a maximizing sequence, i.e. , lim (cid:96) ↑∞ Λ k ( M, g , ω (cid:96) ) → Λ (cid:63)k . Since A is weak*sequentially compact, there exists a ω (cid:63) ∈ A and a weak* convergent sequence { ω (cid:96) } ∞ (cid:96) =1 such that ω (cid:96) → ω (cid:63) [CM90, Hen06]. Since the mapping ω → Λ k ( M, g , ω ) is weak* continuous over A ,Λ (cid:63)k = lim (cid:96) ↑∞ Λ k ( M, g , ω (cid:96) ) = Λ k ( M, g , ω (cid:63) ) [CM90, Hen06].For any (cid:15) >
0, by the definition of supremum in (2), there exists an ¯ ω ∈ C ∞ ( M ) such that0 ≤ Λ ck ( M, [ g ]) − Λ k ( M, ¯ ωg ) ≤ (cid:15). M is a compact surface, there exists ω + ( (cid:15) ) > ω − ( (cid:15) ) > ω ∈ A ( M, g , ω − ( (cid:15) ) , ω + ( (cid:15) )).Using the optimality of Λ (cid:63)k , we have thatΛ ck ( M, [ g ]) − (cid:15) ≤ Λ k ( M, ¯ ωg ) = Λ k ( M, g , ¯ ω ) ≤ Λ (cid:63)k ( M, g , ω − ( (cid:15) ) , ω + ( (cid:15) )) . (cid:3) Consider S = { ( x, y, z ) ∈ R : x + y + z = 1 } and let ι : S (cid:44) → R be the inclusion. Let g := ι ∗ ( dx + dy + dz ) be the Riemannian metric on S induced from the Euclidean metric dx + dy + dz on R . Consider the parameterization x = cos φ sin θ, y = sin φ sin θ, z = cos θ, where θ ∈ [0 , π ] is the colatitude and φ ∈ [0 , π ] is the azimuthal angle. We compute vol( S , g ) =4 π . In these coordinates, the Laplace-Beltrami operator is given by∆ f = 1sin θ ∂ θ (sin θ ∂ θ f ) + sin − θ ∂ φ f. The eigenvalues of the Laplacian on ( S , g ) are of the form (cid:96) ( (cid:96) + 1), (cid:96) = 0 , , . . . , each withmultiplicity 2 (cid:96) + 1. It follows by scaling that the eigenvalues of a sphere of area 1 are Λ( S , g ) =4 π(cid:96) ( (cid:96) + 1). Typically, the spherical harmonic functions , denoted Y (cid:96),m ( θ, φ ), are chosen as a basisfor each eigenspace. Numerical values of the volume-normalized eigenvalues, Λ k ( S , g ), are listedin Table 1 for comparison. Remark 3.1.
We remark that there are other (spatially dependent) metrics on the sphereisometric to g and hence have the same Laplace-Beltrami spectrum. This impacts the uniquenessof optimization results presented later. An example of such a metric is constructed as follows.Let N = (0 , ,
1) and S = (0 , , −
1) be the north pole and south pole of S . There is a C ∞ diffeomorphism (stereographic projection) π : S − { N } −→ R , π ( x, y, z ) = (cid:16) x − z , y − z (cid:17) . Theinverse map is given by π − : R −→ S − { N } , π − ( u, v ) = (cid:18) u u + v , v u + v , − u + v u + v (cid:19) . Let h := ( π − ) ∗ g be the pullback Riemannian metric on R . Then h = 4( du + dv )(1 + u + v ) . For any α ∈ R , define the dilation T α : R −→ R by T α ( u, v ) = ( e α u, e α v ). In particular, T is theidentity map. For each α ∈ R , we define the following Riemannian metric on S − { N } , g α := ( π − ◦ T α ◦ π ) ∗ g = π ∗ T ∗ α h = 1(cosh( α ) + sinh( α ) · z ) g . See http://dlmf.nist.gov/14.30 . § g α extends to a C ∞ Riemannian metric on S with constant sectional curvature +1. Notethat when α = 0, the right hand side recovers g .The diffeomorphism π − ◦ T α ◦ π : S −{ N } −→ S −{ N } extends to a diffeomorphism φ α : S −→ S , and g α = φ ∗ α g . So φ α : ( S , g α ) → ( S , g ) is an isometry and ι ◦ φ α : ( S , g α ) −→ ( R , dx + dy + dz ) is an isometric embedding. The isometric conformal factor for α = is plotted in Figure1. To plot this conformal factor on the sphere in Figure 1 (and again for Figures 9 and 10(left)),we have used the Hammer projection, x = 2 √ φ sin θ (cid:113) φ cos θ , y = √ φ (cid:113) φ cos θ , where θ ∈ [0 , π ] is the azimuthal angle (longitude) and φ ∈ (cid:2) − π , π (cid:3) is the altitudinal angle(latitude). k “kissing” spheres Let ( S , g ) be the sphere embedded in R with the canonical metric. We consider k copies of( S , g ) and bring them together in R , so that they are “barely touching”. (This can be madeprecise by considering a sequence of surfaces degenerating in this configuration [CS03].) We referto this configuration as k kissing spheres . It follows from § k kissing spheres will have k zero eigenvalues (0 = λ = . . . = λ k − ) with corresponding eigenfunctions localized and constanton each sphere. The first nonzero volume-normalized eigenvalue isΛ k = 8 πk ( λ k has multiplicity 3 k ) . (15)The corresponding eigenfunctions can be chosen to be spherical harmonic functions supported oneach single sphere. Numerical values of the k -th eigenvalue of k kissing spheres are listed in Table1 for comparison. 10 .3 Spectrum of flat tori The flat torus is generated by identification of opposite sides of a parallelogram with the sameorientation. Consider the flat torus with corners (0 , t , (1 , t , ( a, b ) t , and (1 + a, b ) t . We referto this torus as the ( a, b )-flat torus. This is isometric to the quotient of the Euclidean plane bythe lattice L , R /L , where L is the lattice generated by the two linearly independent vectors, b = (1 , t and b = ( a, b ) t .The spectrum of the ( a, b )-flat torus can be explicitly computed [Mil64, GT10, LS11]. Define B = ( b , b ) = (cid:18) a b (cid:19) . The dual lattice L ∗ is defined L ∗ = (cid:8) y ∈ R : x · y ∈ Z , ∀ x ∈ L (cid:9) and has a basis given by thecolumns of D = ( B t ) − . For the ( a, b )-flat torus, we compute D = ( d , d ) = ( B t ) − = (cid:18) − ab b (cid:19) . Each y ∈ L ∗ determines an eigenfunction ψ ( x ) = e πıx · y with corresponding eigenvalue λ = 4 π (cid:107) y (cid:107) .Since y ∈ L ∗ = ⇒ − y ∈ L ∗ , each nontrivial eigenvalue has even multiplicity. It follows that theeigenvalues of the ( a, b )-flat torus are of the form λ ( a, b ) = 4 π (cid:2) c (cid:0) a /b (cid:1) − c c a/b + c /b (cid:3) , ( c , c ) ∈ Z . More precisely, we can write a Courant-Fischer type expression for the k -th eigenvalue, λ k ( a, b ) = min E ⊂ Z | E | = k +1 max ( c ,c ) ∈ E π (cid:2) c (cid:0) a /b (cid:1) − c c a/b + c /b (cid:3) . (16)For example, the first eigenvalue of the ( , √ )-torus, λ = π (multiplicity 6), is obtained when( c , c ) = ( ± , , ± (1 , , ±
1) implying Λ = λ b = π √ ≈ .
58. Numerical values ofvolume-normalized Laplace-Beltrami eigenvalues, Λ k ( a, b ) := λ k ( a, b ) · b for the square flat torus,( a, b ) = (0 , a, b ) = ( , √ ) are listed in Table 1 for comparison.It is useful to consider the linear transformation from the [0 , π ] square to the ( a, b )-flat torus, (cid:18) uv (cid:19) = 12 π (cid:18) a b (cid:19) (cid:18) xy (cid:19) and (cid:18) xy (cid:19) = 2 πb (cid:18) b − a (cid:19) (cid:18) uv (cid:19) . (17)See Figure 2. The pullback metric on the square is then given by14 π (cid:18) aa a + b (cid:19) . Using (12), we obtain the Laplace-Beltrami operator on the square∆ a,b = 4 π b (cid:2) ( a + b ) ∂ x − a∂ x ∂ y + ∂ y (cid:3) . (18)By construction, this mapping is an isometry and hence the eigenvalues of the flat Laplacian on the( a, b )-flat torus are precisely the same as the eigenvalues of ∆ a,b on [0 , π ] (with periodic boundaryconditions). 11 π ,0)(2 π ,2 π )(0,2 π ) xy (0,0) (1,0) (1+a,b)(a,b)uv a b −1 −0.5 0 0.5 100.511.52 Figure 2: (left)
Coordinates used in the construction of a flat torus. (right)
The fundamentaldomain for the moduli space of genus γ = 1 Riemannian surfaces. See § § b . In this section, we consider the optimization problemsup ( a,b ) ∈ R Λ k ( a, b ) , where Λ k ( a, b ) := b · λ k ( a, b ) . (19)Up to isometry and homothety (dilation), there is a one-to-one correspondence between the modulispace of flat tori and the fundamental region, F := { ( a, b ) ∈ R : a ∈ ( − / , /
2] and a + b ≥ } , (20)as illustrated in Figure 2(right). It follows that the admissible set in (19) can be reduced to F . Tosee this more explicitly, we prove in the following proposition that there exist three transformationsof the parameters ( a, b ) which preserve the value of Λ k ( a, b ). The first two are isometries and thethird corresponds to a rotation and homothety. The last two are due to the SL (2 , Z ) invarianceof Z [IT92]. Each transformation is illustrated in Figure 3. By composing these transformations,the fundamental domain can be restricted to F and furthermore, on F , eigenvalues are symmetricwith respect to the b -axis. Proposition 3.2.
The value of Λ k ( a, b ) := b · λ k ( a, b ) is invariant under the transformations ( a, b ) (cid:55)→ ( − a, b ) , ( a, b ) (cid:55)→ ( a + 1 , b ) , and ( a, b ) (cid:55)→ (cid:18) − aa + b , ba + b (cid:19) . Proof.
The first transformation is an isometry of the flat torus and leaves the spectrum, and henceΛ k , invariant.Suppose that ψ a,b ( u, v ) is an eigenfunction of the ( a, b )-flat torus. Define the function on the( a + 1 , b )-flat torus, ψ a +1 ,b ( u, v ) = (cid:26) u a,b ( u, v ) if v > ba ( u − ψ a,b ( u − , v ) if v ≤ ba ( u − . − a,b) (1 − a,b)uv (0,0) (1,0)(1+a,b)(a,b) uv (0,0) (1,0) (1+a,b)(a,b) uv (1+a ,b )(a ,b ) Figure 3: An illustration of the transformations of flat tori in Proposition 3.2. See § ψ a,b ( u, v ) is periodic, ψ a +1 ,b ( u, v ) is periodic too. The function constructed is an eigenfunctionof the flat tori ( a + 1 , b ) with the same eigenvalue.To check invariance with respect to the third transformation, we consider the mapping(˜ x, ˜ y ) = ( − y, x ) , and (cid:16) ˜ a, ˜ b (cid:17) = (cid:18) − aa + b , ba + b (cid:19) . We then have∆ ˜ x, ˜ y (˜ a, ˜ b ) ˜ u = 4 π (cid:16) ba + b (cid:17) (cid:34)(cid:32)(cid:18) − aa + b (cid:19) + (cid:18) ba + b (cid:19) (cid:33) ˜ u ˜ x ˜ x − (cid:18) aa + b (cid:19) ˜ u ˜ x ˜ y + ˜ u ˜ y ˜ y (cid:35) = λ ˜ u = ⇒ π b (cid:2) u yy − au xy + ( a + b ) u xx (cid:3) = λ (cid:18) ba + b (cid:19) u = λ ˜ bu Thus, the spectrum scales by the factor a + b , but Λ k is invariant.Proposition 3.2 allows us to reduce the optimization problem (19) toΛ (cid:63)k = max { Λ k ( a, b ) : ( a, b ) ∈ F } . (21)The following proposition shows that (21) has a solution and gives a local maximum. We denoteby (cid:100)·(cid:101) the ceiling function, (cid:100) x (cid:101) for x > x . Proposition 3.3.
Fix k ≥ . There exists a flat torus represented by a point ( a (cid:63)k , b (cid:63)k ) ∈ F attainingthe supremum in (21) . Furthermore, the maximal value ˜Λ k = max (cid:40) Λ k ( a, b ) : ( a, b ) ∈ F with a + b ≥ (cid:18)(cid:24) k (cid:25) − (cid:19) (cid:41) (22) has the following analytic solution ˜Λ k = 4 π (cid:6) k (cid:7) (cid:113)(cid:6) k (cid:7) − , (23)13 hich is attained by the ( a, b ) -flat torus with ( a, b ) = (cid:18) , (cid:113)(cid:6) k (cid:7) − (cid:19) . The optimal value in (23) is obtained only for the integer lattice values ( c , c ) = (1 , , ( − , , (1 , , ( − , − , (0 , (cid:108) k (cid:109) ) , and (0 , − (cid:108) k (cid:109) ) and thus the maximal eigenvalue has multiplicity 6.Proof. By Proposition 3.2, we may restrict to the set F as defined in (20). Since every eigenvalueof a flat torus has even multiplicity, without loss of generality, we assume k to be even, k = 2 m for m ∈ Z . We consider the Courant-Fischer type expression for the k -th eigenvalue (16) with a trial subspaceof the form E k = { (0 , , (0 , ± , . . . (0 , ± m ) } . (This is equivalent to using (8) and a trial subspace of the form E k = span (cid:8) , e ± ı(cid:96)y (cid:9) m(cid:96) =1 on thesquare.) For each k , we obtain Λ k ( a, b ) = b λ k ( a, b ) ≤ π m b . Let λ (cid:3) k denote the eigenvalues of the flat tori with ( a, b ) = (0 , k , define ˜ b k := C k λ (cid:3) k .Thus, for b > ˜ b k , b λ k ( a, b ) ≤ λ (cid:3) k . This implies that for each k we can further restrict the admissible set to cl( F ) ∩ { ( a, b ) : b ≤ ˜ b k } ,where cl( · ) denotes closure. Since this is a compact set, the supremum is attained.To show (23), we rewrite the optimization problem using the expression for Laplace-Beltramieigenvalues of flat tori in (16),max ( a,b ) min E ⊂ Z | E | = k +1 max ( c ,c ) ∈ E Λ( a, b ; c , c ) where Λ( a, b ; c , c ) := 4 π (cid:20) ( c a − c ) b + c b (cid:21) . (24)In (24), we can rewriteΛ( a, b ; c , c ) = c t A ( a, b ) c where A ( a, b ) = 4 π b (cid:18) a + b − a − a (cid:19) and c = (cid:18) c c (cid:19) . Furthermore, for every ( a, b ) ∈ F , we compute(tr A ) − A ) = 16 π b (cid:2) ( a + b + 1) − b (cid:3) = 16 π b (cid:2) a + ( b − (cid:3) (cid:2) a + ( b + 1) (cid:3) ≥ , which shows that each sublevel set of the quadratic form can be viewed as an ellipse, circu-lar for ( a, b ) = (0 , a, b )-flat torus (16) canbe interpreted as follows. We consider increasingly large sub-level-sets of the ( a, b )-ellipse, i.e. , { ( x, y ) : Λ( a, b ; x, y ) ≤ γ } for increasing γ . Eigenvalues occur every time the sub-level-sets of the14llipse enclose a new integer lattice point. We thus interpret (24) as finding the ( a, b )-parameterizedellipse for ( a, b ) ∈ F whose k -th smallest enclosed value on the integer lattice is maximal.When k = 1 (or equivalently, k = 2), we have from (24) that˜Λ = max ( a,b ) ∈ F (cid:26) min c ∈ E \ (0 , c t A ( a, b ) c (cid:27) ≤ max ( a,b ) ∈ F (0 , A ( a, b )(0 , t = max ( a,b ) ∈ F π b = 8 π √ . However, if we choose ( a, b ) = (cid:16) , √ (cid:17) , and solve the inner optimization problem in (24) to findthe normalized eigenvalue, we obtain Λ ( a, b ) = π √ . This implies that ˜Λ = π √ .Thus we can assume k >
2. Let m > k = 2 m . Observe that for b > m , the first k nontrivial eigenvalues are obtained from (24) by choosing c = 0 and c = ± , ± , . . . , ± m . In thiscase, we find that Λ k = 4 π m /b ≤ π m , attained in the case where ( a, b ) = (cid:0) , m (cid:1) . We concludethat ˜Λ m ≥ π m and that we can restrict the admissible set to b ≤ m .We consider candidate subsets E jk ⊂ Z , j = 1 , E k = { (0 , , (0 , ± , . . . (0 , ± m ) } E k = { (0 , , (0 , ± , . . . (0 , ± ( m − , ( ± , } From (24), we have that ˜Λ k ≤ max ( a,b ) ∈ Fb ≤ m min j =1 , max c ∈ E jk Λ( a, b, c , c )We see that for n < m ,Λ( a, b, , m ) = 4 π m b ≥ π n b = Λ( a, b, , ± n )and so the elements in E k are dominated by ( c , c ) = (0 , m ). Thus,max c ∈ E k Λ( a, b, c , c ) = 4 π m b . Looking at E k , we have to compare the functions Λ( a, b, ,
0) = 4 π (cid:16) a + b b (cid:17) and Λ( a, b, , m −
1) = π ( m − b . If a + b ≥ ( m − then the first term dominates. Thus, we have shown that if a + b ≥ ( m − thenΛ k ( a, b ) ≤ π · max ( a,b ) ∈ Fb ≤ m min (cid:26) a b + b, m b (cid:27) ≤ π · max √ ≤ b ≤ m min (cid:26) / b + b, m b (cid:27) The first term is increasing for b ≥ √ . The second term is decreasing in b . The optimal value of b is found by setting the two terms equal to each other. They are equal at b = (cid:112) m − / π m √ m − / . Thus, for all ( a, b ) ∈ F , with a + b ≥ ( m − we have thatΛ k ( a, b ) ≤ π m (cid:112) m − / . a, b ) = (cid:16) , (cid:112) m − / (cid:17) . Equation (23) then follows from the substitution m (cid:55)→ (cid:6) k (cid:7) . Remark 3.4.
We note that the admissible sets in (22) and (21) agree for k = 1 , , , π √ is the largest first eigenvalue for any flat torus of volumeone.In Figure 4, we plot Λ k ( a, b ) for k = 1 . . .
16 and ( a, b ) ∈ F . Each eigenvalue has multiplicitytwo, so only odd values of k are shown. Note that Λ k ( a, b ) has local maxima which are not globallymaxima. We tabulate the values of the maximum of Λ k ( a, b ) in Table 1 for k = 1 , . . . , Remark 3.5.
We conjecture that the solutions to the optimization problems in (22) and (21)agree. According to the proof of Proposition 3.3, this conjecture is equivalent to the statement: for a + b < ( m − with m ≥
3, the ellipse E ( a, b ) = (cid:40) c ∈ R : c t A ( a, b ) c ≤ π m (cid:112) m − / (cid:41) contains at least 1 + 2 m integer points.The maximal value for k = 2, Λ (cid:63) = 45 .
58, is less than the value for the 2-kissing spheres,Λ = 50 .
26. Generally, for all k (cid:54) = 1 ,
3, the maximum value for Λ (cid:63)k is less than the value for k kissing spheres. Since the topological spectrum is a non-decreasing function of the genus [CS03], thisimplies that flat tori do not attain the genus γ = 1 topological spectrum for k (cid:54) = 1 ,
3. Since, by (10),Λ t (1) ≥ Λ t (1)+4 π ≈ .
85, a flat tori also does not attain the genus γ = 1 topological spectrum for k = 3. Thus, for k ≥
2, to study the topological spectrum, we require an inhomogeneous conformalfactor.
To provide another comparison, we consider the torus embedded in R with parameterization, x ( u, v ) = (( r cos u + R ) cos v, ( r cos u + R ) sin v, r sin u ) , u, v ∈ [0 , π ] . Here r >
R > r is the major radius, u is the poloidal coordinate, and v isthe toroidal coordinate. See Figure 5. We consider the metric induced from R , g ( u, v ) = (cid:18) r
00 ( r cos u + R ) (cid:19) . From (12), we obtain the Laplace-Beltrami operator∆ f = r − ( r cos u + R ) − ∂ u ( r cos u + R ) ∂ u f + ( r cos u + R ) − ∂ v f. Noting that the Laplace-Beltrami eigenvalue problem − ∆ ψ = λψ is separable, we take ψ ( u, v ) = φ ( u ) e ımv for m ∈ N to obtain the periodic eigenvalue problem on the interval [0 , π ], − r − ∂ u φ + r − sin u ( r cos u + R ) − ∂ u φ + m ( r cos u + R ) − φ = λφ. (25)16igure 4: The first 16 volume-normalized eigenvalues, Λ k ( a, b ), of flat tori plotted as a function ofthe tori parameters ( a, b ). Each eigenvalue has multiplicity two, so only odd eigenvalues are shown.See § Figure 5: (left)
A diagram of the coordinates used for the embedded tori. (right)
The eigenvaluesof an embedded torus with volume one as the aspect ratio is varied. See § m > Chebfun
Matlab toolbox [DHT14]. Let T a denote the torus with volume(2 π ) Rr = 1 and (squared) aspect ratio a = R/r >
1. In Figure 5, we plot the volume-normalizedLaplace-Beltrami eigenvalues, Λ k ( a ) := λ k ( T a , g ) · vol( T a , g ), as a function of the aspect ratio, a .We remark that a similar figure appears in [GS08], where the eigenvalues are computed using afinite difference method. Numerical values of the eigenvalues for the horn torus ( a = 1) are listedin Table 1 for comparison.Now, consider the problem of maximizing the k -th Laplace-Beltrami eigenvalue over the aspectratio, a , sup a ∈ [1 , ∞ ) Λ k ( a ) . (26)As a → ∞ , for fixed k , it is straightforward to show using the Courant-Fischer formula thatΛ k ( a ) →
0, so there exists an a (cid:63)k which attains the supremum in (26). From Figure 5, we observe that a (cid:63)k is an increasing sequence, corresponding to a sequence of tori with increasingly large aspect ratio.The numerical values of the optimal eigenvalues are listed in Table 1. The maximal eigenvalueshave multiplicity greater than one. Each of the corresponding optimal eigenspaces contain aneigenfunction which is non-oscillatory in the poloidal coordinate and increasingly oscillatory in thetoroidal coordinate ( i.e. , the first eigenfunction of (25) for an increasing sequence in m ). Comparedto, e.g. , the flat tori studied in § Computational methods
In this section, we introduce a numerical method for approximating the conformal and topologicalspectra of a Riemannian surface (
M, g ), as given in (2) and (3). Our method is an adaption of themethods found in [Oud04, Ost10, AF12, OK13, OK14] for shape optimization problems involvingextremal eigenvalues of the Laplacian to the setting of Laplace-Beltrami eigenvalues of Riemanniansurfaces using the computational tools developed in [LWY +
14, SLG + § M, g ),we evolve ω within A ( M, g , ω − , ω + ) to increase Λ k ( M, g , ω ). At each iteration, the variation ofΛ k ( M, g, ω ) with respect to the conformal factor is computed using Proposition 2.2, as describedbelow in § L ∞ ( M ) constraints. The process is iterated until a metric g satisfyingconvergence criteria is obtained. Metrics obtained by this approach are (approximately) localmaxima of Λ k ( M, g ), not necessarily global maxima. We repeat this evolution for many differentchoices of initial metric and choose the conformal factor which yields the largest value of Λ k ( M, g ).For the solution of the optimization problem in (6), we additionally must consider a parameter-ization of the conformal classes. For genus γ = 1, this parameterization ( a, b ) ∈ F is described in § a and b to increase Λ k ( M, g, ω ). The derivatives of λ k ( M, g, ω ) with respect to theparameters a and b are computed in § λ k ( M, ωg ) with respect to the conformal factor, ω , and this formula is only validfor simple eigenvalues. It is well-known that eigenvalues λ k ( M, g ) vary continuous with the metric g , but are not differentiable when they have multiplicity greater than one. In principle, for an an-alytic deformation g t , left- and right-derivatives of λ k ( M, g t ) with respect to t exist [SI08, Pen13a]and could be computed numerically. However, in practice, eigenvalues computed numerically thatapproximate the Laplace-Beltrami eigenvalues of a surface are always simple. This is due to dis-cretization error and finite precision. Thus, we are faced with the problem of maximizing a functionthat we know to be non-smooth, but whose gradient is well-defined at points in which we sam-ple. For a variety of such non-smooth problems, the BFGS quasi-Newton method with an inexactline search has proven to be very effective [LO13], but the convergence theory remains sparse. Inparticular, for this problem, a gradient ascent algorithm will generate a sequence of conformalfactors where the k -th and ( k + 1)-th eigenvalues will converge towards each other. The sequencewill become “stuck” at this point and the objective function values will be relatively small com-pared to the optimal value. As reported in other computational studies of extremal eigenfunctions19Ost10, AF12, OK14], for this problem we observe that a BFGS approximation to the Hessianavoids this phenomena.Finally, in Proposition 1.1, we introduced two constants ω + and ω − which provide point-wisebounds on the conformal factor ω ( x ) for x ∈ M . An approximate solution to (2) can be obtainedby computing the solution to (5) for a sequence of values ω + and ω − such that ω + ↑ ∞ and ω − ↓ ω + and ω − to be large and small constants respectively. Taking sequencestending to ±∞ would be a poor idea as conformal factors with very large or small values reducecomputational accuracy.In the following subsections, we describe the methods used for the computation of the Laplace-Beltrami eigenpairs, as well as compute the variation of Laplace-Beltrami eignenvalues with respectto the conformal factor and moduli space parameters. In this section, we describe the finite element and spectral methods for computing Laplace-Beltramieigenpairs.
Finite Element Method
For some of our eigenpair computations, we use the finite element method (FEM) [RWP06, QBM06,LT05, Bof10], which we briefly describe here. The finite element method is based on the weakformulation of (1), given by (cid:90) M ∇ M ψ · ∇ M η = λ (cid:90) M ψη, ∀ η ∈ C ∞ ( M ) . (27)Numerically, we represent M ⊂ R as a triangular mesh { V = { v i } Ni =1 , T = { T l } Ll =1 } , where v i ∈ R is the i-th vertex and T l is the l-th triangle. We use piecewise linear elements to discretize thesurface, so that the triangular mesh approaches the smooth surface in the L -sense as the meshis refined. We choose linear conforming elements { e i } Ni =1 satisfying e i ( v j ) = δ i,j , where δ i,j is theKronecker delta symbol, and write S = span { e i } Ni =1 . The discrete Galerkin version of (27) is tofind a φ ∈ S , such that (cid:88) l (cid:90) T l ∇ M φ · ∇ M η = λ (cid:88) l (cid:90) T l φ η, ∀ η ∈ S. We define φ = N (cid:88) i x i e i A ij = (cid:88) l (cid:90) T l ∇ M e i ∇ M e j B ij = (cid:88) l (cid:90) T l e i e j , where the stiffness matrix , A , is symmetric and the mass matrix , B , is symmetric and positivedefinite. Both A and B are sparse N × N matrices. The finite element method approximates20 − − − Num. of vertices R e l a t i v e E rr o r Figure 6: Relative error of the finite element method for computing the first 50 Laplace-Beltramieigenvalues on the unit sphere. Each curve in this figure represents one eigenvalue. (Lower eigen-values are more accurate.) See § Ax = λBx, φ = N (cid:88) i x i e i . (28)There are a variety of numerical packages to solve (28). We use Matlab’s built-in function eigs withdefault convergence criteria. This eigenvalue solver is based on Arnoldi’s method [Sor92, LS96].Figure 6 demonstrates the 2nd order of convergence in the mesh size h ( ∼ √ N − ) for the Laplace-Beltrami eigenvalues of the unit sphere; see § R , equipped with the inducedmetric. This mesh has 21,161 vertices. In Figure 7, we plot the first 8 nontrivial eigenfunctions.Note that in Figure 7 and later three-dimensional plots (Figures 11, 12, 13, 17, and 18), we use aMatlab visualization effect, achieved by the command, lighting phong . Although the reflectionmakes it easier to see the three-dimensional structure, it also slightly distorts the color. Numericalvalues of the corresponding volume-normalized eigenvalues are listed in Table 1 for comparison.We use this mesh again in § maximization problems. Lower bounds onthe eigenvalues could also be obtained numerically using non-conforming elements [AD04], however21 = 7 . λ = 16 . λ = 20 . λ = 21 . λ = 42 . λ = 71 . λ = 87 . λ = 95 . § Spectral Method
For eigenvalue computations on the torus, we use a spectral method [Tre00], which we briefly discusshere. We use the transformation, given in (17) and illustrated in Figure 2, that takes the ( a, b )-flattorus to the [0 , π ] square. The Laplace-Beltrami operator, ∆ a,b , on the square is defined in (18).Thus, we seek solutions to the eigenvalue problem∆ a,b ψ = λωψ (29)defined on the [0 , π ] square with periodic boundary conditions. The discrete operators obtainedby spectral collocation for the first and second derivatives on a one-dimensional periodic grid on220 , π ] with (even) N points are represented by the Toeplitz matrices D = − cot h − cot h . . . . . . + cot h + cot h . . . − cot h − cot h . . . ...... . . . . . . + cot h + cot h D (2) = . . . .... . . − csc (cid:0) h (cid:1) . . . + csc (cid:0) h (cid:1) − π h − + csc (cid:0) h (cid:1) . . . − csc (cid:0) h (cid:1) . . .... . . . . Here, h = πN . See, for example, [Tre00, Ch. 3]. The two-dimensional operators are then easilyobtained from D and D (2) using the Kronecker product, ⊗ . That is, if I represents the N × N identity matrix, then D (2) x,x = D (2) ⊗ I, D (2) x,y = 12 (( I ⊗ D ) ∗ ( D ⊗ I ) + ( D ⊗ I ) ∗ ( I ⊗ D )) , and D (2) y,y = I ⊗ D (2) , are N × N discrete approximations to ∂ x , ∂ x,y , and ∂ y respectively. A discrete approximation to(29) is then given by4 π b (cid:104) ( a + b ) D (2) x,x − aD (2) x,y + D (2) y,y (cid:105) v = λ Ω v, v ∈ R N . Here Ω is a diagonal matrix with entries given by the values of ω . This generalized eigenvalueproblem is then solved using Matlab’s built-in function eigs with default convergence criteria. InFigure 8, we give a log-log plot of the relative error of the first 16 eigenvalues for the conformalfactor given by ω ( x, y ) = e cos x +cos y on the equilateral torus. The method is seen to be spectrallyconvergent. Here, we apply Proposition 2.2 to the eigenvalues of the sphere and ( a, b )-flat torus. The resultsare stated as propositions for reference. First, consider the mapping ω (cid:55)→ λ k ( ω ) satisfying − ∆ ψ = ω λ ( ω ) ψ on S . − − − Number of Points, N R e l a ti v e E rr o r Figure 8: Relative error of the spectral method for computing Laplace-Beltrami eigenvalues on atorus. Similar to Figure 6, each of the 16 curves in this figure represents one eigenvalue. (Lowereigenvalues are more accurate.) See § Proposition 4.1.
Let λ ( ω ) be a simple eigenvalue of ( S , ωg ) and corresponding eigenfunction ψ normalized such that (cid:104) ψ, ψ (cid:105) ωg = 1 . Then, δλδω · δω = − λ (cid:104) ψ ω − , δω (cid:105) ωg , where (cid:104) f, h (cid:105) ωg = (cid:82) S f hωdµ g . We next compute the gradient of a Laplace-Beltrami eigenvalue on the ( a, b )-flat tori withrespect to both the conformal factor ω and the parameters a and b . Recall the linear transformationintroduced in § , π ] square to the ( a, b )-flat torus (see Figure 2). Considerthe mapping ( a, b, ω ) (cid:55)→ λ k ( a, b, ω ) satisfying − ∆ a,b ψ = ω λ ( a, b, ω ) ψ on [0 , π ] . (30)where ∆ a,b is defined in (18). Proposition 4.2.
Let λ ( a, b, ω ) be a simple eigenvalue of an ( a, b ) -flat torus with conformal factor ω and corresponding eigenfunction ψ normalized such that (cid:104) ψ, ψ (cid:105) ωg = 1 . Then, ∂λ∂a = −(cid:104) ψ, ω − ∆ a ψ (cid:105) ωg , ∆ a := 4 π b (cid:2) a∂ x − ∂ x ∂ y (cid:3) ∂λ∂b = −(cid:104) ψ, ω − ∆ b ψ (cid:105) ωg , ∆ b := 2 λω ( x, y ) b + 8 π b ∂ x δλδω · δω = − λ (cid:104) ψ ω − , δω (cid:105) ωg , here (cid:104)· , ·(cid:105) ωg is the inner product induced by the metric, (cid:104) f, h (cid:105) ωg = (cid:90) M f hdµ g = (cid:90) [0 , π ] f h (cid:112) | g | dxdy = b π (cid:90) [0 , π ] f ( x, y ) h ( x, y ) ω ( x, y ) dxdy. All computations for the flat torus using the spectral method are done on the domain [0 , π ] (with periodic boundary conditions). Eigenvalue derivatives are computed numerally using theformulae in Proposition 4.2. The operators ∆ a and ∆ b are implemented using the Toeplitz matrices, D and D (2) , and the Kronecker product as discussed in § ∂ x = 12 π ∂ u and ∂ y = a π ∂ u + b π ∂ v , to push these derivatives forward from the square to the flat torus (see Figure 2). We obtain thefollowing result, which is used in the finite element computations on flat tori. Proposition 4.3.
Let λ ( a, b, ω ) be a simple eigenvalue of an ( a, b ) -flat torus with conformal factor ω and corresponding eigenfunction ψ normalized such that (cid:104) ψ, ψ (cid:105) ωg = 1 . Then, ∂λ∂a = −(cid:104) ψ, ω − ∆ a ψ (cid:105) ωg , ∆ a := − b ∂ u ∂ v ∂λ∂b = −(cid:104) ψ, ω − ∆ b ψ (cid:105) ωg , ∆ b := 2 λωb + 2 b ∂ u δλδω · δω = − λ (cid:104) ψ ω − , δω (cid:105) ωg , where (cid:104) f, h (cid:105) ωg = (cid:82) T f hωdµ g is the inner product on the flat torus. In this section, we compute the conformal spectrum for several manifolds and topological spectrumfor genus γ = 0 ,
1. Numerical values of volume-normalized eigenvalues, Λ k , are given in Table 1 forcomparison. By the Uniformization Theorem, any closed genus-0 Riemannian surface (
M, g ) is conformal to S with the canonical metric of constant sectional curvature, g [IT92]. In other words, the modulispace of closed Riemannian surfaces consists of one point and the conformal spectrum for any genus γ = 0 Riemannian surface is identical. In particular, for any genus zero Riemannian surface, ( M, g ),Λ ck ( M, [ g ]) = Λ ck ( S , [ g ]) = Λ tk (0) . In this section, we approximate Λ ck ( S , g ) using the computational methods described in § ,
962 vertices. The optimization problem is solved using a quasi-Newton method,where the gradient of the eigenvalues is computed via Proposition 4.1.25 =1, Λ =25.13 − − − − − Figure 9: A Hammer projection of the best conformal factors found for Λ k , k = 1 , . . . , § (left) A sequence of conformal factorson the sphere to maximize Λ . See § (right) A sequence of tori to maximize Λ . See § k = 1 , , · · · , k = 2, the 1st, 6th, 10th, and 26th iterates of the conformal factor. The meshof the sphere used here has 10,242 vertices. The optimization code is only able to achieve a valueof Λ = 47 .
77 for this grid size and initial condition, however the general pattern of the conformalfactor having two localized maxima is clearly observed.27ersch’s result that the standard metric on S is the only metric up to isometry attaining Λ t (0) issupported in the computational results [Her70]. This numerical result gives just one representativefrom the isometric class; see Remark 3.1, where a conformal factor on the sphere, isometric to theuniform conformal factor, is constructed that gives the same first topological eigenvalue. For k = 2,it was shown in [Nad02] that the maximum is approached by a sequence of surfaces degeneratingto a union of two identical round spheres, a configuration we refer to as two kissing spheres, withsecond eigenvalue Λ t (1) = 16 π ≈ .
26. The value Λ (cid:63) = 50 .
78, obtained numerically is slightlylarger. As discussed in § S corresponding to “two kissing spheres” is the one shown in the topright panel of Figure 9.From Figure 9, we further observe that the k -th eigenvalue is large precisely when the metrichas k localized regions with large value. This corresponds to the “ k -kissing spheres” surface asdescribed in § k , the regions where the metric is localizedis increasingly small. Since it is possible for the eigenfunctions to become very concentrated atthese regions of concentrated measure, we reason that for larger values of k , to improve accuracywe should use a finer mesh at these regions or, equivalently, deform the surface at these points tolocally enlarge the volume. We choose the later option, and consider a mesh consisting of k spheres“glued” together which approximates k kissing spheres. For example, to construct the mesh for k = 2, we remove one element (triangle) from the mesh representing each sphere and then identifythe edges associated with the missing faces of the two punctured balls. On those glued meshes, weagain maximize Λ as a function of the conformal factor, ω . The best conformal factor found for k = 1 , . . . , πk .To further test these optimal conformal factors, we consider configurations of spheres withdifferent sizes; see Figure 12. For Λ , we consider a mesh approximating a sphere with radius 1 / to Λ , we consider meshes approximating glued spheres. The larger spheres have radius 1and the smaller spheres have radius 1 /
2. In each case, we verified that the constructed surface hasgenus γ = 0 using the Euler characteristic of the mesh. In each case, the conformal factor is veryflat on each sphere and the optimal values obtained are very close to 8 πk .As another computational experiment, we again consider the mesh of “Homer Simpson”, asdiscussed in § ω (cid:63) which attains Λ c and plotthe function u = log( ω (cid:63) ) / .
01, 2 .
01, 2 .
01, 5 .
93, 6 .
03, 6 .
04, 6 .
12 and 6 .
17. We see that the firstthree eigenvalues are close to the first three eigenvalues of the unit sphere ( λ = 2 . λ = 6 . In this section, we study the first conformal eigenvalue of the ( a, b )-flat tori for various values of( a, b ). For all computations, we use the spectral method described in §
4. For a comparison, we firstcompute the first non-trivial eigenvalue of the ( a, b )-flat tori.28 = 25 .
13 Λ = 50 .
28 Λ = 75 . = 100 .
52 Λ = 125 .
66 Λ = 150 . k , k = 1 , . . . k -kissing spheres. See § a, b )-flat torusis λ ( a, b ) = π b . Thus, the volume normalized eigenvalue is given by Λ ( a, b ) = π b . Note thatΛ ( a, b ) is monotone decreasing in b and does not depend on a . When b = √ , we recover theoptimal value Λ (cid:63) = π √ , as discussed in § ( a, b ) for ( a, b ) ∈ F is given in Figure14(left). Note that this is the same as the top left panel of Figure 4, except the range of values of b is smaller. Values of Λ ( a, b ) for a small selection of parameters ( a, b ) are also tabulated Figure14. The parameters ( a, b ) chosen are indicated by crosshairs, ‘+’, in Figure 14(left).We abbreviate the first conformal eigenvalue of the flat torus, Λ c ( T a,b , [ g ]), by Λ c ( a, b ). Werecall from (10) that Λ c ( a, b ) > π ≈ .
13. Clearly we have that Λ c ( a, b ) ≤ π √ ≈ .
58 withequality only at ( a, b ) = ( , √ ). Proposition 5.1.
For fixed a , Λ c ( a, b ) is a non-increasing function in b . = 25 .
13 Λ = 50 .
26 Λ = 75 . = 100 .
50 Λ = 125 .
63 Λ = 150 . k , k = 1 , . . . k − /
2. See § Proof.
The Rayleigh quotient for the first nonzero eigenvalue can be written λ ( a, b, ω ) = min (cid:82) ψω =0 (cid:82) ψ ω =1 π (cid:90) [0 , π ] b ( aψ x − ψ y ) + ψ x dxdy. Let a and ω fixed and let b ≥ b . Let ψ be an eigenfunction corresponding to λ ( a, b , ω ) (whichcould have multiplicity greater than one). Then we have that λ ( a, b , ω ) ≤ π (cid:90) [0 , π ] b ( aψ x − ψ y ) + ψ x dxdy ≤ π (cid:90) [0 , π ] b ( aψ x − ψ y ) + ψ x dxdy = λ ( a, b , ω )We conclude that for a and ω fixed, λ ( a, b, ω ) is a non-increasing function in b .30igure 13: A plot of the function u = log( ω (cid:63) ) /
2, where ω (cid:63) is the conformal factor corresponding tothe first conformal eigenvalue Λ c , for a “Homer Simpson” mesh. See § a . Take b > b and let ω be a conformal factor attaining Λ c ( a, b ). Then,Λ c ( a, b ) ≥ λ ( a, b , ω ) by optimality ≥ λ ( a, b , ω ) by the monotonicity of λ ( a, b, ω ) in b = Λ c ( a, b ) . In Figure 14(right), we plot values of Λ c ( a, b ) for ( a, b ) ∈ F , computed on a 40 ×
40 mesh. Aneigenvalue optimization problem was solved for the values of ( a, b ) indicated by crosshairs, ‘+’; theother values were obtained by interpolation. Values of Λ c ( a, b ) for a small selection of parameters( a, b ) are also tabulated. We observe that for fixed a , the value of Λ c ( a, b ) is non-increasing in b ,as proved in Proposition 5.1. We also observe that Λ c ( a, b ) varies smoothly with ( a, b ). In Figure5.2, the optimal conformal factors are plotted on the ( a, b )-tori for these values of ( a, b ). The flatmetric attains the maximal value obtained for the square torus, ( a, b ) = (0 , a, b ) = (1 / , √ / b increases and the torus becomes long and thin, the best conformalfactors found have structure which have higher density along a thin strip. We observe that theoptimal conformal factor continuously deforms as the parameters ( a, b ) change. This is in contrastwith other eigenvalue optimization problems where the optimizing structure can be discontinuouswith changing objective function parameters [OK13, OK14]. In this section, we approximate Λ tk (1) using the computational methods described in §
4. We proceedwith several numerical studies. First we use a spectral method to identify approximate maximizers31 a, b ) Λ ( a, b ) Λ c ( a, b )A ( , √ ) 45.58 45.58B (0 ,
1) 39.48 39.48C ( ,
1) 39.48 40.33D ( , ) 26.32 33.45E (0 ,
2) 19.74 30.97F ( ,
2) 19.74 30.95Figure 14: (left)
The first eigenvalue of ( a, b )-flat tori, Λ ( a, b ), for values of ( a, b ) ∈ F . Selectedvalues of ( a, b ), indicated by crosshairs, ‘+’, are tabulated below for reference. (right) The firstconformal eigenvalue of ( a, b )-flat tori, Λ c ( a, b ), for values of ( a, b ) ∈ F . Selected values are tabulatedbelow for reference. An eigenvalue optimization problem was solved for values of ( a, b ) indicatedby crosshairs, ‘+’; the other values were obtained by interpolation. (bottom) Tabulated values ofΛ ( a, b ) and Λ c ( a, b ) for selected values of ( a, b ). The conformal factors attaining the given valuesof Λ c ( a, b ) plotted in Figure 5.2. See § a, b, ω ) on a flat torus. By examining the structure of the minimizers, we recognizethat the minimizer is obtained by a configuration consisting of the union of an equilateral flat torusand k − ×
60 mesh. As discussed in § γ = 1, as shown by the shaded area in Figure 2(right), parameterizes the conformal classes ofmetrics [ g ]. Thus, any genus γ = 1 surface can be described by a triple ( a, b, ω ) where ( a, b ) ∈ F asin (20) and ω a smooth positive function. The optimization problem is solved using a quasi-Newtonmethod, where the gradient of the eigenvalues with respect to the triple ( a, b, ω ) is computed viaProposition 4.2.Using this computational method, the best triples ( a, b, ω ) found for k = 1 , , and 3 are presentedin Figure 16. To obtain these triples, we chose many different initializations. The initial conditionsused for Figure 1 were the sum of localized Gaussians located at distributed points on the torus. Tofurther illustrate our computational method, we consider a randomly initialized conformal factor.32 conformal factor, Λ =45.59 conformal factor, Λ =39.48 (A) (B) conformal factor, Λ =40.33 conformal factor, Λ =33.45 (C) (D) conformal factor, Λ =30.97 conformal factor, Λ =30.95 (E) (F)Figure 15: A plot of the function u = log( ω (cid:63) ) /
2, where ω (cid:63) is the conformal factor attaining Λ c ( a, b )for the values of ( a, b ) in the Table in Figure 14. See § k = 2, the 0th, 5th, 24th, and 30th iterates of the conformal factoron the ( a, b )-torus. 33 k=1, a=0.5, b=0.866, Λ =45.59 − − − − − k=2, a=0.5, b=0.866, Λ =68.2 k=3, a=0.4601, b=0.8891, Λ =86.91 Figure 16: Maximal triples ( a, b, ω ) obtained for k = 1 (left), k = 2 (center), and k = 3 (right).The color represents the conformal factor, ω . See § t (1) = π √ ≈ .
58 is attained only by the flat metric on the equilateral flat torus, ( a, b ) = (cid:16) , √ (cid:17) [Nad96]. For k = 2, the optimal conformal factor found is mostly flat with one localized maximum. The valueobtained (Λ = 68 .
2) is very close to the value found for the disconnected union of an equilateralflat torus and a sphere of appropriate volumes, Λ = Λ t (1) + Λ t (0) ≈ .
72 (see § k = 3,the optimal conformal factor found is mostly flat with two localized maximum. The value obtained(Λ = 86 .
91) is not as close to 95.85, the value for the disconnected union of an equilateral flattorus and two spheres. For larger values of k , we observe that optimal metrics are mostly flat, buthave k − γ = 0 case described in § k because thelocalized regions are increasingly small. It is thus very difficult to realize metrics which correspondto this configuration using this method.To compute optimal configurations for larger values of k , we proceed as follows as in § k − k − k = 2, we remove one face from the mesh representing the flat tori andone face from the mesh representing the sphere. We then identify the edges associated with themissing faces of these two punctured surfaces. As discussed in § a, b ) = (cid:16) , √ (cid:17) , and spheres of appropriate size, we can obtain k -th eigenvalue at least as large asΛ k = 8 π √ π ( k − . (31)On this mesh, we use the finite element method to compute the Laplace-Beltrami eigenvalues andinitialize a quasi-Newton optimization method using a random conformal factor. We observe thatthe maximal eigenvalue is achieved when the conformal factor is nearly constant over the mesh.See Figure 17, where the optimal values are given by Λ = 70 .
70, Λ = 95 .
80, Λ = 120 .
94, andΛ = 146 .
06 which are indeed very close to those given in (31).To further test these optimal solutions, we consider several other configurations of spheres andflat tori. As shown in Figure 18, we take k = 2 and consider a torus glued to a sphere withradius a factor of 0.7 of the optimal radius. Initializing the optimization method with a constantuniform conformal factor, an optimal conformal factor is achieved where the sphere has a relatively34igure 17: To maximize Λ k , for k = 2, 3, 4, and 5, we consider a mesh of a flat torus glued to 1, 2,3, and 4 kissing spheres. The optimal conformal factors found, displayed here, are nearly constant.See § λ is plotted in Figure 18 (top right). For k = 5, we consider atorus glued to 4 spheres which have radii a factor of 0.75, 0.9, 1.1, and 1.25 of the optimal radius.Again initializing the optimization method with a constant uniform conformal factor, we obtainthe conformal factor in Figure 18 (bottom left). An eigenfunction associated to λ is plotted inFigure 18 (bottom right). In these two experiments, the optimal numerical values Λ = 70 . = 146 . and Λ respectively. The figures on the left display the optimal conformal factorand the figures on the right display an eigenfunction corresponding to λ k . See § § § γ = 1, as verified numerally using the Euler characteristic of the mesh. The first feweigenvalues of this configuration are given in Table 1. The value of Λ is very small as comparedto (31) with k = 2. 36igure 19: Kissing equilateral flat tori. Edges with the same color are glued together. See § We have presented a computational method for approximating the conformal and topological spec-tra, as defined in (2) and (3). Our method is based on a relaxation, given in (5), for which we proveexistence of a minimizer (see Proposition 1.1). Based on the results of extensive computations, wemake the following conjecture.
Conjecture 6.1.
The following hold for the topological spectrum: • Λ tk (0) = 8 πk , attained by a sequence of surfaces degenerating to a union of k identical roundspheres. • Λ tk (1) = π √ + 8 π ( k − , attained by a sequence of surfaces degenerating into a union of anequilateral flat torus and k − identical round spheres. The first part of this conjecture was also stated by Nadirashvili in [Nad02]. A proof of theconjecture involving Λ tk (0) would imply that the lower bound, Λ tk (0) ≥ πk , proven in [CS03,Corollary 1], is tight. This conjecture is proven for k = 1 and k = 2 in [Her70] and [Nad02]respectively. The conjecture involving Λ tk (1) agrees with the eigenvalue gap estimate (10), provenin [CS03], and the result of [Nad96] for k = 1. The relatively large value of Λ k for the configurationconsisting of a union of an equilateral flat torus and k − n = 2, Weyl’s law states that for any fixed surface ( M, g ), Λ k ( M, g ) ∼ πk as k →∞ . The conjectured topological spectrum for genus γ = 0 , tk ( γ ) ∼ πk .Thus, the conjecture implies that for fixed k , there exist surfaces with k -th eigenvalue which exceedthe asymptotic estimate given by Weyl’s law by no more than a factor of two. As a comparison, weproved in § k has a local maximum with value 4 π (cid:6) k (cid:7) (cid:16)(cid:6) k (cid:7) − (cid:17) − .For k large, we obtain Λ k ∼ π k . Noting that 4 π < π < π , this rate lies between Weyl’sestimate and the conjectured topological spectrum for genus γ = 1.In § e.g. γ = 2) could in principle be treated in37he same way [IT92, Bus10], although we do not attempt this here. For genus γ = 2, (9) and thespectral gap (10) together imply that Λ tk (2) ≥ π ( k + 1)where the lower bound is attained by attaching k − k sphere flat torus flat torus torus Simpson tori1 25.13 39.47 45.58 23.21 7.464 34.212 25.13 39.47 45.58 23.21 16.45 34.213 25.13 39.47 45.58 30.63 19.94 43.984 75.39 39.47 45.58 66.58 20.89 43.985 75.39 78.95 45.58 66.58 41.23 78.226 75.39 78.95 45.58 78.80 69.83 78.227 75.39 78.95 136.7 83.71 85.40 78.228 75.39 78.95 136.7 83.71 92.32 78.22 k kissing best flat best embedded Equil. torusspheres torus torus and k − k (15) (23) (26) spheres (31)1 25.13 45.58 23.47 45.582 50.26 45.58 23.47 70.713 75.39 81.55 65.09 95.854 100.5 81.55 65.09 120.95 125.6 120.1 108.34 146.16 150.7 120.1 108.34 171.27 175.9 159.2 150.25 196.38 201.0 159.2 150.25 221.5 k Λ tk (0) Λ tk (1)1 25.13 45.582 50.26 70.713 75.39 95.854 100.5 120.95 125.6 146.16 150.7 171.2Table 1: A comparison of various volume-normalized eigenvalues, Λ k ( M, g ) = λ k ( M, g ) · vol( M, g ).This is equivalent to λ k ( M, g ) after the metric has been normalized to have unit volume. The firsttable are the Laplace-Beltrami eigenvalues of the sphere, square flat torus ( a, b ) = (0 , a, b ) = (cid:16) , √ (cid:17) , horn embedded torus, Homer Simpson, and kissing equilateral flattori as discussed in § § § § § § k kissing spheres, best flat tori,best embedded tori, and the disjoint union of an equilateral torus and k − γ = 0and γ = 1 surfaces. 39 cknowledgements We would like to thank Ahmad El Soufi, Alexandre Girouard, Richard Laugesen, Peter Li, andStan Osher for useful conversations. We would like to thank Melissa Liu for help with the examplein Remark 3.1. We would also like to thank the referees for their valuable comments.
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