High-Energy Eigenfunctions of the Laplacian on the Torus and the Sphere with Nodal Sets of Complicated Topology

HIGH-ENERGY EIGENFUNCTIONS OF THE LAPLACIAN ON THE TORUS AND THE SPHERE WITH NODAL SETS OF COMPLICATED TOPOLOGY A. ENCISO, D. PERALTA-SALAS, AND F. TORRES DE LIZAUR Abstract. Let Σ be an oriented compact hypersurface in the round sphere Sn or in the flat torus Tn,n⩾3. In the case of the torus, Σ is further assumed to be contained in a contractible subset of Tn. We show that for any sufficiently large enough odd integer Nthere exists an eigenfunctions ψof the Laplacian on Snor Tnsatisfying ∆ψ=−λψ (with λ=N(N+n−1) or N2on Snor Tn, respectively), and with a connected component of the nodal set of ψgiven by Σ, up to an ambient diffeomorphism. 1. Introduction Let Mbe a closed manifold of dimension n⩾3 endowed with a smooth Riemannian metric g. The Laplace eigenfunctions of Msatisfy the equation ∆uk=−λkuk, where 0 = λ0< λ1⩽λ2⩽. . . are the eigenvalues of the Laplacian. The zero set u−1 k(0) is called the nodal set of the eigenfunction. The study of the nodal sets of the eigenfunctions of the Laplacian in a compact Riemannian manifold is a classical topic in geometric analysis with a number of important open problems [18, 19]. When the Riemannian metric is not fixed, the nodal set is quite flexible. Indeed, it has been recently shown that [9], given a separating hypersurface Σ in M, there is a metric gon the manifold for which the nodal set u−1 1(0) of the first eigenfunction is precisely Σ. This result has been extended to the class of metrics conformal to a metric g0prescribed a priori [10], and to higher codimension submanifolds arising as the joint nodal set of several eigenfunctions corresponding to a degenerate eigenvalue [5]. For a fixed Riemannian metric, the problem is much more rigid than when one can freely choose a metric adapted to the geometry of the hypersurface that one aims to recover from the nodal set of the eigenfunctions. In this case, the techniques developed in [9, 5, 10] do not work. Nevertheless, since the Hausdorff measure of the nodal sets of the eigenfunctions grows as the eigenvalue tends to infinity [13, 14], one expects that the nodal set may become topologically complicated for high-energy eigenfunctions. Our goal in this paper is to establish the existence of high-energy eigenfunctions of the Laplacian on the round sphere Snand the flat torus Tnwith nodal sets diffeomorphic to a given submanifold. All along this paper, Sndenotes the unit sphere in Rn+1 and Tnis the standard flat n-torus, (R/2πZ)n. 1 2 A. ENCISO, D. PERALTA-SALAS, AND F. TORRES DE LIZAUR More precisely, our main theorem shows that for a sequence of high enough eigenvalues, there exist meigenfunctions of the Laplacian on Snor Tnwith a joint nodal set diffeomorphic to a given codimension msubmanifold Σ. For the construction we need to assume that the normal bundle of Σ is trivial. This means that a small tubular neighborhood of the submanifold Σ must be diffeomorphic to Σ ×Rm. In the statement, structural stability means that any small enough perturbation of the corresponding eigenfunction (in the Cknorm with k⩾1) still has a union of connected components of the nodal set that is diffeomorphic to the submanifold Σ under consideration. Throughout, diffeomorphisms are of class C∞ and submanifolds are C∞and without boundary. Theorem 1.1. Let Σbe a finite union of (disjoint, possibly knotted or linked) codimension m⩾1compact submanifolds of Snor Tn,n⩾3, with trivial normal bundle. In the case of the torus, we further assume that Σis contained in a contractible subset. If m= 1, we also assume that Σis connected. Then for any large enough odd integer Nthere are meigenfunctions ψ1, . . . , ψmof the Laplacian with eigenvalue λ=N(N+n−2) (in Sn) or λ=N2(in Tn), and a diffeomorphism Φsuch that Φ(Σ) is the union of connected components of the joint nodal set ψ−1 1(0) ∩···∩ψ−1 m(0). Furthermore, Φ(Σ) is structurally stable. An important observation is that the proof of this theorem yields a reasonably complete understanding of the behavior of the diffeomorphism Φ, which is, in particular, connected with the identity. Oversimplifying a little, the effect of Φ is to uniformly rescale a contractible subset of the manifold that contains Σ to have a diameter of order 1/N. In particular, the control that we have over the diffeomorphism Φ allows us to prove an analog of this result for quotients of the sphere by finite groups of isometries (lens spaces). Notice that Φ(Σ) is not guaranteed to contain all the components of the nodal set of the eigenfunction. The proof of the main theorem involves an interplay between rigid and flexible properties of high-energy eigenfunctions of the Laplacian. Indeed, rigidity appears because high-energy eigenfunctions in any Riemannian n-manifold behave, locally in sets of diameter 1/√λ, as monochromatic waves in Rndo in balls of diameter 1. We recall that a monochromatic wave is any solution to the Helmholtz equation ∆φ+φ= 0. The catch here is that, in general, one cannot check whether a given monochromatic wave in Rnactually corresponds to a high-energy eigenfunction on the compact manifold. To prove the converse implication, what we call an inverse localization theorem (see Sections 2 and 3), it is key to exploit some flexibility that arises in the problem as a consequence of the fact that large eigenvalues of the Laplacian in the torus or in the sphere have increasingly high multiplicities (for this reason the proof does not work in a general Riemannian manifold). The inverse localization is a powerful tool to ensure that any monochromatic wave in a compact set of Rncan be reproduced in a small ball of the manifold by a high-energy eigenfunction. This allows us to transfer any structurally stable nodal set that can be realized in Euclidean space to high-energy eigenfunctions on Snand Tn. The inverse localization was first introduced in [11] to construct high-energy Beltrami fields on the torus and the sphere with topologically complicated vortex structures, and was also exploited in [6, 7] to solve a problem of M. Berry [2] on knotted nodal lines of high-energy NODAL SETS OF HIGH-ENERGY EIGENFUNCTIONS 3 eigenfunctions of the harmonic oscillator and the hydrogen atom, and in [17] to analyze the nodal sets of the eigenfunctions of the Dirac operator. One should notice that the techniques introduced in [8] to prove the existence of solutions to second-order elliptic PDEs in Rn(including the monochromatic waves) with a prescribed nodal set Σ do not work for compact manifolds. The reason is that the proof is based on the construction of a local solution in a neighborhood of Σ, which is then approximated by a global solution in Rnusing a Runge-type global approximation theorem. For compact manifolds the complement of the set Σ is precompact, so we cannot apply the global approximation theorem obtained in [8]. In fact, as is well known, this is not just a technical issue, but a fundamental obstruction in any approximation theorem of this sort. This invalidates the whole strategy followed in [8] and makes it apparent that new tools are needed to prove the existence of Laplace eigenfunctions with geometrically complex nodal sets in compact manifolds. We finish this introduction with two corollaries. It is known that an oriented codimension one or two submanifold in Snor Tnhas trivial normal bundle [15], therefore the main theorem implies the following: Corollary 1.2. Let Σbe an oriented, compact, connected hypersurface in Snor Tn,n⩾3. In the case of the torus, we further assume that Σis contained in a contractible subset. Then for any large enough odd integer Nthere is an eigenfunction ψof the Laplacian with eigenvalue λ=N(N+n−2) (in Sn) or λ=N2 (in Tn), and a diffeomorphism Φsuch that Φ(Σ) is a structurally stable connected component of the nodal set ψ−1(0). Corollary 1.3. Let Σbe a finite union of (disjoint, possibly knotted or linked) codimension two compact submanifolds in Snor Tn,n⩾3. In the case of the torus, we further assume that Σis contained in a contractible subset. Then for any large enough odd integer Nthere is a complex-valued eigenfunction ψof the Laplacian with eigenvalue λ=N(N+n−2) (in Sn) or λ=N2(in Tn), and a diffeomorphism Φsuch that Φ(Σ) is a union of structurally stable connected components of the nodal set ψ−1(0). The paper is organized as follows. In Sections 2 and 3 we prove an inverse localization theorem for the eigenfunctions of the Laplacian on Snand Tn, respectively. Theorem 1.1 is then proved in Section 4. Finally, in Section 5, we prove a refinement of the inverse localization Theorem on Snthat allows us to approximate several given monochromatic waves by a single eigenfunction of the Laplacian in different small regions of Sn. 2. An inverse localization theorem on the sphere In this section we prove an inverse localization theorem for eigenfunctions of the Laplacian on Snfor n⩾2. We recall that the eigenvalues of the Laplacian on the n-sphere are of the form N(N+n−1), where Nis a nonnegative integer, and the corresponding multiplicity is given by d(N, n) := N+n−1 N2N+n−1 N+n−1. 4 A. ENCISO, D. PERALTA-SALAS, AND F. TORRES DE LIZAUR For the precise statement of the theorem, let us fix an arbitrary point p0∈Sn and take a patch of normal geodesic coordinates Ψ : B→Bcentered at p0. Here and in what follows, Bρ(resp. Bρ) denotes the ball in Rn(resp. the geodesic ball in Sn) centered at the origin (resp. at p0) and of radius ρ, and we shall drop the subscript when ρ= 1. For the ease of notation, we will use the Rm-valued functions φ:= (φ1,··· , φm) and ψ:= (ψ1,··· , ψm), and the action of the Laplacian on such functions is understood componentwise. Theorem 2.1. Let φbe an Rm-valued monochromatic wave in Rn, satisfying ∆φ+ φ= 0. Fix a positive integer rand a positive constant δ0. For any large enough integer N, there is an Rm-valued eigenfunction ψof the Laplacian on Snwith eigenvalue N(N+n−1) such that    φ−ψ◦Ψ−1· N   Cr(B) ⩽δ0. To prove Theorem 2.1, we will proceed in two successive approximation steps. First, we will approximate the function φin Bby an Rm-valued function ϕthat can be written as a finite sum of terms of the form cj |x−xj|n 2−1Jn 2−1(|x−xj|) with cj∈Rmand xj∈Rn,j= 1, ..., N0, for N0large enough (Proposition 2.2 below). Notice that any function of this form is a monochromatic wave. In the second step, we show that there is a collection of meigenfunctions (ψ1, ..., ψm) =: ψ in Snwith eigenvalue N(N+n−1) such that, when considered in a ball of radius N−1, they approximate ϕ:= (ϕ1, ..., ϕm) in the unit ball, provided that Nis large enough. Proposition 2.2. Given any δ > 0, there is a constant R > 0and finitely many constant vectors {cj}N0 j=1 ⊂Rmand points {xj}N0 j=1 ⊂BRsuch that the function ϕ:= N0 X j=1 cj |x−xj|n 2−1Jn 2−1(|x−xj|) approximates the function φin the unit ball as kφ−ϕkCr(B)< δ . Proof. It is more convenient to work with complex-valued functions, so we set e φ:= φ+iφ. First, we notice that, since e φis also a solution of the Helmholtz equation, it can be written in the ball B2as an expansion (2.1) e φ= ∞ X l=0 d(l,n−1) X k=1 blk jl(r)Ylk(ω), where r:= |x| ∈ R+and ω:= x/r ∈Sn−1are spherical coordinates in Rn,Ylk is a basis of spherical harmonics of eigenvalue l(l+n−2), jlare n-dimensional hyperspherical Bessel functions and blk ∈Cmare constant coefficients. NODAL SETS OF HIGH-ENERGY EIGENFUNCTIONS 5 The series in (2.1) is convergent in the L2sense, so for any δ0>0, we can truncate the sum at some integer L (2.2) φ1:= L X l=0 d(l,n−1) X k=1 blk jl(r)Ylk(ω) so that it approximates e φas (2.3) kφ1−e φkL2(B2)< δ0. The Cm-valued function φ1decays as |φ1(x)|⩽C/|x|n−1 2for large enough |x| (because of the decay properties of the spherical Bessel functions). Hence, Herglotz’s theorem (see e.g. [12, Theorem 7.1.27]) ensures that we can write (2.4) φ1(x) = ZSn−1 f1(ξ)eix·ξdσ(ξ), where dσ is the area measure on Sn−1:= {ξ∈Rn:|ξ|= 1}and f1is a Cm-valued function in L2(Sn−1). We now choose a smooth Cm-valued function f2approximating f1as kf1−f2kL2(Sn−1)< δ0, which is always possible since smooth functions are dense in L2(Sn−1). The function defined as the inverse Fourier transform of f2, (2.5) φ2(x) := ZSn−1 f2(ξ)eix·ξdσ(ξ), approximates φ1uniformly: by the Cauchy–Schwarz inequality, we get |φ2(x)−φ1(x)|=ZSn−1 (f2(ξ)−f1(ξ)) eix·ξdσ(ξ)⩽Ckf2−f1kL2(Sn−1)< Cδ0 (2.6) for any x∈Rn. Our next objective is to approximate the function f2by a trigonometric polynomial: for any given δ0, we will find a constant R > 0, finitely many points {xj}N0 j=1 ⊂BRand constants {cj}N0 j=1 ⊂Cmsuch that the smooth function in Rn f(ξ) := 1 (2π)n 2 N0 X j=1 cje−ixj·ξ, when restricted to the unit sphere, approximates f2in the C0norm, (2.7) kf−f2kC0(Sn−1)< δ0. In order to do so, we begin by extending f2to a smooth function g:Rn→Cm with compact support, g(ξ) := χ(|ξ|)f2ξ |ξ|, where χ(s) is a real-valued smooth bump function, being 1 when, for example, |s−1|<1 4, and vanishing for |s−1|>1 2. The Fourier transform bgof gis Schwartz, 6 A. ENCISO, D. PERALTA-SALAS, AND F. TORRES DE LIZAUR so it is easy to see that, outside some ball BR, the L1norm of bgis very small, ZRn\BR|bg(x)|dx < δ0, and therefore we get a very good approximation of gby just considering its Fourier representation with frequencies within the ball BR, that is, (2.8) sup ξ∈Rng(ξ)−ZBRbg(x)e−ix·ξdx< δ0/2. Next, let us show that we can approximate the integral ZBRbg(x)e−ix·ξdx by the sum (2.9) f(ξ) := 1 (2π)n 2 N0 X j=1 cje−ixj·ξ with constants cj∈Cmand points xj∈BR, so that we have the bound (2.10) sup ξ∈Sn−1ZBRbg(x)e−ix·ξdx −f(ξ)< δ0/2. Indeed, consider a covering of the ball BRby closed sets {Uj}N0 j=1, with piecewise smooth boundaries, pairwise disjoint interiors, and diameters not exceeding δ00. Since the function e−ix·ξbg(x) is smooth, we have that for each x, y ∈Uj sup ξ∈Sn−1bg(x)e−ix·ξ−bg(y)e−iy·ξ|< Cδ00 , with the constant Cdepending on bg(and therefore on δ0) but not on δ00. If xjis any point in Ujand we set cj:= (2π)n 2bg(xj)|Uj|in (2.9), we get sup ξ∈Sn−1ZBRbg(x)e−ix·ξdx −f(ξ)⩽ N0 X j=1 ZUj sup ξ∈Sn−1bg(x)e−ix·ξ−bg(xj)e−ixj·ξdx ⩽Cδ00 , with Cdepending on δ0and Rbut not on δ00 nor N0. By taking δ00 so that Cδ00 < δ0/2, the estimate (2.10) follows. Now, in view of (2.8) and (2.10), one has kf−gkC0(Sn−1)< δ0, so the estimate (2.7) follows upon noticing that the function f2is the restriction to Sn−1of the function g. To conclude, set eϕ(x) := ZSn−1 f(ξ)eix·ξdσ(ξ) = N0 X j=1 cj (2π)n 2ZSn−1 ei(x−xj)·ξdσ(ξ) = = N0 X j=1 cj |x−xj|n 2−1Jn 2−1(|x−xj|), NODAL SETS OF HIGH-ENERGY EIGENFUNCTIONS 7 then from Equation (2.7) we infer that keϕ−φ2kC0(Rn)⩽ZSn−1|f(ξ)−f2(ξ)|dσ(ξ)< Cδ0, and from Equations (2.3) and (2.6) we get the L2estimate (2.11) ke φ−eϕkL2(B2)⩽Ckeϕ−φ2kC0(Rn)+Ckφ2−φ1kC0(Rn)+ +kφ1−e φkL2(B2)< Cδ0. Furthermore, both eϕand e φare Cm-valued functions satisfying the Helmholtz equation in Rn(note that the Fourier transform of eϕis supported on Sn−1), so by standard elliptic regularity estimates we have ke φ−eϕkCr(B)⩽Cke φ−eϕkL2(B2)< Cδ0. This in particular implies that kφ−Re eϕkCr(B)< Cδ0, and taking δ0small enough so that Cδ0< δ, resetting cj:= Re cj, and defining ϕ:= Re eϕ, the proposition follows.  The second step consists in showing that, for any large enough integer N, we can find an Rm-valued eigenfunction ψof the Laplacian on Snwith eigenvalue N(N+n−1) that approximates, in the ball B1/N , when appropriately rescaled, the function ϕin the unit ball. The proof is based on asymptotic expansions of ultraspherical polynomials, and uses the representation of ϕas a sum of shifted Bessel functions which we obtained in the previous proposition as a key ingredient. It is then straightforward that Theorem 2.1 follows from Propositions 2.2 and 2.3, provided that Nis large enough and δis chosen so that 2δ < δ0. Proposition 2.3. Given a constant δ > 0, for any large enough positive integer Nthere is an Rm-valued eigenfunction ψof the Laplacian on Snwith eigenvalue N(N+n−1) satisfying    ϕ−ψ◦Ψ−1· k   Cr(B) < δ . Proof. Consider the ultraspherical polynomial of dimension n+ 1 and degree N, Cn N(t), which is defined as (2.12) Cn N(t) := Γ(N+ 1)Γ(n 2) Γ(N+n 2)P(n 2−1,n 2−1) N(t), where Γ(t) is the gamma function and P(α, β) N(t) are the Jacobi polynomials (see e.g [16, Chapter IV, Section 4.7]). We have included a normalizing factor so that Cn N(1) = 1 for all N. Let p, q be two points in Sn, considered as vectors in Rn+1 with |p|=|q|= 1. The addition theorem for ultraspherical polynomials ensures that Cn N(p·q) (where p·qdenotes the scalar product in Rn+1 of the vectors pand q) can be written as (2.13) Cn N(p·q) = 2πn+1 2 Γ(n+1 2) 1 d(N, n) d(N,n) X k=1 YNk(p)YNk(q), 8 A. ENCISO, D. PERALTA-SALAS, AND F. TORRES DE LIZAUR with {YNk}d(N,n) k=1 being an arbitrary orthonormal basis of eigenfunctions of the Laplacian on Sn(spherical harmonics) with eigenvalue N(N+n−1). The function ϕis written as the finite sum ϕ(x) = N0 X j=1 cj |x−xj|n 2−1Jn 2−1(|x−xj|), with coefficients cj∈Rmand points xj∈BR. With these cjand xjwe define, for any point p∈Sn, the function ψ(p) := N0 X j=1 cj 2n 2−1Γ(n 2)Cn N(p·pj), where pj:= Ψ−1(xj N). As long as N > R,pjis well defined. In view of Equation (2.13) it is clear that ψis an Rm-valued eigenfunction of the Laplacian on Sn with eigenvalue N(N+n−1). Our aim is to study the asymptotic properties of the eigenfunction ψ. To begin with, note that if we consider points p:= Ψ−1(x N) with N > R and x∈BR, we have (2.14) p·pj= cos distSn(p, pj)= cos |x−xj|+O(N−1) N, as N→ ∞. The last equality comes from Ψ : B→Bbeing a patch of normal geodesic coordinates (by distSn(p, pj) we mean the distance between pand pj considered on the sphere Sn). From now on we set (2.15) e ψ(x) := ψ◦Ψ−1x N. When Nis large, one has Γ(N+ 1) Γ(N+n 2)=N1−n 2+O(N−n 2), so from Equation (2.14) we infer Cn N(p·pj) = Γn 2N1−n 2+O(N−n 2)P(n 2−1,n 2−1) Ncos |x−xj|+O(N−1) N. (2.16) By virtue of Darboux’s formula for the Jacobi polynomials [16, Theorem 8.1.1], we have the estimate 1 Nn 2−1P(n 2−1,n 2−1) Ncos t N= 2n 2−1Jn 2−1(t) tn 2−1+O(N−1), uniformly in compact sets (e.g., for |t|⩽2R). Hence, in view of Equation (2.16), the function e ψcan be written as e ψ(x) = N0 X j=1 cj 2n 2−1Γ(n 2)Cn Ncos |x−xj|+O(N−1) N = N0 X j=1 cj |x−xj|n 2−1Jn 2−1(|x−xj|) + O(N−1), NODAL SETS OF HIGH-ENERGY EIGENFUNCTIONS 9 for Nbig enough and x, xj∈BR. From this we get the uniform bound (2.17) kϕ−e ψkC0(B)< δ0 for any δ0>0 and all Nlarge enough. It remains to promote this bound to a Crestimate. For this, note that, since the eigenfunction ψhas eigenvalue N(N+n−1), the rescaled function e ψverifies on Bthe equation ∆e ψ+e ψ=1 NAe ψ , with Ae ψ:= −(n−1) e ψ+G1∂e ψ+G2∂2e ψ , where ∂ke ψis a matrix whose entries are k-th order derivatives of e ψ, and Gk(x, N) are smooth matrix-valued functions with uniformly bounded derivatives, i.e., (2.18) sup x∈B|∂α xGk(x, N)|⩽Cα, with constants Cαindependent of N. Since ϕsatisfies the Helmholtz equation ∆ϕ+ϕ= 0 , the difference ϕ−e ψsatisfies ∆(ϕ−e ψ)+(ϕ−e ψ) = 1 NAe ψ , and, considering the estimates (2.17) and (2.18), by standard elliptic estimates we get kϕ−e ψkCr,α(B)< Ckϕ−e ψkC0(B)+C NkAe ψkCr−2,α(B) < Cδ0+C Nkϕ−e ψkCr,α(B)+C NkϕkCr,α(B), so we conclude that, for Nbig enough and δ0small enough, kϕ−e ψkCr(B)⩽Cδ0+CkϕkCr,α N< δ . The proposition then follows.  3. An inverse localization theorem on the torus In this section we prove an inverse localization theorem for eigenfunctions of the Laplacian on Tnfor n⩾3. We recall that the eigenvalues of the Laplacian on the n-torus are the integers of the form λ=|k|2 for some k∈Zn. In particular, the spectrum of the Laplacian in Tncontains the set of the squares of integers. As in the previous section, we fix an arbitrary point p0∈Tnand take a patch of normal geodesic coordinates Ψ : B→Bcentered at p0. Theorem 3.1. Let φbe an Rm-valued function in Rn, satisfying ∆φ+φ= 0. Fix a positive integer rand a positive constant δ0. For any large enough odd integer N, there is an Rm-valued eigenfunction ψof the Laplacian on Tnwith eigenvalue N2 such that    φ−ψ◦Ψ−1· N   Cr(B) ⩽δ0.