Arnold's problem on the topology of eigenfields: spectral bounds for the components, Betti numbers, and singularities of zero sets that do not separate the domain

A 2005 note of V. I. Arnold initiated the study of eigenfields: m-tuples of Laplace–Beltrami eigenfunctions on a closed d-dimensional Riemannian manifold sharing a single eigenvalue $\Lambda$, whose common zero set has codimension $m$ and therefore does not cut the oscillating domain into pieces, so Courant's nodal domain theorem gives no information. Arnold proved that on a surface the number $N$ of positive components of $G=\{z:u(z)v(z)\ne0\}$ for a same-eigenvalue pair satisfies $N\le n(2\Lambda)$, where $n$ is the eigenvalue counting function, and he left open whether the total number $\tilde N$ of components of $G$ is bounded in terms of the eigenvalue for arbitrary metrics on $S^2$ or $T^2$, whether dimension 3 admits bounds for the numbers and multiplicities of zero-surface singularities, whether Milnor-number-type multiplicities of nodal singular points can be defined in dimension at least 3, and whether Sturm-type bounds hold for the linking numbers of the field graph with the zero section when $m>d$. In the subsequent literature, the sub-question of whether the first Betti number of a nodal surface in dimension 3 is unbounded at small eigenvalue number was resolved (it is unbounded, by the prescribed-nodal-set construction of Enciso and Peralta-Salas, 2015), while the fine two-dimensional question and the three-dimensional, multiplicity, and $m>d$ facets remain open; only coarse, persistence-homology versions of the bounds are known.

Background

Let $(M,g)$ be a closed (compact, without boundary) Riemannian manifold and let $\Delta$ be its Laplace–Beltrami operator, whose eigenfunctions describe the oscillation modes of $M$. The eigenvalues of $-\Delta$ form a nondecreasing sequence $0=\lambda_0\le\lambda_1\le\lambda_2\le\cdots$, each repeated according to its multiplicity; the eigenvalue number $n(\lambda)$ is the position of $\lambda$ in this list, i.e. the number of eigenvalues not exceeding $\lambda$, counted with multiplicity. Courant's nodal domain theorem states that the $k$-th eigenfunction has at most $k$ nodal domains (connected components of the complement of its zero set), so spectral data controls the topology of a single eigenfunction's zero set only through the domains into which that set divides the manifold.

Arnold's note 'On the topology of the eigenfields' (2005) studies the situation in which this division fails. An eigenfield is an $m$-tuple $u=(u_1,\dots,u_m)$ of eigenfunctions of $-\Delta$ sharing a single eigenvalue $\Lambda$, viewed as a section of the trivial bundle $M\times\mathbb R^m$ (physically, coupled oscillation modes of one frequency). Generically the common zero set $\{u_1=\cdots=u_m=0\}$ has codimension $m$ (ambient dimension minus set dimension); for $m\ge2$ it does not separate $M$, and Courant's theorem is silent. In the model case $m=d=2$ on a closed surface, the zeros of the field are the intersection points of the nodal curves $\{u=0\}$ and $\{v=0\}$ of two same-eigenvalue eigenfunctions. Arnold considers $G=\{z:u(z)v(z)\ne0\}$, calls a connected component of $G$ positive if the integral of $uv\,(\nabla u,\nabla v)$ over it is nonnegative (equivalently, the integral of the scalar product of the gradients of $u^2$ and $v^2$), and proves $N\le n(2\Lambda)$ for the number $N$ of positive components. A topological lemma relates the number $\alpha_0$ of transversal intersections of the two nodal curves to the total number $\tilde N$ of components of $G$: $\alpha_0=(\sum_i\chi_i)-\chi\le\tilde N-\chi$, where $\chi_i$ are the Euler characteristics of the components of $G$ and $\chi$ that of $M$. Thus a bound on $\tilde N$ in terms of $\Lambda$ would bound the number of zeros of the eigenfield. For $m\le d$ eigenfunctions with the same eigenvalue on a $d$-manifold the analogous bound is $N\le n(m\Lambda)$, and in dimension 3, for a pair, the number of intersection curves of the two nodal surfaces is at most $\tilde N+b_1/2$, where $b_1$, the first Betti number (the rank of the first homology group), is that of the nodal surface of one of the eigenfunctions.

For the case $m>d$, Arnold points to Sturm theory in one dimension (where the zeros of solutions of second-order equations interlace, bounding the number of zeros of a linear combination by the oscillation counts of its terms) and to the linking number of two disjoint closed oriented curves in 3-space (the signed intersection count of one curve with an oriented surface spanning the other), expecting analogous Sturm-type bounds for the linking of the graph of the field, $x\mapsto(u_1(x),\dots,u_m(x))\subset M\times\mathbb R^m$, with the zero section $M\times\{0\}$; he notes these invariants are homotopy classes rather than numbers when $m>d+1$.

The neighboring results delimit what is known. In dimension 2, Karpushkin (1995) associated to each singular point of the zero level line of an eigenfunction a multiplicity similar to the Milnor number of an isolated hypersurface singularity (the algebraic measure of the degeneracy, computable as the dimension of the local algebra generated by the partial derivatives) and bounded the sums in terms of the eigenvalue number; no analogue is published in dimension 3 or higher. For linear combinations of eigenfunctions with different eigenvalues, the naive extension of Courant's theorem is false in dimension $d>1$ (a counterexample for spherical functions on $S^2$ is recorded in Arnold's note, and Bérard–Helffer (2018, 2020) constructed systematic families); on suitable tori the failure is extreme, since Buhovsky–Logunov–Sodin (2020) built a smooth metric on $T^2$ whose eigenfunctions at infinitely many eigenvalues have infinitely many isolated critical points, and noted that a minor modification of the construction yields eigenfunctions with level sets having infinitely many connected components. In the opposite, flexibility direction, Colin de Verdière (1987) proved that a finite part of the spectrum can be prescribed by the choice of metric; Enciso and Peralta-Salas (2015) showed that in dimension $n\ge3$ any closed orientable separating hypersurface is the exact nodal set of the first nontrivial eigenfunction of some metric; and Enciso–Hartley–Peralta-Salas (2016) realized any knot as a component of the zero set of a same-eigenvalue pair at the first eigenvalue. Most recently, Buhovsky–Payette–Polterovich–Polterovich–Shelukhin–Stojisavljević (2026) introduced a coarse nodal count based on persistence barcodes (the interval decomposition of the homology groups of the sublevel sets $\{f\le t\}$ as $t$ varies; the coarse Betti number $m^r(f,\delta)$ counts homology classes surviving over parameter ranges longer than $\delta$), proving Courant-type bounds for linear combinations, for products, and for eigensections of general elliptic operators, and showing through the examples above that the coarse count is necessary because the usual counts can be infinite. The fine questions raised by Arnold survive precisely in the gaps between these threads.

Problem Statement

Let $M$ be a closed connected $d$-dimensional Riemannian manifold, $d\ge2$, and let $u_1,\dots,u_m$ be Laplace–Beltrami eigenfunctions on $M$ sharing a single eigenvalue $\Lambda$ (an eigenfield). Generically the common zero set has codimension $m$ (ambient dimension minus set dimension), so for $m\ge2$ it does not divide $M$ into nodal domains and Courant's theorem is silent about its topology. The objective, following Arnold (2005), is to determine to what extent spectral data — the eigenvalue $\Lambda$ or the eigenvalue number $n(\Lambda)$, the number of eigenvalues not exceeding $\Lambda$ counted with multiplicity — nevertheless controls the topology of the zero set and of the complement of the union of the individual nodal hypersurfaces, uniformly over all choices of the Riemannian metric. The concrete open facets are:

(i) (Two-dimensional fine bound.) For two same-eigenvalue eigenfunctions $u,v$ on a closed surface, let $\tilde N$ be the total number of connected components of $G=\{z:u(z)v(z)\ne0\}$. Is $\tilde N$ bounded by a function of the eigenvalue number $n(\Lambda)$ for all Riemannian metrics on, say, $S^2$ or $T^2$; or can $\tilde N$ be made arbitrarily large while $n(\Lambda)$ stays small? Arnold phrases the question as boundedness of $\tilde N$ 'in terms of $\Lambda$' and frames the potential failure through a small eigenvalue number; since $\tilde N$ is unchanged under a constant rescaling of the metric while $\Lambda$ rescales, a bound literally in terms of $\Lambda$ over arbitrarily rescaled metrics would amount to a uniform constant bound, so the scale-invariant content of the question is the bound in terms of $n(\Lambda)$ (or in terms of $\Lambda$ within a fixed normalization such as unit area). Arnold proved $N\le n(2\Lambda)$ for the positive components only; a complete answer must control all components, including those adjacent to degenerate (non-transversal or critical) pieces of the nodal curves, and must be consistent with Arnold's topological lemma $\alpha_0\le\tilde N-\chi$, where $\alpha_0$ is the number of transversal intersection points of the two nodal curves and $\chi$ is the Euler characteristic of the surface.

(ii) (Three-dimensional singularities.) Establish, or disprove, universal bounds in terms of the eigenvalue for the number and the multiplicities of the singularities of the zero surfaces $\{u=0\}$ of eigenfunctions on manifolds of dimension at least 3. One sub-question of Arnold's note — whether the first Betti number $b_1$ (the rank of the first homology group) of a nodal surface can be arbitrarily large at a small eigenvalue number — was resolved by Enciso and Peralta-Salas (2015): it can, even for the first nonconstant eigenfunction; any surviving bound must be consistent with this fact and is expected to concern the singular points themselves.

(iii) (Multiplicity theory in dimension at least 3.) Define Milnor-number-type multiplicities for the isolated singular points of nodal sets of eigenfunctions in dimension at least 3 — Karpushkin's theory covers surfaces only — and bound their sums in terms of the eigenvalue. (The Milnor number of an isolated hypersurface singularity is the algebraic measure of its degeneracy, computable as the dimension of the local algebra generated by the partial derivatives; Karpushkin's two-dimensional analogue associates such an integer to each singular point of a nodal curve.)

(iv) (More components than dimensions.) For $m>d$, prove or disprove Sturm-type bounds, in terms of the eigenvalue, for the linking numbers of the graph of the eigenfield with the zero section — more generally, for the corresponding homotopical invariants, which need not be numerical when $m>d+1$. Here the graph of the eigenfield is the image of $x\mapsto(u_1(x),\dots,u_m(x))$ in $M\times\mathbb R^m$, the zero section is $M\times\{0\}$, the linking number is the signed intersection count of one closed oriented submanifold with a chain spanning the other, and 'Sturm-type' refers, by analogy with the one-dimensional theory where zeros of solutions of second-order equations interlace, to a bound in terms of the oscillation counts of the components.

The facets are bound together by the common objective of extending Courant-type spectral control of nodal topology to eigenfields whose zero sets do not separate the domain.

References

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