Pólya's 1954 conjecture predicts that the Dirichlet eigenvalue counting function of any bounded domain is bounded above by the leading Weyl term; it is proved for plane-tiling domains and a few special families, and remains open for general planar domains. Freitas and Wang (arXiv:2506.04341, 2025) showed that on the flat infinite cylinder S1×R\mathbb{S}^1\times\mathbb{R} the conjecture fails for explicitly characterized families of cylindrical strips, with the smallest counterexamples occurring at area 2π(8−64−4π2)≈19.152\pi(8-\sqrt{64-4\pi^2})\approx 19.15, and posed Conjecture 1.1: there exists a positive number A0A_0 such that every bounded domain Ω⊂S1×R\Omega\subset\mathbb{S}^1\times\mathbb{R} with ∣Ω∣≤A0|\Omega|\le A_0 satisfies Pólya's conjecture, proposing as a possible quantitative version of the conjecture the value A0=2π(8−64−4π2)A_0=2\pi(8-\sqrt{64-4\pi^2}). This problem asks for a proof or refutation of that conjecture.

Background

For a bounded domain Ω\Omega with Dirichlet boundary conditions, the eigenvalues 0<λ1(Ω)≤λ2(Ω)≤⋯0<\lambda_1(\Omega)\le\lambda_2(\Omega)\le\cdots of the Dirichlet Laplacian (the vibrating-membrane problem) satisfy Weyl's asymptotic law: the counting function NΩ(λ)=#{k:λk(Ω)≤λ}N_\Omega(\lambda)=\#\{k:\lambda_k(\Omega)\le\lambda\} grows like ∣Ω∣λ/(4π)|\Omega|\lambda/(4\pi) in two dimensions, where ∣Ω∣|\Omega| is the area. Pólya's conjecture asserts that the leading Weyl term is already an upper bound for all λ>0\lambda>0, equivalently that λk(Ω)≥4πk/∣Ω∣\lambda_k(\Omega)\ge 4\pi k/|\Omega| for every kk. Pólya proved the conjecture in 1961 for domains that tile the plane; Urakawa extended this to higher-dimensional Euclidean tiling domains, Laptev proved it for Cartesian products of intervals, Filonov–Levitin–Polterovich–Sher proved it for Euclidean balls in all dimensions, and Freitas–Lagacé–Payette and Freitas–Salavessa obtained it for families of domains tiling a given domain, including some non-Euclidean settings. The conjecture remains open for general bounded planar domains. On curved manifolds the situation is genuinely different: Coleman showed that Pólya's conjecture is false on round spheres, while Pólya-type inequalities do hold on spheres and hemispheres (Freitas–Mao–Salavessa).

The present problem lives on the flat infinite cylinder S1×R\mathbb{S}^1\times\mathbb{R}, the product of the unit circle (circumference 2π2\pi) with the line, equipped with the product metric — a flat surface with one compact direction. Its isoperimetric domains, i.e. domains of given area with least perimeter, are known (Howards–Hutchings–Morgan): geodesic disks for areas up to 4π≈12.574\pi\approx 12.57, and cylindrical strips Ch=S1×IhC_h=\mathbb{S}^1\times I_h (where IhI_h is an interval of length hh) for larger areas. The Dirichlet spectrum of ChC_h is explicit, λ=m2+n2π2/h2\lambda=m^2+n^2\pi^2/h^2 with m∈Zm\in\mathbb{Z} and n∈Nn\in\mathbb{N}, so Pólya's conjecture on strips becomes a lattice-point counting question. Freitas and Wang (2025) resolved this completely: Pólya's conjecture holds on ChC_h exactly for hh in an explicit union of intervals, failing in two sporadic windows h∈(3.0481,3.2380)h\in(3.0481,3.2380) (through the eighth eigenvalue λ8\lambda_8) and h∈(4.0460,4.0824)h\in(4.0460,4.0824) (through λ13\lambda_{13}), and for all h>π2/2h>\pi^2/2 through λ1\lambda_1; they also proved that the weaker averaged (Li–Yau) inequalities 1k∑i=1kλi≥k/h\frac1k\sum_{i=1}^k\lambda_i\ge k/h hold on ChC_h if and only if h≤π2h\le\pi^2, and that geodesic disks of radius RR fail the conjecture beyond R≈3.76085R\approx 3.76085 (they are known to satisfy it up to R=πR=\pi). Combining the strip characterization with the disk results, the isoperimetric domains of a given area satisfy Pólya's conjecture precisely when the area lies in an explicit union of intervals, the lowest of which ends at A0=2π(8−64−4π2)≈19.1516A_0=2\pi(8-\sqrt{64-4\pi^2})\approx 19.1516 — and this value is also the infimum of the areas of all counterexamples currently known. All known counterexamples on the cylinder are non-simply-connected and violate the Euclidean isoperimetric inequality, suggesting that area, rather than topology alone, controls failure; this motivated Freitas and Wang to conjecture that sufficiently small domains always satisfy Pólya's conjecture.

Problem Statement

Prove or disprove Conjecture 1.1 of Freitas and Wang (arXiv:2506.04341): does there exist a positive number A0A_0 such that every bounded domain Ω⊂S1×R\Omega\subset\mathbb{S}^1\times\mathbb{R} (Dirichlet Laplacian on the flat infinite cylinder of circumference 2π2\pi) with area ∣Ω∣≤A0|\Omega|\le A_0 satisfies Pólya's conjecture, that is,

λk(Ω)  ≥  4πk∣Ω∣for all k=1,2,…,\lambda_k(\Omega)\;\ge\;\frac{4\pi k}{|\Omega|}\quad\text{for all }k=1,2,\dots,

equivalently NΩ(λ)≤∣Ω∣λ/(4π)N_\Omega(\lambda)\le|\Omega|\lambda/(4\pi) for all λ>0\lambda>0? In the affirmative case, the source proposes, as a possible quantitative version of the conjecture (a refinement discussed in the source, not part of Conjecture 1.1 itself), taking A0A_0 to be the upper end of the lowest interval of its Corollary B, namely 2π(8−64−4π2)≈19.15162\pi(8-\sqrt{64-4\pi^2})\approx 19.1516; determining the optimal threshold A0∗=sup⁡{A: every bounded Ω⊂S1×R with ∣Ω∣≤A satisfies Poˊlya’s conjecture}A_0^*=\sup\{A:\text{ every bounded }\Omega\subset\mathbb{S}^1\times\mathbb{R}\text{ with }|\Omega|\le A\text{ satisfies Pólya's conjecture}\} and whether it equals this suggested value would settle that quantitative version. A refutation must exhibit bounded domains of arbitrarily small area violating the inequality, since a single counterexample of area aa only rules out thresholds A0≥aA_0\ge a.

References

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