Pólya's conjecture for spherical n-lunes: the exact threshold p*(n) and the spectra of rational lunes

A spherical n-lune is the intersection of the unit sphere with a wedge between two totally geodesic boundary faces, and its Dirichlet eigenvalues can be compared with the first Weyl term, giving the Pólya-type inequality $\lambda_k^{n/2}\geq n!pk$ for a lune of opening angle $\pi/p$. Freitas and Salavessa (2026) determined the full Dirichlet and Neumann spectra of these dihedral lunes, proved that lunes whose opening angle is smaller than $\pi$ and is an irrational multiple of $\pi$, or is of the form $\pi/p$ with $p\geq 2$, satisfy Pólya's conjecture eventually in every dimension, and computed the exact threshold $p^*(n)$ — the minimal integer such that, for every $p\geq p^*(n)$, the lune $L^n_{\pi/p}$ satisfies the inequality for all eigenvalues — for dimensions $n\leq 50$, together with the general bounds $0.120754\,n\lesssim p^*(n)\leq n-1$ for $n\geq 9$. This problem asks to complete that program: determine $p^*(n)$ exactly in all dimensions, together with its sharp asymptotic growth and a characterization of the low-eigenvalue failures that make the first-eigenvalue criterion insufficient; and determine the spectra and the Pólya status of the remaining rational lunes $L^n_{m\pi/p}$ with $m>1$, for which neither the invariant-theory method nor the billiard nonperiodicity route applies.

Background

For a bounded domain $\Omega$ with piecewise smooth boundary, the Dirichlet eigenvalues $0<\lambda_1\leq\lambda_2\leq\cdots$ and Neumann eigenvalues $0=\mu_1\leq\mu_2\leq\cdots$ of the Laplacian govern vibrating-membrane problems, and Weyl's law states that $\lambda_k\sim 4\pi^2k^{2/n}/(\omega_n|\Omega|)^{2/n}$ as $k\to\infty$, where $n$ is the dimension, $|\Omega|$ the volume, and $\omega_n$ the volume of the unit ball in $\mathbb{R}^n$. Pólya (1961) proved for planar domains that tile the plane that each individual Dirichlet eigenvalue lies above this first Weyl term, $\lambda_k\geq 4\pi^2k^{2/n}/(\omega_n|\Omega|)^{2/n}$ (with the reverse inequality for Neumann eigenvalues), and conjectured the same for all bounded Euclidean domains; this remains one of the classical open problems of spectral theory, open even for general planar domains. The same comparison can be posed for domains on curved surfaces such as the sphere.

The unit sphere $\mathbb{S}^n\subset\mathbb{R}^{n+1}$ has Laplacian eigenvalues $K(K+n-1)$ for integers $K\geq 0$, with multiplicities given by binomial coefficients in $K$ and $n$ (the spherical harmonics of degree $K$). A spherical n-lune $L^n_\theta$ is the intersection of $\mathbb{S}^n$ with a wedge of dihedral angle $\theta\in(0,\pi]$ between two half-hyperplanes through the origin: in coordinates $\mathbb{R}^{n+1}=\mathbb{R}^{n-1}\times\mathbb{R}^2$ it consists of the points whose $\mathbb{R}^2$-component has polar angle in $[0,\theta]$. Its two boundary faces are totally geodesic $(n-1)$-hemispheres meeting along an equatorial $(n-2)$-sphere, and its volume is $|L^n_\theta|=(\theta/2\pi)|\mathbb{S}^n|$. The case $\theta=\pi/p$ is called a dihedral lune, since the two face reflections generate the dihedral reflection group $D_p$ of order $2p$ whose chamber is precisely $L^n_{\pi/p}$. For the dihedral lune, Pólya's inequality takes the explicit form $\lambda_k^{n/2}\geq n!\,p\,k$, and for a general rational opening $\theta=m\pi/p$ (with $m,p$ coprime) it reads $\lambda_k^{n/2}\geq (n!\,p/m)\,k$.

Two classical threads motivate the problem. First, the hemisphere $L^n_{\pi}$: Bérard and Besson (1980), through spectra of spherical domains obtained from crystallographic-type group actions, and later Buoso, Luzzini, Provenzano and Stubbe (2023) through semiclassical eigenvalue means, showed that the situation there is rigid — the Neumann Pólya inequality holds in all dimensions, while Freitas, Mao and Salavessa (2025) proved that for the Dirichlet problem in dimensions $n\geq 3$ the hemisphere violates Pólya's inequality for infinitely many eigenvalues, with sharp two-sided bounds measuring the deviation from the Weyl term. Second, the two-term Weyl asymptotics: under a nonperiodicity condition on the geodesic billiard (the set of periodic billiard trajectories has measure zero) and a nonblocking condition, the eigenvalues admit a two-term asymptotic formula whose second term has a definite sign, which forces Pólya's inequality to hold eventually, i.e. for all $k$ beyond some order. Freitas and Salavessa (2023) additionally showed that any domain satisfying the conjecture eventually passes this property to fine enough isometric tilings, and Freitas and Wang (2025) characterized exactly when the cylinder $\mathbb{S}^1\times(0,h)$ fails the conjecture at low eigenvalues.

The source paper (Freitas and Salavessa, arXiv:2608.09557, 2026) brings these threads together for lunes. They first characterize the geodesic billiards (Theorem A): on $L^n_\theta$ all billiard trajectories are periodic exactly when $\theta$ is a rational multiple of $\pi$ (of minimal period $2p$ for $\theta=m\pi/p$), so rational lunes are convex domains failing the nonperiodicity condition. For dihedral lunes they fully determine the Dirichlet and Neumann spectra (Theorem E) via Hilbert–Poincaré series of algebras of anti-invariant homogeneous polynomials under $D_p$: the Dirichlet eigenvalues are exactly the sphere eigenvalues $K(K+n-1)$ with $K\geq p$, organized in chains $K=(m+1)p+r$, $r=0,\dots,p-1$, with explicit multiplicities. They obtain a generalized two-term asymptotic formula for the Dirichlet eigenvalues with a sharp bounded oscillating coefficient in place of the usual constant second term (Theorem B), and conclude (Corollary C) that lunes with opening $\theta<\pi$ and $\theta/\pi$ irrational, or $\theta=\pi/p$ with $p\geq 2$, satisfy Pólya's conjecture eventually in every dimension — in striking contrast with hemispheres.

For the full (all-orders) conjecture on dihedral lunes they reduce Pólya's inequality to the nonnegativity of explicit polynomial functions $M_{n,p,r}(m)$ in elementary parameters (here $m$ is the chain index in $K=(m+1)p+r$; the same letter will also denote the numerator of a rational opening $m\pi/p$), define $p^*(n)$ as the minimal integer such that, for every $p\geq p^*(n)$, the lune $L^n_{\pi/p}$ satisfies the inequality for all eigenvalues, and prove (Theorem D) that for $n\geq 9$ the conjecture holds if $p\geq n-1$ and fails if $p<\bigl((-1+\sqrt{1+4e^{-2}})/2\bigr)n\approx 0.120754\,n$ (the failure bound comes from the first eigenvalue alone). By exact computation of the polynomials they determine the optimal values $p^*(n)$ for all $n\leq 50$ (Theorem 6.5 and Appendix B): $p^*(n)$ equals $1$ at $n=2$, $2$ for $3\leq n\leq 8$, then increases by one roughly every eight dimensions, reaching $8$ for $48\leq n\leq 50$ — far below $n-1$ and only slightly above the first-eigenvalue threshold. They also observe (Remark 5.1) that the first eigenvalue is not the governing obstruction: in the lune $L^{24}_{\pi/4}$ the first and third eigenvalues satisfy Pólya's inequality while the second does not, so any threshold based only on $\lambda_1$ is not optimal in general.

Problem Statement

Let $L^n_\theta\subset\mathbb{S}^n$ denote the spherical n-lune of opening angle $\theta\in(0,\pi]$, whose Dirichlet eigenvalues $\lambda_k$ are compared with the first Weyl term through Pólya's inequality; for a rational opening $\theta=m\pi/p$ with $m,p$ coprime this reads $\lambda_k(L^n_{m\pi/p})^{n/2}\geq(n!\,p/m)\,k$ for all $k$. The problem, which completes the program initiated by Freitas and Salavessa (arXiv:2608.09557), has two linked parts.

(1) Dihedral lunes: determine $p^*(n)$ — as in Definition 5.2 of the source, the minimal integer such that, for every $p\geq p^*(n)$, the lune $L^n_{\pi/p}$ satisfies Pólya's inequality for all eigenvalues (equivalently, inequality (5.7) of the source holds for all $r=0,\dots,p-1$ and all $m\geq 0$) — exactly for every dimension $n\geq 2$; in particular close the gap left by the source, which only gives $p^*(n)\leq n-1$ and $p^*(n)>0.120754\,n(1-o(1))$ for large $n$, with exact values known solely for $n\leq 50$. This includes determining the sharp asymptotic behavior of $p^*(n)$ as $n\to\infty$, and characterizing the structure of the failing low eigenvalues — for which chains $(m,r)$ in the decomposition $K=(m+1)p+r$ and which orders the inequality first fails — since the first eigenvalue alone does not govern the threshold (in $L^{24}_{\pi/4}$ it is the second eigenvalue that fails).

(2) General rational lunes: determine the Dirichlet spectrum (eigenvalues and multiplicities) of $L^n_{m\pi/p}$ for integers $m>1$ with $\gcd(m,p)=1$ and $m\pi/p<\pi$, for which no explicit description is known; and decide whether these lunes satisfy Pólya's inequality eventually — a question left fully open because they are covered neither by the eventual-Pólya corollary (only irrational openings and $\pi/p$ are), nor by the reflection-group spectral method (the lune is not a chamber of $D_p$), nor by two-term billiard asymptotics (all their billiard orbits are periodic). Identify for which parameters $(n,m,p)$ low-eigenvalue failures occur.

A resolution requires all of the following, and each is required in its own right unless noted: for part (1), the exact value of $p^*(n)$ for every $n\geq 2$ and the characterization of the failing low eigenvalues (which chains $K=(m+1)p+r$ and which orders fail first), where the sharp asymptotic law of $p^*(n)$ follows from the exact values; for part (2), the Dirichlet spectrum of $L^n_{m\pi/p}$ for $m>1$ and the eventual-Pólya verdict for these lunes, where the classification of the parameters $(n,m,p)$ at which low-eigenvalue failures occur follows from the spectrum together with the verdict.

References

1. P. Freitas and I. Salavessa, Spherical n-lunes: billiards and eigenvalues, arXiv:2608.09557 (2026), doi:10.48550/arxiv.2608.09557, https://arxiv.org/abs/2608.09557 (primary source of this statement)
2. G. Pólya, On the eigenvalues of vibrating membranes, Proc. London Math. Soc. (3) 11 (1961) 419–433, doi:10.1112/plms/s3-11.1.419, https://doi.org/10.1112/plms/s3-11.1.419
3. P. Bérard and G. Besson, Spectres et groupes cristallographiques. II. Domaines sphériques, Ann. Inst. Fourier (Grenoble) 30 (1980) 237–248, doi:10.5802/aif.800, https://doi.org/10.5802/aif.800
4. D. Gromes, Über die asymptotische Verteilung der Eigenwerte des Laplace-Operators für Gebiete auf der Kugeloberfläche, Math. Z. 94 (1966) 110–121, doi:10.1007/BF01118974, https://doi.org/10.1007/BF01118974
5. P. Freitas, J. Mao and I. Salavessa, Pólya-type inequalities on spheres and hemispheres, Ann. Inst. Fourier (Grenoble) 75 (2025) 979–1051, doi:10.5802/aif.3657, https://doi.org/10.5802/aif.3657
6. P. Freitas and I. Salavessa, Families of non-tiling domains satisfying Pólya's conjecture, J. Math. Phys. 64 (2023) 121503, doi:10.1063/5.0161050, https://doi.org/10.1063/5.0161050
7. P. Freitas and R. Wang, Pólya's conjecture on $\mathbb{S}^1\times\mathbb{R}$, arXiv:2506.04341 (2025), https://arxiv.org/abs/2506.04341
8. D. Buoso, P. Luzzini, L. Provenzano and J. Stubbe, Semiclassical estimates for eigenvalue means of Laplacians on spheres, J. Geom. Anal. 33 (2023) article 280, doi:10.1007/s12220-023-01326-6, https://doi.org/10.1007/s12220-023-01326-6