Enciso and Gómez-Serrano (2022) proved that semi-regular polygons — circumscriptible polygons whose interior angles are all equal but possibly one, which is larger — are determined by their Dirichlet or Neumann spectrum within the class of convex piecewise smooth planar domains with straight corners, and left three open issues: computing the t\sqrt{t} heat-trace coefficient for domains with curved corners, extending spectral determination beyond convex domains, and the classical question of whether smooth or convex planar domains that are isospectral but non-congruent exist at all. The first issue is now resolved in the Dirichlet case: Looi and Sher (2025) computed the curved-corner coefficient at order t\sqrt{t}, proved a sign law for its angular factor, and thereby removed the straight-corners hypothesis from the polygonalization step. The surviving open core is the convexity restriction: no result establishes, or refutes, spectral determination of semi-regular polygons in the full class of bounded piecewise smooth (curvilinear-polygonal) domains, where the known extremizer characterization fails due to non-convex 'hair' constructions and where sign cancellation between ∫κ2 ds\int\kappa^2\,ds and curved-corner terms obstructs polygonalization; and for Neumann conditions the curved-corner coefficient is known to exist in the small-time expansion, but its explicit factorization and sign law remain unproved. The smooth/convex isospectral existence question remains open for the Laplacian, its Steklov analogue (the eigenvalue problem that places the spectral parameter in the boundary condition) having been answered negatively in 2026.

Background

Kac's 1966 question 'Can one hear the shape of a drum?' asks whether a bounded planar domain Ω\Omega is determined, up to congruence (rigid motion composed possibly with a reflection), by the eigenvalues of the Laplacian on Ω\Omega with Dirichlet (clamped-boundary) or Neumann (free-boundary) conditions; two domains with identical eigenvalue lists are called isospectral. Gordon, Webb and Wolpert answered negatively in 1992 with a pair of non-congruent isospectral polygonal drums, but all known planar isospectral pairs are non-convex and have corners, and it is unknown whether a smooth drum or a convex drum can be 'heard'.

The main tool for positive results is the small-time heat trace hΩ(t)=Tr etΔΩh_\Omega(t)=\mathrm{Tr}\,e^{t\Delta_\Omega}, whose asymptotic coefficients as t→0+t\to 0^+ are spectral invariants. For a curvilinear polygon — a bounded planar domain whose boundary is smooth except at finitely many corner points with interior angles αj∈(0,2π)\alpha_j\in(0,2\pi), angles above π\pi being called reflex — the Dirichlet expansion begins hΩDir(t)=∣Ω∣4πt−∣∂Ω∣8πt+112π(∫∂Ωκ ds+∑j=1nπ2−αj22αj)+t (1256π∫∂Ωκ2 ds+curved-corner terms)+⋯ ,h_\Omega^{\mathrm{Dir}}(t)=\frac{|\Omega|}{4\pi t}-\frac{|\partial\Omega|}{8\sqrt{\pi t}}+\frac{1}{12\pi}\Big(\int_{\partial\Omega}\kappa\,ds+\sum_{j=1}^{n}\frac{\pi^2-\alpha_j^2}{2\alpha_j}\Big)+\sqrt{t}\,\Big(\frac{1}{256\sqrt{\pi}}\int_{\partial\Omega}\kappa^2\,ds+\text{curved-corner terms}\Big)+\cdots, where κ\kappa is the curvature of the boundary: area, perimeter, total curvature, and the multiset of corner angles are all audible. The corner terms go back to unpublished work of Ray and to van den Berg and Srisatkunarajah (Dirichlet) and to Mazzeo and Rowlett (Neumann); Lu and Rowlett showed that the corners of a planar domain are determined by its spectrum.

A polygon is called semi-regular if it is circumscriptible (admits an inscribed circle tangent to every side) and its interior angles are all equal except possibly one, which is larger; regular polygons are the special case. Enciso and Gómez-Serrano proved that every semi-regular polygon PnP_n is spectrally determined in the class of convex piecewise smooth domains with straight corners (corners near which the boundary is flat): any convex such domain isospectral to PnP_n is congruent to it. The proof has two parts. Polygonalization: the t\sqrt{t} coefficient forces ∫∂Ωκ2 ds=0\int_{\partial\Omega}\kappa^2\,ds=0, so the boundary consists of line segments. Polygon identification: semi-regular polygons are characterized as the extremizers of a constrained isoperimetric-type inequality among convex polygons whose number of sides is not fixed, which is what allows PnP_n to be recognized among polygons with an arbitrary number of sides. Both hypotheses are load-bearing. The straight-corners assumption was needed because the t\sqrt{t} coefficient had not been computed when the boundary is curved near a corner; convexity is exactly what makes the extremizer characterization work, since appending a thin straight 'hair' (a long spike of negligible area) to a convex polygon produces non-semi-regular extremizers, so the variational argument fails in any non-convex class. Looi and Sher (2025) computed the missing Dirichlet coefficient: the curved-corner contribution at order t\sqrt{t} is C1/2(α,κ+,κ−)=c1/2(α)κ++κ−4sin⁡(α/2)C_{1/2}(\alpha,\kappa_+,\kappa_-)=c_{1/2}(\alpha)\frac{\kappa_++\kappa_-}{4\sin(\alpha/2)}, where κ±\kappa_\pm are the one-sided limiting curvatures of the arcs meeting at the corner and the explicit factor c1/2(α)c_{1/2}(\alpha) is positive exactly for convex angles α<π\alpha<\pi and negative for reflex angles. This yields polygonalization for 'admissible' curvilinear polygons — those whose corner terms are all nonnegative, a condition strictly weaker than convexity — and so removes the straight-corners hypothesis in the Dirichlet case. What no result addresses is determination with no convexity-type restriction at all: for non-admissible domains the corner terms can be negative and can cancel ∫κ2 ds\int\kappa^2\,ds, and among non-convex polygons the extremizer characterization is false. Whether semi-regular polygons are determined by their spectrum in the full class of bounded piecewise smooth domains — alongside the classical question of whether any smooth or convex non-congruent isospectral planar pair exists — remains open.

Problem Statement

Let PnP_n be a semi-regular nn-sided polygon, i.e. a circumscriptible polygon whose interior angles are all equal except possibly one, which is larger than the rest (regular nn-gons are the special case). Enciso and Gómez-Serrano proved that PnP_n is determined by its Dirichlet or Neumann spectrum among convex piecewise smooth planar domains with straight corners, and Looi and Sher removed the straight-corners hypothesis in the Dirichlet case, at the cost of an 'admissibility' condition on corner curvatures that is strictly weaker than convexity. The problem is to settle the determination question in the natural maximal class, with no convexity, admissibility, or straightness restriction:

Prove, or refute by an explicit counterexample, the following statement. If Ω\Omega is any bounded piecewise smooth planar domain — a curvilinear polygon whose boundary is smooth except at finitely many corners of arbitrary interior angles in (0,2π)(0,2\pi), allowing reflex angles and arbitrary non-convex shapes — and the Dirichlet spectrum of Ω\Omega coincides with that of PnP_n, or the Neumann spectrum of Ω\Omega coincides with that of PnP_n, then Ω\Omega is congruent to PnP_n.

The current state of the art leaves two documented obstructions: (a) polygonalization without admissibility — at order t\sqrt{t} the curved-corner terms C1/2(αj,κj,+,κj,−)C_{1/2}(\alpha_j,\kappa_{j,+},\kappa_{j,-}) may be negative at reflex corners and cancel the nonnegative ∫∂Ωκ2 ds\int_{\partial\Omega}\kappa^2\,ds; and (b) identification among arbitrary, possibly non-convex polygons — the constrained-isoperimetric extremizer characterization used by Enciso and Gómez-Serrano fails in the non-convex class (straight 'hair' appendages produce non-semi-regular extremizers). For the Neumann branch, the explicit factorization and sign law of the Neumann curved-corner analogue of the Dirichlet coefficient C1/2(α,κ+,κ−)C_{1/2}(\alpha,\kappa_+,\kappa_-) remain open: Looi and Sher derive the existence of this coefficient in the Neumann small-time expansion, but the explicit factored form and sign behavior that polygonalization would need are expected in the literature and not proved.

References

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