Area-Spectral Rigidity of Convex Planar Domains

Definitions and mathematical statement

Let $\Omega,\Omega'\subseteq{\mathbb{R}}^2$ be convex planar domains, and write $|\Omega|$ for the two-dimensional Lebesgue measure of $\Omega$. The supplied record does not specify boundedness, finiteness of area, boundary regularity, or strict convexity. These hypotheses must be recovered before fixing a precise billiard model.

Write $\mathcal L_{\mathrm{bill}}(\Omega)$ provisionally for the invariant called the "billiard length spectrum" in the supplied cleaned statement, and let $G$ provisionally denote the class of transformations called "unitary affine transformations" there. The extracted question is whether, for every pair of domains in the intended class, \[ \left(|\Omega|=|\Omega'|\ \text{and}\ \mathcal L_{\mathrm{bill}}(\Omega)=\mathcal L_{\mathrm{bill}}(\Omega')\right) \quad\Longrightarrow\quad \exists\,T\in G\ \text{such that}\ T(\Omega)=\Omega'. \] An affine transformation has the form $T(x)=Mx+b$, where $M\in\mathrm{GL}_2({\mathbb{R}})$ and $b\in{\mathbb{R}}^2$. The restriction on $M$ intended by "unitary" is not supplied. In particular, an orthogonal linear part and a determinant-one linear part give different transformation classes; neither interpretation is selected here.

Unresolved source formulation. The retained original passage instead asks whether \[ A(\Omega)=A(\Omega') \quad\Longrightarrow\quad \Omega\text{ and }\Omega'\text{ are equal up to a unitary affine transformation}. \] It does not define $A$ or explicitly state the additional spectral hypothesis appearing in the cleaned version. Consequently, $A$ cannot safely be identified with scalar area, an area spectrum, or the conjunction of invariants in the cleaned statement. The background references area spectral rigidity and symplectic billiards, whereas the cleaned statement names a billiard length spectrum. The billiard model, spectral invariant, use of multiplicities, treatment of iterated periodic trajectories, and intended transformation group therefore remain unresolved. The displayed implication is a faithful conditional transcription of the cleaned question, not a fully specified mathematical formulation.

The supplied dated curation reports partial results for axial or selected centrally symmetric strictly convex domains near ellipses, including a 2026 strengthening for the axial class, and labels the unrestricted equality question open. These are supplied literature assessments, not independently verified facts or additional hypotheses imposed on the question.

Short English statement

The cleaned record asks whether equal area and equal billiard length spectra determine a convex planar domain up to the stated affine equivalence. A precise formulation still requires identification of the billiard model, spectrum, admissible domains, transformation group, and the original notation $A$. The supplied partially solved label and dated special-case results require independent mathematical review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30006364$, title "Area-Spectral Rigidity of Convex Planar Domains," exact source problem code catalogue entry; collection "Oberwolfach Reports Open Problems." The proposed year is recorded as 2025; no proposer is supplied.
• Supplied report citation: Dynamische Systeme (2026), Oberwolfach Reports, DOI 10.4171/owr/2025/31, pp. 1647--1706. https://doi.org/10.4171/owr/2025/31. The record attributes the extracted question to the workshop report and its open-problem discussions, without identifying an individual author of the question.
• Baracco--Bernardi--Nardi, Area spectral rigidity. https://arxiv.org/abs/2410.12644.
• Area spectral rigidity for axial symplectic billiards. https://doi.org/10.3934/jmd.2026012.
• Oberwolfach Report 31/2025. https://ems.press/content/serial-article-files/52240.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved, reports rigidity for axial or selected centrally symmetric strictly convex domains near ellipses and a 2026 strengthening for the axial class, and describes the unrestricted equality question as open. This is supplied dated curation, not independently verified literature assessment.
• The registry labels the statement corrected_verified and says that the official report was checked, the question isolated, and its notation repaired. These are supplied source-verification claims; no independent source visit or verification is asserted in this draft.