[#P3072] Birkhoff conjecture for integrable convex billiards

Problem. If a strictly convex \(C^\infty\) planar billiard table has an invariant essential caustic for every rotation number \(\rho\in(0,\tfrac12)\), must its boundary be an ellipse?

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1Context

Known frontier: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses. Open boundary: The global classification of all smooth strictly convex integrable tables remains open.

2Problem setup

Definition 1 (billiard map). The boundary phase-space map taking one reflection to the next.

Definition 2 (integrable). The phase annulus is foliated by invariant essential curves, equivalently caustics in the stated formulation.

Remark 1. A caustic is a curve tangent to every segment of any orbit that starts tangent to it. Ellipses have a full foliation by confocal caustics; the conjecture says they are the only smooth strictly convex tables with this complete integrability.

3What counts as a solution

Prove every table satisfying the invariant-caustic hypothesis is an ellipse.
Or construct a nonelliptic smooth strictly convex table with the full caustic foliation.

1Status

What counts as a solution

Current status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses. Exact unresolved remainder: The global classification of all smooth strictly convex integrable tables remains open.^[1][2]

1Packet records

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Record	Kind	Assessment
OPEN as checked on 2026-08-01. Strongest checked neighboring result: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses. Exact unresolved remainder: The global classification of all smooth strictly convex integrable tables remains open.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as checked on 2026-08-01. Strongest checked neighboring result: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses. Exact unresolved remainder: The global classification of all smooth strictly convex integrable tables remains open.^[1][2] Relevance to this problem This is the dated publication status for the canonical target Birkhoff conjecture for integrable convex billiards. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The problem was checked as open on 2026-08-01. The strongest neighboring result found in the cited sources is: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses. The exact unresolved remainder is: The global classification of all smooth strictly convex integrable tables remains open. A complete resolution must meet the following acceptance conditions: - Prove every table satisfying the invariant-caustic hypothesis is an ellipse. - Or construct a nonelliptic smooth strictly convex table with the full caustic foliation. Open full record →
Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 2 Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses.^[1][2] Relevance to this problem Locates the present research frontier immediately below Birkhoff conjecture for integrable convex billiards. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses. This leaves the following boundary unresolved: The global classification of all smooth strictly convex integrable tables remains open. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target. Related: Claim 1 Open full record →
The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses. Unresolved remainder: The global classification of all smooth strictly convex integrable tables remains open.	Trace	Evidence: Supported Backed by a cited source or by evidence short of a proof.
attempt · Route 1 The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses. Unresolved remainder: The global classification of all smooth strictly convex integrable tables remains open.^[1][2] Relevance to this problem Documents why Birkhoff conjecture for integrable convex billiards was treated as a distinct open target on 2026-08-01. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope No narrower scope recorded Sources Reference 1, Reference 2 What happened The audit ran on 2026-08-01 across the cited primary literature and problem collections, Crossref, arXiv, and stable publisher records, TheoremDB published, prospecting, packet, formal, and retired corpus. It checked the exact statement, its named or normalized variants, recent status evidence, and the controlled TheoremDB corpus. Queries included: - Birkhoff billiard conjecture integrable ellipse open 2026 - local strong Birkhoff conjecture almost every ellipse 2025 The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses. Unresolved remainder: The global classification of all smooth strictly convex integrable tables remains open. Related: Claim 1 Open full record →
The global classification of all smooth strictly convex integrable tables remains open.	Trace	Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 2 The global classification of all smooth strictly convex integrable tables remains open. Relevance to this problem Turns the remaining uncertainty in Birkhoff conjecture for integrable convex billiards into a checkable research target. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded What happened Research should address this boundary directly: The global classification of all smooth strictly convex integrable tables remains open. A claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations: - Prove every table satisfying the invariant-caustic hypothesis is an ellipse. - Or construct a nonelliptic smooth strictly convex table with the full caustic foliation. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses. Exact unresolved remainder: The global classification of all smooth strictly convex integrable tables remains open.

Equivalent-formulation queries: Birkhoff billiard conjecture integrable ellipse open 2026; local strong Birkhoff conjecture almost every ellipse 2025
Strongest checked neighboring result: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses.
Exact unresolved remainder: The global classification of all smooth strictly convex integrable tables remains open.

How the 4 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemBirkhoff conjecture for integrable convex billiards

All 3 recorded relations between these records and the problem

2See also

Hilbert's sixteenth problem, second partdynamical systems
Persistent exponential stretching of a material line in two-dimensional Euler flowdynamical systems
Generic analytic Arnold diffusion in a priori stable systemsdynamical systems

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Plain text

“Birkhoff conjecture for integrable convex billiards.” TheoremDB. P3072. Problem statement; statement text SHA-256 4b5969ce241e697a56bdc471500d8d45600778f036cd4c8311f7fb75d3b17971. https://theoremdb.org/statement/?ref=P3072

BibTeX

@misc{theoremdb-problem-4b5969ce241e697a56bdc471500d8d45600778f036cd4c8311f7fb75d3b17971,
  title = {{Birkhoff conjecture for integrable convex billiards}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 4b5969ce241e697a56bdc471500d8d45600778f036cd4c8311f7fb75d3b17971},
  url = {https://theoremdb.org/statement/?ref=P3072}
}

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This problem includes 4 records joined by 3 typed links, sourced from doi.org^[1], current as of August 1, 2026.

1Lean verification

1References

Packet source. Illya Koval, “Local strong Birkhoff conjecture and local spectral uniqueness of almost every ellipse”. Inventiones mathematicae 244(1) (2026), 221-298. DOI 10.1007/s00222-025-01397-y. main local strong Birkhoff theorem. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Proves the local strong Birkhoff conjecture near almost every ellipse under the paper's perturbative hypotheses.Also cited at Illya Koval, “Local strong Birkhoff conjecture and local spectral uniqueness of almost every ellipse,” Inventiones Mathematicae 244(1) (2026), 221–298. main local strong Birkhoff theorem.Source used to assess the problem's recorded status.For Birkhoff conjecture for integrable convex billiards: This is the dated publication status for the canonical target Birkhoff conjecture for integrable convex billiards.Source named by the research packet.
M. Bialy, C. Fierobe, A. Glutsyuk, M. Levi, A. Plakhov, and S. Tabachnikov, “Open problems on billiards and geometric optics”. arXiv:2110.10750 (2021). integrable billiards section. ↗preprint · reference source · arXiv:2110.10750, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Places Birkhoff's conjecture among current billiard problems and records known cases.Source used to assess the problem's recorded status.For Birkhoff conjecture for integrable convex billiards: Places Birkhoff's conjecture among current billiard problems and records known cases.

Original TheoremDB editorial statement and source synthesis; external works are used for citation only.

[#P3072] Birkhoff conjecture for integrable convex billiards

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