Counterexamples to Schiffer's conjecture and the Pompeiu problem
Infinitely many planar domains that are not balls admit a Neumann eigenfunction of the Laplacian that is constant on the boundary, disproving Schiffer's conjecture and, through a classical equivalence, the 1929 Pompeiu problem.
- Lab
- Independent
- Model
- GPT-5.6, Claude Opus 4.8, Claude Fable 5
- Field
- Mathematics
- Date
- 2026-08-05
- Human collaborators
- Gonzalo Cao-Labora, Jaume de Dios Pont
- Problem posed
- 1929 · open 97 yrs
What was found
Pompeiu asked in 1929 whether a bounded domain over which some nonzero function integrates to zero under every rigid motion must be a ball. Schiffer's 1957 reformulation asks the same thing through Neumann eigenfunctions of the Laplacian that are constant on the boundary, and appears as Problem 80 on Yau's 1982 list; Williams proved the two formulations equivalent for simply connected domains in 1976. Cao-Labora and de Dios Pont construct infinitely many N-fold symmetric planar domains, with N large, that are not balls and carry such an eigenfunction. The strategy is to relax the problem so that N may be any real number, which corresponds to the Schiffer problem only at integers, apply bifurcation theory there, and show the size of the local bifurcation branch can be taken independently of N. Their Corollary 1.2 applies Williams' equivalence to the same domains, so a single construction settles both problems.
Novelty check
Schiffer's conjecture is Problem 80 on Yau's list, with a partial-results literature running since the 1970s, and the Pompeiu problem has stood since 1929. Both are recorded as open in the standard references and in DeepMind's formal-conjectures repository, which carries a Lean statement of the Pompeiu problem as an unsolved challenge that this work's formalization closes. It is not the first counterexample. On 2026-08-03, two days before this paper, Matthew Colbrook and George Stepaniants posted an independent computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures, a single ten-fold symmetric domain; this paper acknowledges it and takes a different route, bifurcation theory, to infinitely many domains. The construction is new; the disproof is not the first.
Caveats and known objections
A preprint, not peer-reviewed. The Lean 4 verification was written by GPT-5.6 from an early draft of the paper and lives in an author's own repository; it had no independent audit at entry time, and the same group produced both the proof and its formalization. Autonomy graded ai-assisted: the paper is explicit that the novel construction strategy is the authors' own, and that the models were used to verify the asymptotic estimates numerically, to produce first drafts of the proofs of the Bessel function estimates, and to help with exposition.
Independent checks
Matthew J. Colbrook and George Stepaniants (independent earlier counterexample): Constructed, two days earlier and by a different method, a ten-fold symmetric planar domain with real-analytic boundary that is a counterexample to both Schiffer's conjecture and the planar Pompeiu conjecture, established by a computer-assisted contraction argument. Corroborates that planar counterexamples exist; does not check this paper's construction. · link ↗
The open problem
315903
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Also recorded at
Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.
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History of this entry
- CorrectedCorrected the novelty check, which said no prior counterexample appears: Colbrook and Stepaniants posted an independent one two days earlier, which this paper itself acknowledges. No grade moved. source ↗
- CorrectedRecorded the MathDB record for the Pompeiu problem, the second of the two questions this entry settles, as a registration. No grade moved.
- CorrectedRecorded the vibemathed record for this result as a registration. The vibemathed link it replaces was classified as commentary, which a registry listing is not.
- AddedEntered the registry graded Formally verified and AI-assisted.
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Graded formally verified for verification and ai-assisted for autonomy. What these mean.
Cite this entry
whataifound.org. (2026). Counterexamples to Schiffer's conjecture and the Pompeiu problem. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-05-schiffer-conjectureBibTeX
@misc{whataifound-independent-2026-conjecture,
title = {Counterexamples to Schiffer's conjecture and the Pompeiu problem},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Formally verified. Autonomy: AI-assisted.},
url = {https://whataifound.org/finding/2026-08-05-schiffer-conjecture}
}