Local smooth isometric immersion of surfaces in three-space
Given a smooth Riemannian metric on a surface, can a neighbourhood of each point be realised as a smooth surface in , i.e. is there a smooth with near the point? The analytic case is the Janet-Cartan theorem (1926 to 1927); smooth local existence holds where the Gaussian curvature is nonzero, and Lin, Nakamura-Maeda, Han-Khuri and others treated nonnegative curvature and various degenerate zero sets. Nash-Kuiper gives realisations, and Pogorelov and Nadirashvili-Yuan gave metrics with no realisation. Does every smooth surface metric admit a smooth local isometric immersion into ?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Differential geometry, isometric embedding
- Posed by
- Discussed by S.-T. Yau, Review of geometry and analysis (Asian J. Math., 2000, p. 236), and stated as Problem 1.9 in M. Ghomi, Open problems in geometry of curves and surfaces (2019)
- Year posed
- 2000
- Years open
- 26y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 45 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem A: there is a positive definite metric on such that no open neighbourhood of the origin admits a isometric immersion into ; the metric can have the full Taylor jet of the Euclidean metric at the origin, so local realisability is not decided by the jet. The curvature takes both signs. Not settled: the smooth local problem under or , finite-regularity versions, and realisation in .
What the AI did
Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscripts are authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues. The main theorem has a Lean formalization in the release.
Verification
No independent mathematician has checked this yet. Checked here: the abstract and Theorem A of 'A Smooth Metric with No Local Isometric Immersion into Three-Space', read against the problem as quoted from Yau and Ghomi. The proof (a Darboux-equation obstruction on caps, an oscillating perturbation, Baire category and the Nadirashvili-Yuan boundary-saddle argument) was not refereed. Lean: formalization.yaml lists OAI.SmoothLocal.Geometry.exists_local_metric_without_local_immersion (lean/OAI/Geometry/IsometricImmersion/Main.lean, comparator ComparatorChallenges/IsometricImmersion.lean). Its statement was read: a smooth positive-definite matrix field g on the open square (-1,1)^2 such that for every open U inside the square containing 0, no smooth F : U -> R^3 has inner products of its derivatives equal to g. That is the headline. Not rebuilt here. A reader should know that a 2002 arXiv preprint of Nadirashvili and Yuan stated a stronger counterexample, which their 2008 paper and later accounts treat as leaving the question open; the manuscript notes this.