Uniqueness of tangent flows at the first singular time of mean curvature flow of closed embedded surfaces
For the mean curvature flow of a smooth closed embedded surface in , parabolic rescalings about a singular point subconverge to self-shrinking tangent flows (Huisken, Ilmanen), and Bamler-Kleiner proved the limits have multiplicity one. Uniqueness was known for compact shrinkers (Schulze), round cylinders (Colding-Minicozzi) and asymptotically conical shrinkers (Chodosh-Schulze), but a general shrinker may have both conical and cylindrical ends. Haslhofer formulates tangent-flow uniqueness as Conjecture 5.2. Is the tangent flow at every singular point unique, independent of the rescaling sequence, including its position and axes?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Geometric analysis; mean curvature flow singularities
- Posed by
- Robert Haslhofer, Mean curvature flow through singularities, arXiv:2510.01355 (2025), Conjecture 5.2; the question is older (Ilmanen and others)
- Year posed
- 2025
- Years open
- 1y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 40 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Claims uniqueness of tangent flows, without mean-convexity or a prescribed tangent model, at every singular point of the first singular time for smooth compact connected embedded surfaces in : all fixed-center rescalings converge to one multiplicity-one self-shrinker flow in the original coordinates, and every backward Brakke tangent agrees with it. The key step is a finite-jet Lojasiewicz analysis handling shrinkers with both conical and cylindrical ends. It does NOT cover later singular times, higher dimensions, or moving-center blowups.
What the AI did
The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscripts are credited to OpenAI with no human author named. Single manuscript; no Lean formalization.
Verification
No independent mathematician has checked this yet. Theorem 1.1 was read against Haslhofer's Conjecture 5.2: at every singular point of the FIRST singular time of a closed embedded surface flow in , the full family of fixed-center rescalings converges locally smoothly with multiplicity one to a single shrinker, with fixed position and axes. The paper itself says it establishes the negative-time, fixed-center part of the conjecture at the first singular time, and that moving centers and continuation through singularities are different questions. It uses Bamler-Kleiner's multiplicity-one theory as input. No Lean formalization.