View source for Uniqueness of tangent flows for mean-curvature flow of surfaces

A '''tangent flow''' describes a mean-curvature flow at a singularity after magnifying space and time; compactness gives such limits along subsequences, and the '''uniqueness of tangent flows''' asks whether every magnification sequence gives the same limit, including its position and axes. Haslhofer formulated it in 2025 together with the rigidity of cylinders, as a problem and a conjecture. A manuscript in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), dated 24 September 2026, claims uniqueness at the first singular time of every smooth closed embedded surface in three-dimensional space, without mean-convexity or a prescribed tangent model. The family has no Lean formalization in the repository's catalogue. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.

== Background ==
* As recounted in the introduction of the manuscript:
** Huisken's monotonicity formula identifies self-shrinkers as the models for parabolic blowups. Ilmanen established smoothness of tangent supports at the first singular time of closed embedded surface flows, and Lu Wang showed that each end of a noncompact self-shrinker of finite topology in three-space is asymptotically conical or a round cylinder. The multiplicity-one theory of Bamler and Kleiner, with their structure theorem for tangent ends, gives smooth properly embedded tangents with finitely many conical and cylindrical ends.
** Uniqueness was known for major classes of models: Schulze for compact multiplicity-one shrinkers, Colding and Minicozzi for round cylinders including their axes, and Chodosh and Schulze for asymptotically conical tangents; Bernstein and Wang obtained uniqueness modulo rotations for planar, spherical and cylindrical models, and Ghosh and Bamler–Lai gave recent alternative approaches in the cylindrical class.
** A shrinker with cylinder-like ends need not be an exact round cylinder, so the remaining case requires a stationary analysis of all its ends at once.

== The original result ==
* '''Claim.''' Theorem 1.1 of ''Uniqueness of tangent flows at the first singular time of embedded surface mean-curvature flow'' states that for a smooth compact connected embedded surface without boundary in '''R'''<sup>3</sup>, at every singular point of its first singular time there is a smooth properly embedded self-shrinker such that the full family of fixed-center parabolic rescalings converges to it locally smoothly and with multiplicity one, uniformly on compact negative-time intervals, in the original ambient coordinates including its position and axes. Every backward integral Brakke tangent then equals this self-shrinker flow at every negative time.
* The manuscript states that it assumes neither mean-convexity nor a prescribed tangent model, that moving-center rescalings and continuations through a singularity are different questions, and that it does not use the rigidity of cylinders.
* ''Method, as described:'' the convergence argument goes back to Simon's gradient-inequality method. On a cylindrical end, a Jacobi field that grows quadratically along the end changes the end type, and its effect on the Gaussian area is smaller than any power of the opening, so the usual analytic argument fails in that direction. The manuscript keeps finite Taylor jets, shows that no formal curve of critical points can open an end by comparing two stationary surfaces built on complex contours on opposite sides of the zero radius of a comparison profile, whose actions must agree up to an exponentially small error while a translator equation forces a nonzero jump, and converts this into a finite inequality with real-algebraic certificates.
* '''Source''' (openai/math repository, commit <code>7ef5c0c</code>): [https://github.com/openai/math/tree/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/preprints/Tangent-flow-uniqueness-2026-09-24 Uniqueness of tangent flows at the first singular time of embedded surface mean-curvature flow], dated 24 September 2026.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 30 October 2026)
|-
| 2026-10-07 || [https://lowrank.github.io/latent-seminar/OpenAI-Math/355-tangent-flow-uniqueness ''Unique tangent flows: Gaussian area, cylindrical openings, and finite rescaled length''], an article of the Latent Seminar blog at lowrank.github.io || Reading (exposition) || Explains parabolic magnification, Gaussian area and the finite-length objective, compact observations and cylindrical openings, the exponentially small obstruction from a translator equation, the fixed Taylor polynomial behind the gradient estimate, and the propagation argument, with proof outlines on why the exponential obstruction dominates the approximation errors, why one finite polynomial is fixed, and how delayed radii make the localization errors summable. || Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs.
|-
| 2026-10-16
| J. Jardine, Lattice24, referee review [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_190.json rank_190] in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995])
| Review (AI-assisted audit)
| Reports no gap. Its two minor notes are limits on verification: the end decomposition and bounded geometry are taken from the Bamler–Kleiner citation, on which the result depends entirely, and the existence of contours with a strict margin on a sector of the required angular size, which it calls load-bearing, was not confirmed. The review describes itself as a partial read.
| Part of an audit that its README describes as complete for the collection's manuscripts without a Lean formalization. The review is published under CC BY 4.0.
|-
| 2026-10-19 || ''AI Proved What?'', episode 355 [https://aiprovedwhat.com/episodes/355-tangent-flow-uniqueness/ Zoom in: one picture?], remade version of 23 October 2026 || Reading (exposition) || Video explainer of the claim and its history. The episode states that later singularities, moving centers and higher dimensions are not claimed, that the paper does not say which shrinkers occur and does not settle whether a shrinker with a cylinder-like end must be a cylinder, working around that question, and that it builds on Bamler and Kleiner's multiplicity-one theorem and description of the ends. || Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as a preprint: OpenAI, 100 pages, dated September 24, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: no Lean proof published. The page states that it found no specialist comment, no reported error and no competing claim.
|}

== Notes ==
* The background summarizes the manuscript's own account of earlier work.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]