Result 355, Differential geometry

Unique tangent flows at the first surface singularity

Proves the first-singular-time case of tangent-flow uniqueness for smooth compact connected embedded surfaces without boundary in ℝ3. At every singular point, all fixed-center backward tangent flows agree as area measures at every negative time in the original ambient coordinates, without mean-convexity or a prescribed tangent model.

Proof

The bigger picture

Why it matters

When a surface moves according to its curvature, it can develop singularities where smooth evolution breaks down. The manuscript reports that, at the first such breakdown, zooming in at a fixed singular point reveals one unambiguous shrinking model.

What changes?

The claim concerns smooth, compact, connected surfaces embedded without boundary in three-dimensional Euclidean space. At each singular point at the first singular time, every fixed-center rescaling converges locally smoothly on compact intervals of negative rescaled time to the same multiplicity-one homothetic self-shrinker: a surface evolving only by shrinking, counted once. Uniqueness holds in the original coordinates, including position and axes. It requires neither mean-convexity, a one-sided curvature condition, nor a prescribed tangent model.

What does that help mathematicians do?

The reported uniqueness means that different sequences of magnifications cannot produce different limiting surface area measures at any negative rescaled time, before the singular moment. Nor is agreement merely up to rotation: the claim fixes the model's axes in the ambient space. This removes ambiguity in the local model used to analyze an initial singularity, without itself identifying which shrinking model occurs.

Are there practical applications?

Its immediate value is foundational: it would make the local geometry of a first singularity a well-defined object for further study of surface evolution. It provides a definite target for investigating which shrinking shapes arise and how surfaces approach them. The stated result does not address later singularities or supply a numerical method for finding that target.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Uniqueness of tangent flows at the first singular time of embedded surface mean-curvature flow

September 24, 2026 100 pages

We prove that, at each singular point of the first singular time of the mean-curvature flow of a smooth compact connected embedded surface without boundary in ℝ3, all fixed-center rescalings converge locally smoothly on compact negative-time intervals to one multiplicity-one homothetic self-shrinker flow. The limit is unique in the original ambient coordinates, including its position and axes. No mean-convexity assumption or prescribed tangent model is required.

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Cite (BibTeX) 
@misc{OAI:Tangent-flow-uniqueness-2026-09-24,
  author = {{OpenAI}},
  title = {{Uniqueness of tangent flows at the first singular time of embedded surface mean-curvature flow}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Tangent-flow-uniqueness-2026-09-24/paper.pdf}{OAI:Tangent-flow-uniqueness-2026-09-24}},
  year = {2026}
}

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An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.