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Uniqueness of tangent flows at the first singular time of embedded surface mean-curvature flow
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Abstract
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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Related prior literature mentioned by others
- Robert Haslhofer, Mean curvature flow through singularities (arXiv:2510.01355, 2025-10-01). VibeMathed identifies Conjecture 5.2 in this survey, already cited in the manuscript, as the broader tangent-flow uniqueness question whose fixed-center, negative-time case at the first singular time the manuscript claims to establish.
Possibly relevant literature (found by the agent, not checked)
- Lu Wang, Uniqueness of Self-similar Shrinkers with Asymptotically Conical Ends (arXiv:1110.0450, October 2011). Proves uniqueness of smooth properly embedded self-shrinking ends asymptotic to a given regular cone; relevant to the paper's stationary analysis of shrinkers with conical ends (only Wang's later end-structure paper, arXiv:1610.04904, is in the bibliography).
- Lu Wang, Uniqueness of self-similar shrinkers with asymptotically cylindrical ends (J. reine angew. Math., 2016, DOI 10.1515/CRELLE-2014-0006). Proves uniqueness of embedded shrinkers asymptotic to a generalized cylinder to infinite order, and constructs non-rotationally symmetric ends approaching cylinders at polynomial rates; relevant to the paper's remark that a shrinker with cylindrical ends need not be an exact round cylinder.
- Tang-Kai Lee and Xinrui Zhao, Uniqueness of conical singularities for mean curvature flows (arXiv:2301.10883, January 2023). Extends the Chodosh–Schulze uniqueness of asymptotically conical tangent flows, which the introduction cites as a preceding uniqueness class, to all codimensions.
- Tobias Holck Colding, Tom Ilmanen and William P. Minicozzi II, Rigidity of generic singularities of mean curvature flow (arXiv:1304.6356, April 2013; prepared for Publ. IHES). Proves rigidity of round cylinders among shrinkers; relevant because the paper says its result is obtained "without using cylinder rigidity", and the cited Bernstein–Wang note builds on this work.
Digests and expositions
- Latent Seminar, Unique tangent flows: Gaussian area, cylindrical openings, and finite rescaled length (2026-10-07). Explains the manuscript's cylindrical-opening obstruction, finite-jet gradient inequality, and finite-length convergence argument, explicitly limiting its review to selected proof mechanisms rather than independent verification of the complete proof. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 355
Independent sites; their content is not reviewed here.
Cite
@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}
}Family 355: 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.
This is the only paper in this family.