Paper · Jul 26, 2026

Closed mean curvature flows with prescribed tangent flows

Jingwen Chen, Tang-Kai Lee, Ao Sun, Xinrui Zhao

math.DGarXiv:2607.23879
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Abstract

Given an embedded shrinker in that is either closed, asymptotically conical, or a Cartesian product of such a shrinker with , we construct a closed embedded mean curvature flow whose tangent flow at the first singularity is modeled on . We also prescribe the first-order asymptotics of the tangent flow. This result is a consequence of a more general theorem that allows us to construct mean curvature flows with an additional force whose tangent flow and first-order asymptotics at the first singularity are prescribed.

Comments: 40pages, comments are welcome!

Mean curvature flowRicci solitonsSingularitiesBoundaries & non-compact

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Generic Dynamics of Mean Curvature Flows with Asymptotically Conical Singularities (Ao Sun and Jinxin Xue, 2021)4Sun 2021 Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers (Maxwell Stolarski, 2022)8Stolarski 2022 Uniqueness of conical singularities for mean curvature flows (Tang-Kai Lee and Xinrui Zhao, 2023)3Lee 2023 Mean curvature flow with multiplicity 2 convergence in ℝ³ (Jingwen Chen and Ao Sun, 2023)7Chen 2023 Mean curvature flow with multiplicity 2 convergence in closed manifolds (Jingwen Chen and Ao Sun, 2024)5Chen 2024 Closed mean curvature flows with asymptotically conical singularities (Tang-Kai Lee and Xinrui Zhao, 2024)1Lee 2024 A stable self-shrinking Möbius bundle in ℝ⁴ (Kahnrad Braxton, Tang-Kai Lee and Jonathan J. Zhu, 2026)6Braxton 2026 Genus Increase in Closed Mean Curvature Flows (Jingwen Chen, 2026)2Chen 2026
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  6. A stable self-shrinking Möbius bundle in

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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