Paper · Oct 8, 2026

Generic mean curvature flow has finitely many singularities

J. M. Daniels-Holgate, Or Hershkovits, Gábor Székelyhidi

math.DGWider flowsarXiv:2610.12763
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Abstract

We show that for a generic compact initial surface in , and any generic compact initial mean convex hypersurface in higher dimensions, the mean curvature flow encounters only finitely many singularities. This resolves the full generic Morse regularity conjectures of Ilmanen and Colding-Minicozzi-Pedersen in the above two cases. More generally, we show that if a mean curvature flow of a compact hypersurface has only multiplicity one spherical and cylindrical singularities, then after a small perturbation of the initial condition only non-degenerate cylindrical and spherical singularities appear.

Comments: 60 pages

Mean curvature flowSingularities

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Read around it

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) 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)

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