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The singular set of mean curvature flow with generic singularities

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Abstract

A mean curvature flow starting from a closed embedded hypersurface in \(\mathbf{R}^{n+1}\) must develop singularities. We show that if the flow has only generic singularities, then the space-time singular set is contained in finitely many compact embedded \((n-1)\)-dimensional Lipschitz submanifolds plus a set of dimension at most \(n-2\). If the initial hypersurface is mean convex, then all singularities are generic and the results apply. In \(\mathbf{R}^3\) and \(\mathbf{R}^4\), we show that for almost all times the evolving hypersurface is completely smooth and any connected component of the singular set is entirely contained in a time-slice. For 2 or 3-convex hypersurfaces in all dimensions, the same arguments lead to the same conclusion: the flow is completely smooth at almost all times and connected components of the singular set are contained in time-slices. A key technical point is a strong parabolic Reifenberg property that we show in all dimensions and for all flows with only generic singularities. We also show that the entire flow clears out very rapidly after a generic singularity. These results are essentially optimal.

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Notes

  1. For many of our results (though not all) one can allow the tangent flow to have multiplicity greater than one (cf. [5]), however this higher multiplicity does not occur in the most important cases.

  2. See also Schulze [43], for uniqueness at smooth closed singularities.

  3. The map can be taken to be a graph over a subset of a time-slice; this is connected to Theorem 1.2.

  4. See, for instance [7, 19, 25, 36, 37, 40, 45].

  5. See, for instance [47].

  6. See [10, 18, 28], section 2.3 of [38, 41, 42, 44], and [47].

  7. See, for instance, the remarks on page 258 of [44].

  8. In fact, one can take \(f(r)\approx (\log |\log r|)^{-\alpha }\) (\(\alpha >0\)).

  9. Ilmanen and White extended the monotonicity to the case where \(M_t\) is a Brakke flow.

  10. This is a different convention than in [12] where we used k for the dimension of the spherical factor.

  11. Cf. lemma \({\text {I}}.1.2\) in [13] and section \({\text {III}}.2\) in [14].

  12. The function u may be multi-valued, but the projection from S to \(\{ t=0 \}\) is a finite-to-one covering map.

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Authors and Affiliations

  1. Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA

    Tobias Holck Colding & William P. Minicozzi II

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  1. Tobias Holck Colding
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  2. William P. Minicozzi II
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Correspondence to William P. Minicozzi II.

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The authors were partially supported by NSF Grants DMS 1404540, DMS 11040934, DMS 1206827, and NSF FRG Grants DMS 0854774 and DMS 0853501.

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Colding, T.H., Minicozzi, W.P. The singular set of mean curvature flow with generic singularities. Invent. math. 204, 443–471 (2016). https://doi.org/10.1007/s00222-015-0617-5

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