Generic mean curvature flow I; generic singularities

By T. Colding, W. Minicozzi. 2009

695 citations recorded by the source corpus.

It has long been conjectured that starting at a generic smooth closed embedded surface in R^3, the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or cylinders. That is, the only singularities of a generic flow are spherical or cylindrical. We will address this conjecture here and in a sequel. The higher dimensional case will be addressed elsewhere. The key in showing this conjecture is to show that shrinking spheres, cylinders and planes are the only stable self-shrinkers under the mean curvature flow. We prove this here in all dimensions. An easy consequence of this is that every other singularity than spheres and cylinders can be perturbed away.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/2c7b86c24edab043f2db5b488fd8f0c85ca5225f

How researchers see this paper

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In Applied Mathematics

1. Robert Haslhofer
2. S. Brendle
3. Lu Wang
4. Qing Cheng
5. Y. Xin
6. Kyeongsu Choi
7. Yejuan Peng
8. Or Hershkovits

In Geometry and Topology

1. Robert Haslhofer
2. S. Brendle
3. Lu Wang
4. Qing Cheng
5. Y. Xin
6. Yejuan Peng
7. Kyeongsu Choi
8. Haizhong Li

Citations in the corpus

• Translating graphs by mean curvature flow (2012, background): “2 in [5], we have ∂υ ∂s ∣∣ s=0 = −∇η, (2.”
• A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension (2011, methodology): “Here, Γ is a Abresch-Langer curve and Sm(√m) is a m-dimensional sphere of radius √ m. Recently, T.H. Colding and W.P. Minicozzi [5] showed that G. Huisken’s classification theorem still holds without the assumption that |A| is bounded.”
• Mean Curvature Flow Solitons in the Presence of Conformal Vector Fields (2017, background): “One of the main research topics on mean curvature flow is the study of self-similar solutions particularly as models of singularities after suitable rescalings of the flow [27], [28], [50], [15], [16], [17].”
• Volume growth, eigenvalue and compactness for self-shrinkers (2011, positive): “Minicozzi II gave a comprehensive study for self-shrinking hypersurfaces [4].”
• Complete self-shrinkers of the mean curvature flow (2012, background): “42) in [10]), Le and Sesum [20] proved a gap theorem on the squared norm of the second fundamental form for self-shrinkers of codimension one: Theorem A (Le and Sesum [20]).”
• Stability properties and topology at infinity of $$f$$f-minimal hypersurfaces (2013, methodology): “Remark 2 When Σ is a complete self-shrinker with vol f (Σ) < +∞, the conclusion in Proposition 2 was originally pointed out by Colding an Minicozzi in [10].”
• Complete $$\lambda $$λ-hypersurfaces of weighted volume-preserving mean curvature flow (2014, methodology): “Furthermore, we classify complete λ-hypersurfaces with polynomial area growth andH−λ ≥ 0, which are generalizations of the results due to Huisken [18], Colding-Minicozzi [10].”
• Ancient asymptotically cylindrical flows and applications (2019, methodology): “Combined with a recent result by Colding-Minicozzi [CM20], we also obtain a classification result in higher codimension (Corollary 1.20).”
• Mean curvature self-shrinkers of high genus: Non-compact examples (2011, background): “ominently a smooth compactness theorem (for closed, xed genus surfaces [CM1]) and a theory of generic singularities of ColdingMinicozzi, including classication of all H0 complete hypersurfaces (see [CM2] and [DX]). See also [LS] and [Wa] for other uniqueness results. The self-shrinker equation is a ”
• Self-shrinkers with a rotational symmetry (2010, methodology): “for mean curvature ow can be quite exotic and wildly behaved, and the classication of solutions to (1.1) seems impossible in general, however in dimension 2 the methods of Colding-Minicozzi in [CM1]{[CM7] oer a possibility. However, for a generic initial surface, one expects to nd a rather tame sing”
• Mean Curvature Flow in Higher Codimension: Introduction and Survey (2011, positive): “Recently there has been done quite some work on the formation and classification of singularities in mean curvature flow [5,11,12,23, 25, 39, 47, 54, 56, 58, 59, 61, 71], partially motivated by Hamilton’s and Perelman’s [45, 66–68] work on the Ricci flow that in many ways behaves akin to the mean cu”