Geodesic
Generic mean curvature flow I; generic singularities
Tobias H. Colding1 ; William P. Minicozzi II2
1 Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307
2 Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218
Annals of mathematics, Tome 175 (2012) no. 2, pp. 755-833.

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It has long been conjectured that starting at a generic smooth closed embedded surface in $\mathbf{R}^3$, 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 to 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 singularity other than spheres and cylinders can be perturbed away.
DOI : 10.4007/annals.2012.175.2.7

Tobias H. Colding 1 ; William P. Minicozzi II 2

1 Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307
2 Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218 
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Tobias H. Colding; William P. Minicozzi II. Generic mean curvature flow I; generic singularities. Annals of mathematics, Tome 175 (2012) no. 2, pp. 755-833. doi : 10.4007/annals.2012.175.2.7. http://geodesic.mathdoc.fr/articles/10.4007/annals.2012.175.2.7/

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