Generic mean curvature flow I; generic singularities


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.


Tobias H. Colding

Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307

William P. Minicozzi II

Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218