Singularities of Mean Curvature Flow in R^3
Tom Ilmanen (ETH Zuerich)
We present several new results about singularities of mean curvature flow of surfaces in $R^3$, obtained with Colding, Minicozzi, and White.
Theorem 1 (Isolation of Cylinder): In $R^3$, the cylinder is isolated among self-shrinkers in the $C^2_{loc}$ topology.
As a consequence we obtain a structure theorem for shrinkers in $R^3$ with gaussian density less than 2, as well as
Theorem 2 (Positive Neighborhood Theorem): A cylinder-type singularity in $R^3$ is contained in a space-time neighborhood
that has positive mean curvature (with one small caveat), and
Theorem 3 (Version of the Genericity Conjecture): almost every level-set of a level-set flow in $R^3$ with gaussian density
ratios less than two has only cylinder and sphere type singularities.