Entropy and self-shrinkers of the mean curvature flow

Blackboard

The Colding-Minicozzi entropy is an important tool for understanding the mean curvature flow (MCF). Together with Ilmanen and White, they conjectured that the round sphere minimises entropy amongst all closed hypersurfaces. We will review the basics of MCF and their theory of generic MCF, and then describe the resolution of the above conjecture, due to Bernstein and Wang for dimensions up to six and recently claimed by the speaker for all remaining dimensions. A key ingredient in the latter is the classification of entropy-stable self-shrinkers that may have a small singular set.