Event Details
Title: Mean curvature flow with generic initial data and applications
Abstract: Mean curvature flow is the gradient flow of the area functional and constitutes a natural geometric heat equation on the space of hypersurfaces in an ambient Riemannian manifold. It is believed, similar to Ricci Flow in the intrinsic setting, to have the potential to serve as a tool to approach several fundamental conjectures in geometry. The obstacle for these applications is that the flow develops singularities, which one in general might not be able to classify completely. Nevertheless, a well-known conjecture of Huisken states that a generic mean curvature flow should have only spherical and cylindrical singularities. As a fundamental step in this direction Colding-Minicozzi have shown that spheres and cylinders are the only linearly stable singularity models. In this mini-course we will give an introduction to weak mean curvature flow and discuss recent progress in joint work with Otis Chodosh, Kyeongsu Choi and Christos Mantoulidis in showing that for generic initial hypersurfaces in \(R^3\) and under a suitable low entropy assumption in \(R^4\) their weak evolution has only spherical and (generalised) cylindrical singularities. We will also discuss applications to the low entropy Schoenflies conjecture.
