Research

The authors show that the mean curvature flow of generic closed surfaces in R^3 avoids asymptotically conical and non-spherical compact singularities with multiplicitiy one. Also, the mean curvature flow of generic closed low-entropy hypersurfaces in R^4 is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons.

▶ Point D represents a soliton corresponding to a linearly unstable singularity in the space of surfaces. A flow starting from the initial surface C converges to D. However, if the initial value at C is generically perturbed, the flow moves away from D in finite time.