Felix Schulze

Evolution of convex lens-shaped networks under the curve shortening flow

We consider convex symmetric lens-shaped networks in R2 that evolve under the curve shortening flow. We show that the enclosed convex domain shrinks to a point in finite time. Furthermore, after appropriate rescaling the evolving networks converge to a self-similarly shrinking network, which we prove to be unique in an appropriate class. We also include a classification result for some self-similarly shrinking networks.

No mass drop for mean curvature flow of mean convex hypersurfaces

A possible evolution of a compact hypersurface in R by mean curvature past singularities is defined via the level set flow. In the case that the initial hypersurface has positive mean curvature, we show that the Brakke flow associated to the level set flow is actually a Brakke flow with equality. We obtain as a consequence that no mass drop can occur along such a flow. As a further application of the techniques used above we give a new variational formulation for mean curvature flow of mean convex hypersurfaces.

Consequences of Strong Stability of Minimal Submanifolds

In this note we show that the recent dynamical stability result for small

Self-similarly expanding networks to curve shortening flow

C^1

Stability of translating solutions to mean curvature flow

-perturbations of strongly stable minimal submanifolds of C.-J. Tsai and M.-T. Wang directly extends to the enhanced Brakke flows of Ilmanen. We illustrate applications of this result, including a local uniqueness statement for strongly stable minimal submanifolds amongst stationary varifolds, and a mechanism to flow through some singularities of Lagrangian mean curvature flow which are proved to occur by Neves.