Mean Curvature Flow and Variational Integrators

Abstract: Mean curvature flow is a well-studied geometric flow under which a surface moves in such a way as to decrease its area as fast as possible. Some surfaces, such as spheres and cylinders, evolve under mean curvature by dilations. Such surfaces, called self-shrinkers, are models for the singularities that can occur under mean curvature flow. The first nontrivial example of a self-shrinker was a torus proved to exist by Angenent in 1989. Self-shrinkers can be seen as the critical points of a weighted surface area functional called the entropy. There are simple formulas for the entropies of spheres and cylinders, but there is no such formula for the entropy of the Angenent torus, nor is their a formula describing the surface itself. The best previous result is a 2018 paper showing that the entropy of the Angenent torus is less than two. I numerically estimated the entropy of the Angenent torus to be 1.8512186, with an estimated accuracy of 0.0000019, using a variational numerical approach in order to facilitate future work on proving rigorous error bounds.

In this talk, I will introduce the basics of mean curvature flow and variational integrators and discuss how I used these ideas to numerically estimate the Angenent torus and its entropy. I will discuss numerical evidence for the error bounds of my estimate and describe a future strategy for rigorously proving such error bounds.

Host: Yanli Song