When Mean Curvature Flow Breaks: Singularities, Topology, and Ilmanen's Conjecture

Abstract: Mean curvature flow is a geometric evolution equation that tends to smooth surfaces, but inevitably develops singularities. A fundamental question is how the topology of a surface can change as the flow passes through a singularity. In 1995, Tom Ilmanen conjectured that genus cannot increase through such transitions. In this talk, I will describe recent joint work with David Hoffman and Brian White that provides counterexamples to this conjecture. We construct self-similar mean curvature flows passing through a conical singularity in which the genus can increase by an arbitrarily large amount. The construction combines self-shrinkers and self-expanders asymptotic to the same cone, using ideas from minimal surface theory, symmetry, and degree theory.

differential geometry

Audience: researchers in the topic

CUNY Geometric Analysis Seminar