Wednesday, March 16, 2022 4:15 PM
Wenkui Du (University of Toronto)

In this talk, I will describe my recent joint work with Robert Haslhofer on uniqueness and non-uniqueness of ancient ovals under mean curvature flow.  We confirm the conjecture of Angenent-Daskalopoulos-Sesum that $SO(k)\times SO(n+1-k)$-symmetric ancient ovals are unique up to parabolically rescaling and space-time rigid motion. On the other hand, we also construct a (k-1)-parameter family of uniformly (k+1)-convex ancient ovals that are only $\mathbb{Z}^{k}_{2}\times O(n+1-k)$-symmetric, which gives counterexamples to a conjecture of Daskalopoulos. In the end, I will also discuss a bit about work in progress and future directions.