The geometry-topology seminar continues on September 13, with a talk by Andrew Cooper of the University of Pennsylvania.
Cooper will speak on
Title: Singular time of the Ricci and mean curvature flows
Abstract: The mean curvature flow (MCF) and Ricci flow (RF) are quasilinear parabolic equations; hence solutions are expected to develop singularities in finite time. It is straightforward that in each case, the relevant full curvature tensor (for MCF, the second fundamental form; for RF the Riemann tensor) must blow up at such a singularity.
This talk will address whether it is possible characterise the singular time of these flows by a weaker criterion. I will present an argument of Sesum to show that the Ricci tensor must blow up at a finite-time singularity of the RF, and adapt it to show that in MCF the second fundamental form must blow up, roughly speaking, in the direction of the mean curvature vector. Time permitting, I will give two independent proofs that under a mildness assumption for the singularity, the mean curvature itself must blow up.