Artemis Vogiatzi (Copenhagen) will speak in the geometry seminar on Thursday 22 May at 2pm in the Salle des Profs (9th floor of building NO). Please note the unusual day! Artemis’s title is “Quartically pinched submanifolds for the mean curvature flow in the sphere” and her abstract is below.
We introduce a new sharp quartic curvature pinching for submanifolds in $\mathbb{S}^{n+m}$, $m\ge2$, which is preserved by the mean curvature flow. Using a blow up argument, we prove a codimension and a cylindrical estimate, where in regions of high curvature, the submanifold becomes approximately codimension one, quantitatively, and is weakly convex and moves by translation or is a self shrinker. With a decay estimate, the rescaling converges smoothly to a totally geodesic limit in infinite time, without using Stampacchia iteration or integral analysis.