Geometry Seminar: A. Vogiatzi (Queen Mary Univ London)

Geometry Seminar (Geometric Analysis)

Speaker: Artemis Vogiatzi (Queen Mary Univ London)

Title: High Codimension Mean Curvature Flow in ℂPⁿ.

Abstract: Mean curvature flow is a geometric evolution equation that describes how a submanifold embedded in a higher-dimensional space changes its shape over time. We establish a codimension estimate that enables us to prove at a singular time of the flow, there exists a rescaling that converges to a smooth codimension one limiting flow in Euclidean space, regardless of the original flow's codimension. Under a cylindrical type pinching, we show that this limiting flow is weakly convex and either moves by translation or is a self-shrinker. These estimates allow us to analyse the behaviour of the flow near singularities and establish the existence of the limiting flow. Considering the ℂPⁿ, we go beyond the finite timeframe of the mean curvature flow, by proving that the rescaling converges smoothly to a totally geodesic limit in infinite time. Our approach relies on the preservation of the quadratic pinching condition along the flow and a gradient estimate that controls the mean curvature in regions of high curvature.