Sharp Quartic Pinching for the Mean Curvature Flow in the Sphere

We prove a sharp quartic curvature pinching for the mean curvature flow in ${\mathbb{S}}^{n+m}$, $m\ge 2$, which generalises Pu’s work on the convergence of submanifolds in ${\mathbb{S}}^{n+m}$ to a round point. 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.