Sharp quartic pinching for the mean curvature flow in the sphere

We introduce a new sharp quartic curvature pinching for submanifolds in S n + m \mathbb {S}^{n+m} , m ≥ 2 m\ge 2 , 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.