Quartically pinched submanifolds for the mean curvature flow in the sphere.

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By using a sharp quartic curvature pinching for the mean curvature flow in $\mathbb{S}^{n+m}$, $m\ge2$, we improve the quadratic curvature conditions. 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 any iteration procedures or integral analysis. Our approach relies on the preservation of the quartic pinching condition along the flow and gradient estimates that control the mean curvature in regions of high curvature.