Quantitative long range curvature estimate for mean curvature flow

Abstract

We prove that smooth convex $\alpha$-noncollapsed ancient mean curvature flow satisfies a quantitative curvature estimate $H(y,t)\le CH(x,t)(H(x,t)|x-y|+1{)}^2$ for any pair of $x,y$. In other words, the rescaled curvature grows at most quadratically in terms of the rescaled extrinsic distance.