Quantitative long range curvature estimate for mean curvature flow

TL;DR

This paper establishes a quantitative curvature estimate for smooth convex ancient mean curvature flows, showing that the rescaled curvature grows at most quadratically with respect to the extrinsic distance, providing new bounds on curvature behavior.

Contribution

It introduces a novel quadratic bound on the growth of rescaled curvature in convex ancient mean curvature flows, advancing understanding of their geometric properties.

Findings

Rescaled curvature grows at most quadratically with extrinsic distance.

Provides a quantitative estimate linking curvature and spatial separation.

Enhances understanding of curvature behavior in ancient mean curvature flows.

Abstract

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