On the rate of convergence of the rescaled mean curvature flow

TL;DR

This paper investigates how quickly solutions to the rescaled mean curvature flow approach self-similar shrinkers, establishing that super-exponential convergence implies the solution is a shrinker itself.

Contribution

It provides an upper bound estimate on the convergence rate to shrinkers and characterizes solutions with super-exponential convergence as shrinkers.

Findings

Solutions converging faster than any exponential rate are shrinkers.

The paper establishes an upper bound on convergence speed to self-similar solutions.

Super-exponential convergence implies the solution is a shrinker.

Abstract

We estimate from above the rate at which a solution to the rescaled mean curvature flow on a closed hypersurface may converge to a limit self-similar solution, i.e. a shrinker. Our main result implies that any solution which converges to a shrinker faster than any fixed exponential rate must itself be shrinker itself.