Quantitative uniqueness for mean curvature flow
Tobias Holck Colding, William P. Minicozzi
TL;DR
The paper develops an effective, quantitative framework for blowup uniqueness in mean curvature flow near cylindrical singularities by combining a finite-dimensional model with a discrete Lojasiewicz-type inequality and a rescaled MCF analysis. It proves a discrete effective uniqueness theorem and a continuous Lojasiewicz inequality for rescaled flows, yielding explicit bounds that keep the flow close to a cylinder when the Gaussian area $F$ experiences only small drops. These tools culminate in an effective uniqueness theorem for rescaled MCF and applications showing that ancient flows with bounded entropy have unique cylindrical blow-downs, implying all blow-downs converge to the same cylinder. The results deepen understanding of regularity and singularity structure near cylinders in MCF and provide robust, quantitative control for higher-codimension settings.
Abstract
We show how to use the arguments of [CM2] to get a stronger effective version of uniqueness of blowups that has a number of consequences.