On the dynamics of formation of generic singularities of mean curvature flow

TL;DR

This paper investigates the formation of generic singularities in mean curvature flow by integrating various analytical approaches, providing detailed estimates and decay rates near blowup points.

Contribution

It introduces a unified method combining multiple techniques to analyze singularity formation in mean curvature flow, with precise estimates and parameter analysis.

Findings

Key parameters exhibit favorable signs near singularities

Sharp decay rates are established in certain regimes

Remainder estimates are provided in various norms

Abstract

We study the formation of generic singularities of mean curvature flow by combining the different approaches, specifically the methods in studying blowup of nonlinear heat equations, the techniques used by the author and the collaborators for mean curvature flow, and these invented by Colding and Minicozzi. We study the solution in a neighborhood of the blowup point, in some generic regimes, we find the key parameters take favorable signs and have sharp decay rates. We provide the remainder estimates in different norms.