On the rate of convergence of cylindrical singularity in mean curvature flow

TL;DR

This paper establishes a unique continuation result for global graphical mean curvature flows near cylindrical singularities and demonstrates the failure of such results for local graphs through counter-examples, highlighting the importance of global assumptions.

Contribution

It proves the first unique continuation result for cylindrical singularities in mean curvature flow and constructs counter-examples showing the necessity of global graphical conditions.

Findings

Unique continuation holds for global graphs over cylinders with small gradient.

Counter-examples show failure of unique continuation for local graphs at super-exponential convergence.

Construction of non-product mean curvature flows with prescribed singular sets at super-exponential rates.

Abstract

We prove that if a rescaled mean curvature flow is a global graph over the round cylinder with small gradient and converges super-exponentially fast, then it must coincide with the cylinder itself. We also show that the result is sharp with counter-examples of local graphs at arbitrarily super-exponential convergence rate with the domain expanding arbitrarily fast. The first part provides the first unique continuation result in the cylindrical setting, the generic singularity model in mean curvature flow. In sharp contrast, in the second part we construct infinite-dimensional families of Tikhonov-type examples for nonlinear equations, including the rescaled mean curvature flow, showing that unique continuation fails for local graphical solutions. These examples demonstrate the essential role of global graphical assumptions in rigidity and highlight new phenomena absent in the compact case. We also construct non-product mean curvature flows that develop singular sets as prescribed lower dimensional Euclidean space at arbitrary super-exponential rates. Our construction works in great generality for a large class of non-linear equations.