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

Yiqi Huang, Xinrui Zhao

TL;DR

The paper analyzes unique continuation at cylindrical singularities in mean curvature flow (MCF). It proves a global-graph rigidity result: if a rescaled MCF over a cylinder remains a small global graph and converges to the cylinder at super-exponential rates, then the graph must be identically zero, with the decay rate bound shown to be sharp by local counterexamples. The approach combines spectral decomposition of the cylinder's linearized operator with a Merle-Zaag-type ODE argument to control nonlinear terms, establishing global continuation results and clarifying the necessity of global graphical data. To demonstrate sharpness, the authors construct Tikhonov-type counterexamples for nonlinear backward parabolic equations, including rescaled MCF, giving infinite-dimensional families of local-graph solutions that decay super-exponentially on domains expanding arbitrarily fast. These results reveal new phenomena at cylindrical singularities absent in the compact case and provide broad constructions that illuminate limitations of backward-uniqueness in nonlinear parabolic problems.

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.

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

TL;DR

The paper analyzes unique continuation at cylindrical singularities in mean curvature flow (MCF). It proves a global-graph rigidity result: if a rescaled MCF over a cylinder remains a small global graph and converges to the cylinder at super-exponential rates, then the graph must be identically zero, with the decay rate bound shown to be sharp by local counterexamples. The approach combines spectral decomposition of the cylinder's linearized operator with a Merle-Zaag-type ODE argument to control nonlinear terms, establishing global continuation results and clarifying the necessity of global graphical data. To demonstrate sharpness, the authors construct Tikhonov-type counterexamples for nonlinear backward parabolic equations, including rescaled MCF, giving infinite-dimensional families of local-graph solutions that decay super-exponentially on domains expanding arbitrarily fast. These results reveal new phenomena at cylindrical singularities absent in the compact case and provide broad constructions that illuminate limitations of backward-uniqueness in nonlinear parabolic problems.

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.

Paper Structure

This paper contains 20 sections, 29 theorems, 257 equations.

Table of Contents

  1. Introduction
  2. Unique continuation for global graphs
  3. Failure of unique continuation for local graphs
  4. Rescaled mean curvature flow over cylinders
  5. Unique continuation of global graphs
  6. Global super-exponential convergence
  7. Local super-exponential convergence
  8. Counter-examples
  9. Tikhonov's counter-examples
  10. Quadratic nonlinear term
  11. Intermediate case
  12. General construction
  13. Special region
  14. Most general region
  15. Counter-examples for rescaled mean curvature flow
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let $\tilde{\varphi}:M\times[0,\infty)\to \mathbb R^{n+1}$ be a solution to the rescaled mean curvature flow equation e:recaled mcf such that $M_t \equiv \tilde{\varphi}(M,t)$ can be expressed as a normal graph of a function $u(x,t)$ over the round cylinder $\mathcal{C}_k=\mathbb{S}^{n-k}_{\sqrt{2(n where $d\mu$ is the standard measure on $\mathcal{C}_k$, then we have $u \equiv 0$.

Theorems & Definitions (56)

  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.6
  • Remark 1.11
  • Corollary 1.12
  • Corollary 1.14
  • Remark 1.15
  • Lemma 2.2
  • proof
  • ...and 46 more