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Non-uniqueness in Mean Curvature Flow: Non-canonical solutions via the parabolic Allen--Cahn

J. M. Daniels-Holgate

TL;DR

The paper tackles non-uniqueness in mean curvature flow after singularities (fattening) by linking weak flows to the parabolic Allen–Cahn equation. It constructs interior Brakke flows inside the fattening via a foliated neighborhood around a generalized hypersurface and diagonal limits of solutions to the parabolic $\varepsilon$-Allen–Cahn equation, establishing existence of unit-regular integral Brakke flows started from $M_0$ and supported at any $X_0$ in $LSF(M_0)$. A key contribution is the first systematic production of closed, non-trivial, non-canonical Brakke motions inside fattening, including flows arising from fractal (Reifenberg-type) initial data. This provides a new mechanism to study non-outermost evolutions, expands the scope of weak MCF constructions, and offers a framework to analyze multiplicity and singularity formation in interior flows within fattened level-set sets.

Abstract

When mean curvature flow evolves non-uniquely, the flow is said to fatten. The work of Ilmanen shows that any weak MCF is supported inside the fattening, and work of Hershkovits--White identified canonical weak flows supported on the boundary of the fattening, known as the outermost flows. It is natural to ask, when the flow fattens, are there weak mean curvature flows supported strictly inside the fattening? Outside of some special cases (e.g. flow from cones), this question was entirely open. We show these interior flows exist, providing a general construction for non-outermost flows as limits of solutions to the parabolic $\varepsilon$-Allen--Cahn. This gives the first examples of closed, non-trivial, non-canonical, integral Brakke motions. As part of this construction, we study the $\varepsilon$-Allen--Cahn flow from low regularity initial data, and our results demonstrate the existence of integral Brakke motions from fractal sets. This includes the existence portion of Hershkovits's work on mean curvature flow from Reifenberg sets.

Non-uniqueness in Mean Curvature Flow: Non-canonical solutions via the parabolic Allen--Cahn

TL;DR

The paper tackles non-uniqueness in mean curvature flow after singularities (fattening) by linking weak flows to the parabolic Allen–Cahn equation. It constructs interior Brakke flows inside the fattening via a foliated neighborhood around a generalized hypersurface and diagonal limits of solutions to the parabolic -Allen–Cahn equation, establishing existence of unit-regular integral Brakke flows started from and supported at any in . A key contribution is the first systematic production of closed, non-trivial, non-canonical Brakke motions inside fattening, including flows arising from fractal (Reifenberg-type) initial data. This provides a new mechanism to study non-outermost evolutions, expands the scope of weak MCF constructions, and offers a framework to analyze multiplicity and singularity formation in interior flows within fattened level-set sets.

Abstract

When mean curvature flow evolves non-uniquely, the flow is said to fatten. The work of Ilmanen shows that any weak MCF is supported inside the fattening, and work of Hershkovits--White identified canonical weak flows supported on the boundary of the fattening, known as the outermost flows. It is natural to ask, when the flow fattens, are there weak mean curvature flows supported strictly inside the fattening? Outside of some special cases (e.g. flow from cones), this question was entirely open. We show these interior flows exist, providing a general construction for non-outermost flows as limits of solutions to the parabolic -Allen--Cahn. This gives the first examples of closed, non-trivial, non-canonical, integral Brakke motions. As part of this construction, we study the -Allen--Cahn flow from low regularity initial data, and our results demonstrate the existence of integral Brakke motions from fractal sets. This includes the existence portion of Hershkovits's work on mean curvature flow from Reifenberg sets.

Paper Structure

This paper contains 15 sections, 20 theorems, 78 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Results
  4. Acknowledgments
  5. Background and Preliminary results
  6. Weak Mean Curvature Flow
  7. Parabolic Allen--Cahn
  8. Barriers and Energy Estimates
  9. Barriers
  10. Energy Estimates
  11. Sets with Hausdorff dimension $n+\kappa$
  12. The Family of Allen-Cahn solutions
  13. Examples
  14. The $g$-Wagon-Wheel of Ilmanen--White
  15. The Figure 8

Key Result

Theorem 1.1

Let $M_0$ be a compact (generalised) hypersurface. Suppose that $\mathrm{LSF}(M_0)$ fattens in finite time. Then, for each $X_0\in\mathrm{LSF}(M_0)$, there exists a unit-regular, integral Brakke flow $\{\mu_t\}_{t\in[0,\infty)}$ starting from $M_0$, supported at $X_0$.

Figures (4)

  • Figure 1: The 8-wheel of Ilmanen--White, Image reproduced from IW
  • Figure 2: The Figure 8
  • Figure 3: LSF a short time later
  • Figure 4: The Horizontal Two-Triple junction

Theorems & Definitions (57)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.2: Integral Brakke Flow BrakkeIlmanenER
  • Definition 2.3: Gaussian Density
  • Definition 2.4: Unit-regular White09
  • Definition 2.5: Weak and Level set flow, IlmanenERhershwhite
  • Definition 2.6
  • Definition 2.7: Outermost flows, hershwhite
  • ...and 47 more