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The epsilon-regularity theorem for Brakke flows near triple junctions

Salvatore Stuvard, Yoshihiro Tonegawa

TL;DR

The paper addresses the parabolic analog of Simon’s $\\varepsilon$-regularity for Brakke flows near a static multiplicity-one triple junction, establishing uniqueness of the tangent flow and a $C^{1,\\alpha}$-type regularity in a neighborhood of the junction under a structural condition on 1-dimensional slices. The authors develop a graphical decomposition away from the spine, prove parabolic no-hole and non-concentration estimates, and perform a blow-up analysis to linearize the problem to a heat-equation setting on each branch, yielding decay of the excess and convergence of blow-ups to the stationary triple junction. A key feature is the structural assumption (A6), which is shown to hold automatically in two principal flow models (multiphase BV-Brakke flows and flows of currents mod 3), giving unconditional regularity in these contexts. The results provide a robust parabolic analogue of Simon’s stationary-cone regularity near triple junctions and offer a detailed framework for future extensions to higher codimension and smoother regularity up to the free boundary. The work has significant implications for understanding the local structure of singularities in weak mean curvature flow and for the geometric-measure-theoretic analysis of evolving networks of surfaces.

Abstract

We establish the $\varepsilon$-regularity theorem for $k$-dimensional, possibly forced, Brakke flows near a static, multiplicity-one triple junction. This result provides the parabolic analogue to L. Simon's foundational work on the singular set of stationary varifolds and confirms that the regular structure of triple junctions persists under weak mean curvature flow. The regularity holds provided the flow satisfies a mild structural assumption on its 1-dimensional slices taken orthogonal to the junction's $(k-1)$-dimensional spine, which prohibits certain topological degeneracies. We prove that this assumption is automatically satisfied by two fundamental classes of flows where such singularities are expected: codimension-one multi-phase flows, such as the canonical $\mathrm{BV}$-Brakke flows constructed by the authors, and flows of arbitrary codimension with the structure of a mod 3 integral current, which arise from Ilmanen's elliptic regularization. For such flows, therefore, the Simon type regularity holds unconditionally.

The epsilon-regularity theorem for Brakke flows near triple junctions

TL;DR

The paper addresses the parabolic analog of Simon’s -regularity for Brakke flows near a static multiplicity-one triple junction, establishing uniqueness of the tangent flow and a -type regularity in a neighborhood of the junction under a structural condition on 1-dimensional slices. The authors develop a graphical decomposition away from the spine, prove parabolic no-hole and non-concentration estimates, and perform a blow-up analysis to linearize the problem to a heat-equation setting on each branch, yielding decay of the excess and convergence of blow-ups to the stationary triple junction. A key feature is the structural assumption (A6), which is shown to hold automatically in two principal flow models (multiphase BV-Brakke flows and flows of currents mod 3), giving unconditional regularity in these contexts. The results provide a robust parabolic analogue of Simon’s stationary-cone regularity near triple junctions and offer a detailed framework for future extensions to higher codimension and smoother regularity up to the free boundary. The work has significant implications for understanding the local structure of singularities in weak mean curvature flow and for the geometric-measure-theoretic analysis of evolving networks of surfaces.

Abstract

We establish the -regularity theorem for -dimensional, possibly forced, Brakke flows near a static, multiplicity-one triple junction. This result provides the parabolic analogue to L. Simon's foundational work on the singular set of stationary varifolds and confirms that the regular structure of triple junctions persists under weak mean curvature flow. The regularity holds provided the flow satisfies a mild structural assumption on its 1-dimensional slices taken orthogonal to the junction's -dimensional spine, which prohibits certain topological degeneracies. We prove that this assumption is automatically satisfied by two fundamental classes of flows where such singularities are expected: codimension-one multi-phase flows, such as the canonical -Brakke flows constructed by the authors, and flows of arbitrary codimension with the structure of a mod 3 integral current, which arise from Ilmanen's elliptic regularization. For such flows, therefore, the Simon type regularity holds unconditionally.

Paper Structure

This paper contains 19 sections, 35 theorems, 326 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Plan of the paper and strategy of proof
  3. Notation and main results
  4. General notation
  5. $k$-dimensional triple junctions
  6. Varifolds
  7. Brakke flows with forcing
  8. Flow close to triple junction
  9. A further technical assumption on the flow
  10. The Main Theorem
  11. Graphical parametrization
  12. No-hole property
  13. Non-concentration estimates
  14. Preliminary estimates
  15. Proof of Theorem \ref{['thm:HS']}
  16. ...and 4 more sections

Key Result

Theorem 1.1

Suppose $\{V_t\}_{t \in (-1,0]}$ is a $k$-dimensional Brakke flow in the open ball $U_1(0) \subset \mathbb{R}^n$ satisfying the structural assumption (A6). If a tangent flow at $0$ is a static multiplicity-one triple junction $\mathbf{C}$ for $t\leq 0$, then the following holds: In fact, an $\varepsilon$-regularity statement holds. More precisely, there exists $\varepsilon_0>0$ such that, under (

Figures (2)

  • Figure 1: An illustration of the condition (A6). The depicted slice configuration, despite containing a triple junction point, has a shorter length than that of the triple junction in the red disc and is excluded by the assumption.
  • Figure 2: The figure illustrates the 2D annulus obtained by slicing the space-time toroidal cylinder $T_{\rho, \kappa}(\zeta,s)$ at a fixed spine location $y=\zeta$ and a fixed time $t$. Notice that the width of each toroidal region and its distance from the spine are comparable quantities, as in classical Whitney-type domain decompositions.

Theorems & Definitions (71)

  • Theorem 1.1: Main theorem, informal statement
  • Corollary 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • proof
  • ...and 61 more