Self-Expanding Solutions to the Mean Curvature Flow for Multiphase Surfaces with Regular Junctions

2025/12/02 by Wei-Hung Liao, Liao, Wei-Hung
Mathematics · #49Q20 #53C44 #53E10 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2512.02578

openalex publication_date 2025/12/02 · openalex created_date 2025/12/04 · openalex updated_date 2026/07/28

Abstract

We consider a multiphase surface C₀ in ℝ³ consisting of a finite number of surfaces passing through the origin , where all 1-dimensional junctions are regular triple junctions in which three planes meet at the same angle and each surface scales down homothetically to a limit curve of finite length. We prove the existence of self-similar expanding solutions of the mean curvature flow on the multiphase surface initially given by C₀. For this initial condition, there are multiple solutions that are combinations of the regular triple junctions and regular quadruple points, where four regular triple junctions meet at an angle of approximately 109.5^∘.

Citations

• The space of asymptotically conical self-expanders of mean curvature flow
• On short time existence for the planar network flow
• Generic mean curvature flow I; generic singularities
• Entire solutions to equivariant elliptic systems with variational structure
• Mean Curvature Motion of Triple Junctions of Graphs in Two Dimensions
• Self similar expanding solutions of the planar network flow
• Self-similarly expanding networks to curve shortening flow

Related

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• Finite topology self-translating surfaces for the mean curvature flow in \mathbb R³