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

TL;DR

This paper proves the existence of self-similar expanding solutions to the mean curvature flow for multiphase surfaces with regular junctions, including triple and quadruple points, starting from a specific initial configuration.

Contribution

It establishes the existence of self-expanding solutions for multiphase surfaces with regular junctions under mean curvature flow, including configurations with quadruple points.

Findings

Existence of self-similar expanding solutions for multiphase surfaces.

Multiple solutions include configurations with triple and quadruple junctions.

Quadruple junctions meet at approximately 109.5 degrees.

Abstract

We consider a multiphase surface $\mathcal{C}_0$ in $\mathbb{R}^3$ 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 $\mathcal{C}_0$. 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 $109.5^{\circ}$.