Self-Expanding Solutions to the Mean Curvature Flow for Multiphase Surfaces with Regular Junctions
Wei-Hung Liao
TL;DR
The paper develops a variational framework to construct self-similar expanding solutions to the multiphase mean curvature flow in $\mathbb{R}^3$ starting from a cone $\mathcal{C}_0$ formed by planes through the origin. By pairing a conformal, weighted-area functional with a flat-chain minimization strategy and a hyperbolic barrier mechanism, it proves the existence of global, area-minimizing multiphase networks asymptotic to $\mathcal{C}_0$ and preserving regular junctions (triple and quadruple points). The main result shows at least one (often finitely many) connected self-expander networks correspond to prescribed asymptotic data, and a parabolic blow-up yields a one-parameter family of such networks. This work extends planar, two-dimensional junction analysis to three dimensions, providing a robust variational and geometric-measure-theory framework for understanding junction persistence and asymptotics in multiphase interface evolution.
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 $109.5^{\circ}$.