Anisotropic curvature flow of immersed networks

TL;DR

This paper studies the evolution of a network of three curves in the plane under anisotropic curvature flow, establishing existence, uniqueness, regularity, and criteria for finite-time singularities.

Contribution

It proves the existence, uniqueness, and regularity of solutions for anisotropic curvature flow of immersed networks with triple junctions, and characterizes finite-time singularities.

Findings

Existence and uniqueness of maximal solutions

Regularity of solutions up to singularity

Finite-time singularity characterized by curve length or curvature blow-up

Abstract

We consider motion by anisotropic curvature of a network of three curves immersed in the plane meeting at a triple junction and with the other ends fixed. We show existence, uniqueness and regularity of a maximal geometric solution and we prove that, if the maximal time is finite, then either the length of one of the curves goes to zero or the $L^2$ norm of the anisotropic curvature blows up.