Type-0 singularities in the network flow -- Evolution of trees

TL;DR

This paper introduces the concept of Type-0 singularities in network flow, showing that when a single curve vanishes and junctions coalesce, curvature remains bounded, allowing for flow continuation.

Contribution

It identifies and characterizes a new type of singularity in network flow, providing a complete description of tree evolution up to the first singularity and conditions for flow restart.

Findings

Curvature remains bounded at Type-0 singularities.

Complete evolution description for tree-like networks.

Flow can be restarted after singularities under certain conditions.

Abstract

The motion by curvature of networks is the generalization to finite union of curves of the curve shortening flow. This evolution has several peculiar features, mainly due to the presence of junctions where the curves meet. In this paper we show that whenever the length of one single curve vanishes and two triple junctions coalesce, then the curvature of the evolving networks remains bounded. This topological singularity is exclusive of the network flow and it can be referred as a Type-0 singularity, in contrast to the well known Type-I and Type-II ones of the usual mean curvature flow of smooth curves or hypersurfaces, characterized by the different rates of blow up of the curvature. As a consequence, we are able to give a complete description of the evolution of tree-like networks till the first singular time, under the assumption that all the tangents flows have unit multiplicity. If the lifespan of such solutions is finite, then the curvature of the network remains bounded and we can apply the results by Ilmanen-Neves-Schulze/Lira-Mazzeo-Pluda-Saez to restart the flow after the singularity.