Calculus of Variations and Geometric Measure Theory

C. Mantegazza - M. Novaga - A. Pluda

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

created by pluda on 08 Apr 2021
modified by root on 02 Feb 2023


Published Paper

Inserted: 8 apr 2021
Last Updated: 2 feb 2023

Journal: J. Reine Angew. Math. (Crelle's Journal)
Volume: 792
Pages: 189-221
Year: 2022


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 and Schulze or Lira, Mazzeo, Pluda and Saez to "restart" the flow after the singularity.