On the uniqueness of nondegenerate blowups for the motion by curvature of networks

TL;DR

This paper proves the uniqueness of nondegenerate compact blowups in the motion by curvature of planar networks, using Lojasiewicz-Simon inequalities to analyze stability and convergence.

Contribution

It introduces a novel application of Lojasiewicz-Simon inequalities to establish the uniqueness of nondegenerate blowups in network curvature flow.

Findings

Uniqueness of nondegenerate compact blowups established

Application of Lojasiewicz-Simon inequalities to network flow

Enhanced understanding of stability in curvature-driven network evolution

Abstract

In this note we prove uniqueness of nondegenerate compact blowups for the motion by curvature of planar networks. The proof follows ideas introduced in "Lojasiewicz-Simon inequalities for minimal networks: stability and convergence" for the study of stability properties of critical points of the length functional and it is based on the application of a Lojasiewicz-Simon gradient inequality.