Self-similarly expanding networks to curve shortening flow

TL;DR

This paper constructs and proves the uniqueness of a self-similarly expanding network of three curves forming 120-degree angles, evolving under the curve shortening flow from a three-half-line initial network.

Contribution

It introduces a new class of self-similarly expanding networks in the plane and establishes their uniqueness under the curve shortening flow.

Findings

Existence of a homothetically expanding network from three half-lines

Uniqueness of the expanding network configuration

Networks form 120-degree angles during expansion

Abstract

We consider a network in the Euclidean plane that consists of three distinct half-lines with common start points. From that network as initial condition, there exists a network that consists of three curves that all start at one point, where they form 120 degree angles, and expands homothetically under curve shortening flow. We also prove uniqueness of these networks.