Motion by curvature of planar networks II

TL;DR

This paper studies the evolution of a planar network of three curves connected at a triple junction under curvature flow, proving long-term existence and convergence to a minimal Steiner connection under certain conditions.

Contribution

It establishes conditions for smooth long-term evolution and convergence of a triple junction network under curvature flow, extending understanding of geometric network evolution.

Findings

Existence of smooth curvature flow until curve lengths approach zero.

Long-term existence and convergence to Steiner minimal connection.

Conditions ensuring the network evolves smoothly and stabilizes.

Abstract

We prove that the curvature flow of an embedded planar network of three curves connected through a triple junction, with fixed endpoints on the boundary of a given strictly convex domain, exists smooth until the lengths of the three curves stay far from zero. If this is the case for all times, then the evolution exists for all times and the network converges to the Steiner minimal connection between the three endpoints.