Evolution of convex lens-shaped networks under curve shortening flow

TL;DR

This paper studies the evolution of convex lens-shaped networks under curve shortening flow, proving finite-time shrinking, convergence to a unique self-similar shrinking network, and classifying certain self-similar solutions.

Contribution

It establishes the finite-time extinction, uniqueness of the self-similar limit, and classifies some self-similarly shrinking networks in the context of convex lens-shaped networks.

Findings

Convex lens-shaped networks shrink to a point in finite time.

Rescaled networks converge to a unique self-similar shrinking network.

Classification results for certain self-similarly shrinking networks.

Abstract

We consider convex symmetric lens-shaped networks in R^2 that evolve under curve shortening flow. We show that the enclosed convex domain shrinks to a point in finite time. Furthermore, after appropriate rescaling the evolving networks converge to a self-similarly shrinking network, which we prove to be unique in an appropriate class. We also include a classification result for some self-similarly shrinking networks.