Evolution of convex lens-shaped networks under the curve shortening flow

Abstract

We consider convex symmetric lens-shaped networks in R 2 \mathbb {R}^2 that evolve under the 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.