Curvature Evolution of Nonconvex Lens-Shaped Domains

TL;DR

This paper investigates the curvature flow of nonconvex lens-shaped domains, demonstrating that such domains become convex in finite time and then shrink to a point, extending classical results to a new class of shapes.

Contribution

It establishes the convexification and shrinking behavior of nonconvex lens-shaped domains under curvature flow, generalizing Grayson's theorem to these special symmetric networks.

Findings

Domains become convex in finite time

Domains shrink homothetically to a point

Extension of Grayson's theorem to lens-shaped domains

Abstract

We study the curvature flow of planar nonconvex lens-shaped domains, considered as special symmetric networks with two triple junctions. We show that the evolving domain becomes convex in finite time; then it shrinks homothetically to a point. Our theorem is the analog of the result of Grayson for curvature flow of closed planar embedded curves.