The area preserving curve shortening flow with Neumann free boundary conditions

TL;DR

This paper investigates a geometric flow that preserves area while shortening curves with free boundary conditions, showing smooth convergence to a circle outside a convex domain without singularities under certain initial conditions.

Contribution

It introduces and analyzes the area-preserving curve shortening flow with Neumann free boundary conditions, proving smooth subconvergence to a circle outside a convex domain.

Findings

Flow avoids singularities under specific initial conditions.

Flow converges smoothly to an arc of a circle outside the domain.

Encloses the same area as the initial curve.

Abstract

We study the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain in the Euclidean plane. Under certain conditions on the initial curve the flow does not develop any singularity, and it subconverges smoothly to an arc of a circle sitting outside of the given fixed domain and enclosing the same area as the initial curve.