# Singularities of the area preserving curve shortening flow with a Neumann free boundary condition

## TL;DR

This paper studies the behavior of area-preserving curve shortening flows with Neumann boundary conditions, identifying conditions for finite-time singularities and characterizing their type and limit flows.

## Contribution

It provides a criterion for finite-time singularity formation and classifies the singularity as type II, also describing the asymptotic limit flows for convex initial curves.

## Key findings

- Finite-time singularities occur under certain initial conditions.
- Singularities are classified as type II.
- Convex initial curves lead to grim reaper or half grim reaper limit flows.

## Abstract

We consider the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain or at a straight line. We give a criterion on initial curves that guarantees the appearance of a singularity in finite time. We prove that the singularity is of type II. Furthermore, if these initial curves are convex, then an appropriate rescaling at the finite maximal time of existence yields a grim reaper or half a grim reaper as limit flow. We construct examples of initial curves satisfying the mentioned criterion.