On the stability of $m$-fold circles and the dynamics of generalized curve shortening flows

TL;DR

This paper investigates the stability and long-term behavior of m-fold circles under generalized p-curve shortening flows, providing insights into their asymptotic and exponential stability properties.

Contribution

It introduces a comprehensive analysis of the stability of m-fold circles within the framework of p-curve shortening flows, extending previous results to a broader class of flows.

Findings

m-fold circles exhibit asymptotic stability under p-curve shortening flows

Exponential stability of these circles is established for certain conditions

The study broadens understanding of curve evolution dynamics in geometric flows

Abstract

In this paper we study the (asymptotic and exponential) stability of the $m$-fold circle as a solution of the $p$-curve shortening flow ($p\geq 1$ an integer).