Asymptotic circularity of immortal area-preserving curvature flows

TL;DR

This paper proves that certain area-preserving curvature flows of closed planar curves become asymptotically circular over time, resolving an open problem and extending results to higher-order flows.

Contribution

It establishes asymptotic circularity of immortal solutions for a broad class of area-preserving curvature flows, including surface diffusion flow, and provides existence results under symmetry conditions.

Findings

Immortal solutions become asymptotically circular

Zero enclosed area solutions blow up in finite time

Extension of results to higher-order surface diffusion flows

Abstract

For a class of area-preserving curvature flows of closed planar curves, we prove that every immortal solution becomes asymptotically circular without any additional assumptions on initial data. As a particular corollary, every solution of zero enclosed area blows up in finite time. This settles an open problem posed by Escher--Ito in 2005 for Gage's area-preserving curve shortening flow, and moreover extends it to the surface diffusion flow of arbitrary order. We also establish a general existence theorem for nontrivial immortal solutions under almost circularity and rotational symmetry.