Star-shaped Centrosymmetric Curves under Gage's Area-preserving Flow

TL;DR

This paper proves that Gage's area-preserving flow transforms star-shaped, centrosymmetric curves into convex shapes and ultimately into circles over time.

Contribution

It establishes the smooth evolution and convergence of star-shaped, centrosymmetric curves under Gage's flow, including convexification and circularization.

Findings

Curves become convex in finite time.

Curves deform into circles as time approaches infinity.

Flow preserves smoothness throughout evolution.

Abstract

It is proved that Gage's area-preserving flow can evolve a centrosymmetric star-shaped initial curve smoothly, make it convex in a finite time and deform it into a circle as time tends to infinity.