A fourth-order area-preserving curve flow in centro-equiaffine geometry

TL;DR

This paper introduces a new fourth-order centro-equiaffine invariant curve flow that preserves area and converges to a circle, extending previous work without smallness restrictions on initial curves.

Contribution

It establishes long-time existence and convergence of the flow to a circle, generalizing prior results in centro-equiaffine geometry without initial curve constraints.

Findings

Flow preserves enclosed area over time.

Evolving curves converge smoothly to a circle.

Long-time existence of the flow is proven.

Abstract

In this paper, inspired by the work of Guan and Li (2015), we introduce a fourth-order centro-equiaffine invariant curve flow via the affine Minkowski formula. Without any smallness assumptions on the initial curve, we establish the long-time existence of the flow and prove that, as $t \to +\infty$, the evolving curve preserves its enclosed area and converges smoothly to a round circle up to the action of $\mathrm{SL}(2)$.