A nonlocal curve flow in centro-affine geometry

TL;DR

This paper studies a specific curve flow in centro-affine geometry, establishing an isoperimetric inequality and analyzing the long-term behavior of convex curves, showing convergence to ellipses under the flow.

Contribution

It introduces a new invariant curve flow in centro-affine geometry and analyzes its long-term behavior, including convergence properties and associated inequalities.

Findings

Established an isoperimetric inequality in centro-affine plane geometry.

Proved that convex curves evolve towards ellipses under the flow.

Analyzed the flow's behavior via a nonlinear parabolic equation.

Abstract

In this paper, the isoperimetric inequality in centro-affine plane geometry is obtained. We also investigate the long-term behavior of an invariant plane curve flow, whose evolution process can be expressed as a second-order nonlinear parabolic equation with respect to centro-affine curvature. The forward and backward limits in time are discussed, which shows that a closed convex embedded curve may converge to an ellipse when evolving according to this flow.