Asymptotic behavior of $\beta$-polygon flows

TL;DR

This paper studies the asymptotic behavior of the $eta$-polygon flow, showing polygons shrink to points, regular polygons are stable, and self-similar solutions exist under certain conditions, extending concepts from curve shortening flow.

Contribution

It introduces new stability and convergence results for the $eta$-polygon flow, including stability of regular polygons and existence of self-similar solutions, extending geometric flow analysis.

Findings

Planar polygons shrink to a point under the flow.

Regular polygons with five or more vertices are asymptotically stable.

Existence of self-similar solutions under angle bounds.

Abstract

In this article we investigate a family of nonlinear evolutions of polygons in the plane called the $\beta$-polygon flow and obtain some results analogous to results for the smooth curve shortening flow: (1) any planar polygon shrinks to a point and (2) a regular polygon with five or more vertices is asymptotically stable in the sense that nearby polygons shrink to points that rescale to a regular polygon. In dimension four we show that the shape of a square is locally stable under perturbations along a hypersurface of all possible perturbations. Furthermore, we are able to show that under a lower bound on angles there exists a rescaled sequence extracted from the evolution that converges to a limiting polygon that is a self-similar solution of the flow. The last result uses a monotonicity formula analogous to Huisken's for the curve shortening flow.