Asymptotic behavior of $\beta$-polygon flows
Abstract
In this article we investigate a family of nonlinear evolutions of polygons in the plane called the -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.
Cite
@article{arxiv.1610.03598,
title = {Asymptotic behavior of $\beta$-polygon flows},
author = {David Glickenstein and Jinjin Liang},
journal= {arXiv preprint arXiv:1610.03598},
year = {2016}
}