Asymptotic Behavior of β-Polygon Flows

David A Glickenstein, Jinjin Liang

Research output: Contribution to journalArticle

Abstract

In this article we investigate a family of nonlinear evolutions of polygons in the plane called the (Formula presented.)-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.

Original languageEnglish (US)
Pages (from-to)1-24
Number of pages24
JournalJournal of Geometric Analysis
DOIs
StateAccepted/In press - Oct 7 2017

Keywords

  • Curve shortening flow
  • Polygon
  • Polygon flow

ASJC Scopus subject areas

  • Geometry and Topology

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