## Abstract

In this article we investigate a family of nonlinear evolutions of polygons in the plane called the β-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 language | English (US) |
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Pages (from-to) | 2902-2925 |

Number of pages | 24 |

Journal | Journal of Geometric Analysis |

Volume | 28 |

Issue number | 3 |

DOIs | |

State | Published - Jul 1 2018 |

## Keywords

- Curve shortening flow
- Polygon
- Polygon flow

## ASJC Scopus subject areas

- Geometry and Topology