### 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 language | English (US) |
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Pages (from-to) | 1-24 |

Number of pages | 24 |

Journal | Journal of Geometric Analysis |

DOIs | |

State | Accepted/In press - Oct 7 2017 |

### Keywords

- Curve shortening flow
- Polygon
- Polygon flow

### ASJC Scopus subject areas

- Geometry and Topology

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## Cite this

*Journal of Geometric Analysis*, 1-24. https://doi.org/10.1007/s12220-017-9940-y