### 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.

Language | English (US) |
---|---|

Pages | 1-24 |

Number of pages | 24 |

Journal | Journal of Geometric Analysis |

DOIs | |

State | Accepted/In press - Oct 7 2017 |

### Fingerprint

### Keywords

- Curve shortening flow
- Polygon
- Polygon flow

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

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

**Asymptotic Behavior of β-Polygon Flows.** / Glickenstein, David A; Liang, Jinjin.

Research output: Contribution to journal › Article

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

}

TY - JOUR

T1 - Asymptotic Behavior of β-Polygon Flows

AU - Glickenstein, David A

AU - Liang, Jinjin

PY - 2017/10/7

Y1 - 2017/10/7

N2 - 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.

AB - 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.

KW - Curve shortening flow

KW - Polygon

KW - Polygon flow

UR - http://www.scopus.com/inward/record.url?scp=85030663558&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85030663558&partnerID=8YFLogxK

U2 - 10.1007/s12220-017-9940-y

DO - 10.1007/s12220-017-9940-y

M3 - Article

SP - 1

EP - 24

JO - Journal of Geometric Analysis

T2 - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

SN - 1050-6926

ER -