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arXiv · 1610.03598

Asymptotic behavior of $β$-polygon flows

David Glickenstein·Jinjin Liang
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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.

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David Glickenstein, Jinjin Liang. 2016-10-12. Asymptotic behavior of $β$-polygon flows. https://arxiv.org/abs/1610.03598

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