https://neckpinch.com/authors/jinjin-liang/New arXiv papers by Jinjin Liang, from Neckpinch Tue, 10 Nov 2026 04:10:28 GMT https://neckpinch.com/papers/1610.03598/arxiv:1610.03598Wed, 12 Oct 2016 00:00:00 GMTDavid Glickenstein, Jinjin Liang — 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.Curve shortening4-manifolds