https://neckpinch.com/authors/laiyuan-gao/New arXiv papers by Laiyuan Gao, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2412.18102/arxiv:2412.18102Tue, 24 Dec 2024 00:00:00 GMTLaiyuan Gao, Shicheng Zhang, Yuntao Zhang — Mayer asks a question what closed, embedded and nonconvex initial curves guarantee that Gage's area-preserving flow (GAPF) exists globally. A folklore conjecture since 2012 says that GAPF evolves smooth, embedded and star-shaped initial curves globally. In this paper, we prove this conjecture by using Dittberner's singularity analysis theory. A star-shaped "flying wing" curve is constructed to show that GAPF may not always preserve the star-shapedness of evolving curves. This example is also a negative answer to Mantegazza's open problem whether the curve shortening flow (CSF) always preserves the star shape of the evolving curves.Curve shorteningVolume-preserving & constrained flowsSingularities https://neckpinch.com/papers/2404.05267/arxiv:2404.05267Mon, 08 Apr 2024 00:00:00 GMTLaiyuan Gao, Horst Martini, Deyan Zhang — A nonlocal curvature flow is introduced to evolve locally convex curves in the plane. It is proved that this flow with any initial locally convex curve has a global solution, keeping the local convexity and the elastic energy of the evolving curve, and that, as the time goes to infinity, the curve converges to a smooth, locally convex curve of constant k-order width. In particular, the limiting curve is a multiple circle if and only if the initial locally convex curve is k-symmetric.Other curvature flowsAnisotropic & nonlocal flowsLong-time existence & convergence