https://neckpinch.com/authors/longzhi-lin/New arXiv papers by Longzhi Lin, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2607.20350arxiv:2607.20350Wed, 22 Jul 2026 00:00:00 GMTYuk Shing Lam, Longzhi Lin — In this paper, we show that the modified mean curvature flow starting from an arbitrary graph in a Fuchsian manifold exists for all time and converges smoothly to an equidistant surface of constant mean curvature as t→ ∞. This result generalizes earlier work to the modified mean curvature flow setting and removes the restrictive global gradient bound initially required for the standard mean curvature flow by Huang, Zhou, and the second author.Mean curvature flowLong-time existence & convergence https://arxiv.org/abs/2311.04298arxiv:2311.04298Tue, 07 Nov 2023 00:00:00 GMTZheng Huang, Longzhi Lin, Zhou Zhang — There has been a conjecture, often attributed to Thurston, which asserts that every almost Fuchsian manifold is foliated by closed incompressible constant mean curvature (CMC) surfaces. In this paper, for a certain class of almost Fuchsian manifolds, we prove the long-time existence and convergence of the modified mean curvature flow ∂F/∂t=−(H-c)vecν, which was first introduced by Xiao and the second named author in. As an application, we confirm Thurston's CMC foliation conjecture for such a subclass of almost Fuchsian manifolds.Mean curvature flowGeometrizationLong-time existence & convergence https://arxiv.org/abs/2107.13127arxiv:2107.13127Wed, 28 Jul 2021 00:00:00 GMTZheng Huang, Longzhi Lin, Zhou Zhang — We consider the dynamic property of the volume preserving mean curvature flow. This flow was introduced by Huisken who also proved it converges to a round sphere of the same enclosed volume if the initial hypersurface is strictly convex in Euclidean space. We study the stability of this flow in hyperbolic space. In particular, we prove that if the initial hypersurface is hyperbolically mean convex and close to an umbilical sphere in the L²-sense, then the flow exists for all time and converges exponentially to an umbilical sphere.Mean curvature flowVolume-preserving & constrained flowsStability & uniquenessLong-time existence & convergence