On the long-time limit of the mean curvature flow in closed manifolds

2024/12/04 by Alexander Mramor, Mramor, Alexander, Ao Sun +1 · 1 citation
Mathematics · #53A10 #53E10 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2412.03475

openalex publication_date 2024/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article we show that generally almost regular flows, introduced by Bamler and Kleiner, in closed 3-manifolds will either go extinct in finite time or flow to a collection of smooth embedded minimal surfaces, possibly with multiplicity. Using a perturbative argument then we construct piecewise almost regular flows which either go extinct in finite time or flow to a stable minimal surface, possibly with multiplicity. We apply these results to construct minimal surfaces in 3-manifolds in a variety of circumstances, mainly novel from the point of the view that the arguments are via parabolic methods.

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• When can minimal hypersurfaces be connected by mean curvature flow?

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