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

Alexander Mramor; Ao Sun

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arXiv · 2412.03475

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

Alexander Mramor·Ao Sun

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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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Alexander Mramor, Ao Sun. 2024-12-04. On the long-time limit of the mean curvature flow in closed manifolds. https://arxiv.org/abs/2412.03475

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