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

TL;DR

This paper studies the long-term behavior of mean curvature flow in closed 3-manifolds, showing flows either end in finite time or converge to minimal surfaces, using novel parabolic methods.

Contribution

It introduces a perturbative approach to construct flows that either go extinct or converge to stable minimal surfaces, expanding the understanding of mean curvature flow in 3-manifolds.

Findings

Flows either go extinct or converge to minimal surfaces

Constructed flows that reach stable minimal surfaces using perturbative methods

Applied results to find minimal surfaces in various 3-manifolds

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.