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

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 that 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.