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

TL;DR

This work develops a parabolic framework for the long-time behavior of almost regular mean curvature flows in closed 3-manifolds, proving that flows either vanish in finite time or converge to a finite union of smoothly embedded minimal surfaces with multiplicities. By introducing piecewise almost regular flows and leveraging Ilmanen's regularity theory, the authors obtain smooth limit surfaces and control over their topology, while avoiding unstable minimal surfaces via finite isotopies. The results yield new, parabolic proofs of the existence of stable minimal surfaces in a range of 3-manifolds (including $S^3$) and connect flow techniques with classical minmax methods. The approach provides both qualitative stability results and constructive perturbation mechanisms, with broad implications for minimal-surface existence and topology in 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.

Theorems & Definitions (45)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture 1.5: Conjecture 1.3 in ChenSun24_Multi2inManifold
• Corollary 1.6
• Corollary 1.7
• Corollary 1.8
• Corollary 1.9
• Corollary 1.10