Lagrangian mean curvature flow in the complex projective plane

TL;DR

This paper proves a conjecture about the long-term behavior of certain Lagrangian tori in the complex projective plane under mean curvature flow with surgery, showing they evolve to minimal Clifford tori and can be constructed with any finite number of surgeries.

Contribution

It establishes a Thomas--Yau-type conjecture for monotone symmetric Lagrangian tori in P^2 and develops new methods applicable to non-Calabi--Yau manifolds.

Findings

Lagrangian tori flow to minimal Clifford tori over infinite time.

Existence of tori with any finite number of surgeries before convergence.

Development of techniques for studying Lagrangian mean curvature flow in non-Calabi--Yau settings.

Abstract

We prove a Thomas--Yau-type conjecture for monotone Lagrangian tori satisfying a symmetry condition in the complex projective plane $\mathbb{CP}^2$. We show that such tori exist for all time under Lagrangian mean curvature flow with surgery, undergoing at most a finite number of surgeries before flowing to a minimal Clifford torus in infinite time. Furthermore, we show that we can construct a torus with any finite number of surgeries before convergence. Along the way, we prove many interesting subsidiary results and develop methods which should be useful in studying Lagrangian mean curvature flow in non-Calabi--Yau manifolds, even in non-symmetric cases.