Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry

Abstract

In this article, we consider Lagrangian mean curvature flow of compact, circle-invariant, almost calibrated Lagrangian surfaces in hyperkähler 4-manifolds with circle symmetry. We show that this Lagrangian mean curvature flow can be continued for all time, through a finite number of finite time singularities, and converges to a chain of special Lagrangians, verifying various aspects of Joyce’s conjectures, Joyce (2015), in this setting. This result provides the first non-trivial setting where Lagrangian mean curvature flow may be used successfully to achieve the desired decomposition of a Lagrangian into a sum of special Lagrangians representing its Hamiltonian isotopy class. We also show that the singularities of the flow are neck pinches in the sense conjectured by Joyce and give examples where such finite time singularities are guaranteed to occur.