Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry - Lotay, JD; Oliveira, G (EMS Press, 2024)
In this article we consider the Lagrangian mean curvature flow of compact, circleinvariant, 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 [Joy15] 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 [Joy15] and give examples where such finite time singularities are guaranteed to occur.