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In this article we consider the Lagrangian mean curvature flow of compact,circle-invariant, almost calibrated Lagrangian surfaces in hyperk\"ahler4-manifolds with circle symmetry. We show that this Lagrangian mean curvatureflow can be continued for all time, through finite time singularities, andconverges to a chain of special Lagrangians, thus verifying various aspects ofJoyce's conjectures in this setting. We show that the singularities of the floware neck pinches in the sense conjectured by Joyce. We also give examples wheresuch finite time singularities are guaranteed to occur.

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