Thomas–Yau stability conjecture for Lagrangian mean curvature flow

Establish a stability condition for a Hamiltonian isotopy class of Lagrangians in a Calabi–Yau manifold that guarantees long-time existence of the Lagrangian mean curvature flow and convergence to a special Lagrangian representative.

Background

The paper frames the search for special Lagrangian representatives of Hamiltonian isotopy classes as a central problem in Calabi–Yau geometry and mirror symmetry. Since Lagrangian mean curvature flow preserves the Lagrangian condition in Kähler–Einstein manifolds, it is a natural proposed mechanism for producing such representatives.

The conjecture remains a broad unresolved problem because Lagrangian mean curvature flow can develop singularities even in Calabi–Yau surfaces. The paper discusses later developments relating possible singularities to stability conditions, but its own results address only a specific perturbative class of flows in the Kummer K3 surface rather than the general conjecture.