Thomas–Yau conjecture for flow-stable Lagrangians

Let $(M^{2n},\omega,J,\Omega)$ be a Calabi–Yau manifold, and let $L\subseteq M$ be a zero-Maslov-class, almost-calibrated Lagrangian. Suppose that $L$ is flow-stable, meaning that, after normalizing $\phi(L)=0$, whenever $L$ is Hamiltonian isotopic to a connected sum $L_1\#L_2$ of almost-calibrated Lagrangians, both the phase and area conditions in the definition of flow-stability hold. Thomas–Yau conjecture. The Lagrangian mean curvature flow of $L$ exists for all time and converges to the unique special Lagrangian in its Hamiltonian isotopy class. The conjecture proposes that flow-stability prevents finite-time singularities by ruling out decomposition into simpler Lagrangians; the stability condition is difficult to verify, and the existence of non-special-Lagrangian flow-stable examples is largely unclear.