Thomas–Yau conjecture for special Lagrangian representatives

Let $(X,\omega,\Omega)$ be a Calabi–Yau manifold, and let $L$ be a compact zero Maslov Lagrangian in $(X,\omega,\Omega)$. A compact zero Maslov Lagrangian is stable if it is not Hamiltonian isotopic to a graded Lagrangian connect sum $L_1\#L_2$ of compact graded Lagrangians $L_1$ and $L_2$ whose variations of Lagrangian angle are less than $2\pi$ and satisfy

$$\operatorname{Arg}\int_{L_1}\Omega\geq \operatorname{Arg}\int_{L_2}\Omega.$$

Thomas–Yau conjecture. There exists a unique special Lagrangian in the Hamiltonian isotopy class of $L$ if and only if $L$ is stable. This conjecture proposes a stability criterion for the existence and uniqueness of special Lagrangian representatives, analogous to slope stability and Hermitian Yang–Mills theory. It was originally proposed by Thomas and remains open in this generality.