The general idea of the continuity method is to work with a 1-parameter family of PDEs, and attempt to deform from an initial given solution, to a solution of the final PDE, provided the deformation encounters no obstruction, and satisfies suitable compactness properties. The hope that the continuity method may be useful here, is based on the foundational fact that compact special Lagrangian submanifolds inside almost Calabi-Yau manifolds have unobstructed deformation theory, before taking brane structures into account.

However, problems immediately ramp up once one attempts to set up a continuity path. The most naïve suggestion, based on the analogy with the HYM equation, is to prescribe the Lagrangian angle as a function on the domain of $L$. This however breaks the domain reparametrisation invariance of $\iota:L\to X$, and the author knows no satisfactory way to make general sense of this approach beyond graphical Lagrangians. Instead we fixed $\omega$, and allow $\Omega$ to vary in an infnite dimensional parameter space subject to the almost Calabi-Yau condition. In noncompact almost Calabi-Yau manifolds, we need to also keep the metric asymptote fixed at infinity. The continuity path is a generic 1-parameter family of $\Omega$. This setup strongly resemble the wall crossing phenomenon studied by Joyce [42] in the context of special Lagrangian enumerative invariants, and indeed the rest of this section liberally borrows from the ideas therein.