The question about finding the initial special Lagrangian is specific to the continuity method, and does not appear in the LMCF approach. A natural suggestion is to look for special Lagrangians near certain degenerate limits, and our relaxation of the complex Monge-Ampère equation ought to give much more flexibility. For instance, conifold degenerations are known to give rise to special Lagrangian spheres appearing as vanishing cycles [37].⁵²⁵² 52 While Hein and Sun’s result is highly nontrivial, the entire difficulty goes into understanding the Calabi-Yau metric near the conifold point. If we are given the license to prescribe arbitrary Kähler metrics, the problem of finding special Lagrangian vanishing spheres near the conifold point becomes easy. Another general source is to work near a suitable large complex structure limit, so that the Kähler metric can be made almost toric outside a small region, such that the torus fibres are much smaller compared to the characteristic length scale of the base. We can then attempt to find special Lagrangians via adiabatic limits, in close analogy with the standard procedure to find holomorphic curves via tropical degenerations [55]. ⁵³⁵³ 53 The large complex structure limit is supposed to correspond to the large volume limit in the mirror, which is related to the $\mu$-stability, thus offering the hope of a mirror calculation of counting invariants. The most accessible special Lagrangians in this approach, should be obtainable by small perturbations of the torus fibres. ⁵⁴⁵⁴ 54 The difficulty in [54] to construct SYZ special Lagrangian fibrations again comes from the Calabi-Yau metrics. If one can freely prescribe Kähler metrics, then finding a special Lagrangian torus is not difficult. The next candidate suggested by the Leray filtration of the torus fibration is already much harder.

One expects that locally along an edge these Lagrangians are perturbations of $T^{n-1}\times\mathbb{R}$, with $T^{n-1}$ contained in the torus fibre direction, so that we obtain $(n-1)$ locally defined closed 1-forms $\int_{S^{1}}\omega$ on the base corresponding to the cycles $S^{1}\subset T^{n-1}$, and the edge is to leading approximation given by requiring these 1-forms to vanish. The local model for the junction where three edges meet, ⁵⁵⁵⁵ 55 This is conceptually related to Matessi’s ‘Lagrangian pair of pants’ [59]. may have the following topological description. In the $n=2$ case, we have a ‘pair of pants’ inside $T^{2}\times\mathbb{R}^{2}$ with three asymptotic ends $S^{1}\times\mathbb{R}$; topologically this is the same as algebraic surface $\{z_{1}+z_{2}=1\}\subset\mathbb{C}^{*}\times\mathbb{C}^{*}$. In higher dimensions, we take a product of the pair of pants with $T^{n-2}$.

In the next order of perturbation, we expect the deformation of the Lagrangian in the base direction to be at least comparable to the length scale of the fibre, and presumably is fixed by the ‘special condition’ $\text{Im}\Omega|_{L}=0$.