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.

There are two fundamental obstacles:

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  How can one find the initial special Lagrangian?

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  How can one guarantee compactness?

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’ $\text{Im}\Omega|_{L}=0$.

The question about compactness largely reflects problems we already encountered in the LMCF approach. The essential issue is that without any further condition on the Kähler metric, special Lagrangians may be too singular, so that the McLean deformation theory for special Lagrangians may fail. The natural answer, closely related to the LMCF viewpoint, is that we should only work with generic Kähler structures, and 1-parameter families thereof. According to the philosophy advocated by Joyce [42], the importance of singularities are ranked according to their genericity. For special Lagrangians with first Betti number $k$, the moduli space of deformations is $k$-dimensional, so in a generic 1-parameter family of Kähler structures, one expects to encounter singularities with genericity index up to $k+1$, and those singularities of index $0,1$ are the most important. ⁵⁶⁵⁶ 56 Understanding the moduli space of special Lagrangians requires singularities up to index $k+1$, but if we restrict to Lagrangian deformations with zero Lagrangian flux, then index $\leq 1$ may suffice in the optimistic view. Before one can seriously pursue the rest of this strategy, it is necessary to have a classification of index $0,1$ special Lagrangian singularities in complex dimension $n$.

Some of the generic singularities in the LMCF are expected to have elliptic analogues in the continuity method approach. We fix $\omega$ and consider a generic 1-parameter family of $\Omega$, and we follow the Lagrangian flux zero deformations of a given special Lagrangian.

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  Under exact isotopy, immersed Lagrangians may lose the unobstructed condition. One expects the surgery of the brane structure suggested by Joyce has an elliptic analogue, involving the same ingredients as the ‘Maslov flow’ studied by Woodward and Palmer [81][82].

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  As already discussed in section 2.7, the Lawlor neck is responsible for the gluing of two immersed special Lagrangians. This corresponds to the ‘Lawlor neck pinching’ singularity, as well as the ‘openning the neck’ surgery in the LMCF.

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  (Stable singularity) Joyce suggests from his work on $U(1)$-invariant special Lagrangians in $\mathbb{C}^{3}$ [41, Example 2.8], that in a continuous 1-parameter family, isolated singular points of special Lagrangian 3-folds with local $T^{2}$-cone singularities can appear and disappear in pairs, by making the two $T^{2}$-cone singularities collide with each other and then smooth out. This is the main motivation for admitting the $T^{2}$-cone singularity in the LMCF, and it seems likely to be a generic singularity in the continuity method as well.

As Bridgeland observed [13, Thm 1.2], the central charge map gives a local homeomorphism between the space of Bridgeland stability conditions, and the hom space from the numerical Grothendieck group to $\mathbb{C}$. The Thomas-Yau-Joyce philosophy then suggests that for fixed $\omega$ and small deformations of $\Omega$, the stability condition only depends on the cohomology class $[\Omega]$.

A possible geometric interpretation consistent with the wall crossing picture, is that in a generic 1-parameter family of deformations of the almost Calabi-Yau structure fixing $\omega$ and $[\Omega]$, the special Lagrangian may undergo surgeries, such that the topology and the Hamiltonian isotopy class may change, but when equipped with the brane structures, the $D^{b}Fuk(X)$ class (or possibly some weaker equivalence class) remains constant. As in the LMCF approach, the role of the unobstructed brane structure is morally to prevent holomorphic discs from having very small area. However, the troubles caused by shrinking discs may be less severe in the continuity method than in the LMCF method, since the continuity method only deals with special Lagrangians, and cannot lose grading in a process like Example 4.3. Instead, a significant amount of difficulty in the continuity method is absorbed into the problem of finding the initial special Lagrangians.

To summarize the pros and cons compared to the LMCF approach, in the continuity method finding an initial special Lagrangian is a significant new difficulty. However, we no longer need to confront the extremely difficult task of long time existence for the flow. On a slightly more technical level, provided one has a classification for low index singularities, the genericity assumption is likely to be easier to use in the continuity method than it is in the LMCF framework.