In [65][66] Thomas and Yau introduced the remarkable philosophy that existence and uniqueness questions of special Lagrangians inside an (almost) Calabi-Yau manifold $(X,\omega,\Omega)$ should be related to stability conditions in the Fukaya category. Turning this intuition into precise mathematical predictions, is however not easy, for at least the following geometric reasons, in addition to the analytic difficulties related to minimal surfaces and mean curvature flows:

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  The Fukaya category as it currently stands is likely inadequate for the purpose: the special Lagrangian representatives of a given derived Fukaya category class in $D^{b}Fuk(X)$, should it exists, is by no means guaranteed to be smooth and embedded. Thus one would like to enlarge the objects of the Fukaya category to include immersed and possibly singular Lagrangians. From the viewpoint of symplectic topology, the lack of Lagrangian objects is a basic difficulty, which is usually hidden in the non-geometric step of taking the twisted complexes in the construction of $D^{b}Fuk(X)$, and the idempotent completions in the construction of $D^{\pi}Fuk(X)$ (cf. the Appendix 6.2).

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  The knowledge of the stability conditions is severely deficient. Thomas-Yau’s paper predates the ingredient of Bridgeland stability³³ 3 What was available at the time, was the famous $\mu$-stability related to Hermitian Yang-Mills connections, of which Thomas and Yau were of course leading experts.⁴⁴ 4 There are current debates whether Bridgeland stability is the ultimately correct framework for formalising the physical intuition of stability conditions controlling BPS particle decay [31]. We regard Bridgeland stability as a working definition, to be modified in case future evidence arises., but the basic problem remains open: how to construct a stability condition on the derived Fukaya category $D^{b}Fuk(X)$ from the information of a holomorphic volume form $\Omega$ on $X$? In contrast, in analogous problems such as the existence of Hermitian-Yang-Mills connections, the stability condition is known a priori before the more serious endeavour to solve the PDE. One of the goals of this paper is to explain some modest progress on this issue, namely that there are nontrivial Floer theoretic obstructions to the existence of special Lagrangians.

Thomas and Yau primarily restricted attention to the case of almost calibrated Lagrangians, meaning the Lagrangian angle function $\theta$ satisfies ⁵⁵ 5 The important thing is that the upper and lower bounds on $\theta$ differ by $\pi$. Shifting the interval by a constant is inconsequential.

$$-\frac{\pi}{2}<\theta<\frac{\pi}{2}.$$

Notice this restriction removes any $\pi\mathbb{Z}$ ambiguity of the Lagrangian angle, so the Lagrangian is graded. Since we are focusing on compact Lagrangians, it makes sense to restrict to a more quantitative version, for some fixed small $\epsilon>0$:

$$-\frac{\pi}{2}+\epsilon\leq\theta\leq\frac{\pi}{2}-\epsilon.$$

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One immediate consequence is an a priori volume bound.

Keeping our narrative closer to the historical development, Thomas and Yau were inspired by $\mu$-stability for Hermitian Yang-Mills connections. They assumed that the principal mechanism which can forbid the Hamiltonian isotopy class of $L$ from admitting a special Lagrangian representative, is related to a distinguished triangle in the Fukaya category, the primary geometric source being that $L$ is Hamiltonian isotopic to a graded Lagrangian connected sum $L=L_{1}\#L_{2}$,⁶⁶ 6 It is very important for Thomas and Yau that the Lagrangian connected sum is asymmetric in $L_{1}$ and $L_{2}$. Our notation for the Lagrangian connected sum agrees with Thomas and Yau, but is opposite to Joyce [44] and a large number of symplectic geometry texts. Our convention is compatible with the distinguished triangle $L_{1}\to L_{1}\#L_{2}\to L_{2}\to L_{1}[1]$. such that $\phi(L_{1})\geq\phi(L_{2})$. Based on this intuition, Thomas made an attempt to define a notion of stability (cf. [65, Definition 5.1], and Definition 3.32 below).⁷⁷ 7 We will not repeat their definition verbatim here because the author thinks its focus on the Hamiltonian isotopy class, instead of the Fukaya category class, is largely a limitation of its time. It is the spirit rather than the letter of their definition which matters. Thomas then made the important prediction:

Thomas and Yau [66] analyzed the problem again from the Lagrangian mean curvature flow (LMCF) perspective, and made a somewhat more cautious prediction, which roughly amounts to the following. When a destabilising decomposition into Lagrangian sums is forbidden, either by a smallness assumption on the oscillation of the phase function $\theta$, or because the volume of $L$ is smaller than $\text{Vol}(L_{1})+\text{Vol}(L_{2})$ for any putative decomposition, (notice both conditions are preserved under the flow), then they conjectured that the mean curvature flow starting from $L$ will converge into a special Lagrangian [66, section 7].