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.

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 $\epsilon>0$:

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].

To trace the subsequent development, and move beyond the almost calibrated setting, we need to recall an important piece of homological algebra known as Bridgeland stability conditions on a triangulated category [13].

With the hindsight of Bridgeland stability condition, and more than one decade of progress on the mean curvature flow as well as symplectic geometry, Joyce [41] produced a major update of the Thomas-Yau picture. We shall discuss more about the LMCF considerations in section 4.1, but it suffices here to give away its main punchline (cf. [41, Conj 3.34]):

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  Let $(X,\omega,\Omega)$ be a Calabi-Yau manifold ⁹⁹ 9 The complex Monge-Ampère equation is convenient but not indispensable, see section 4.1., either compact or Stein. There should be an enlarged version of the derived Fukaya category $D^{b}Fuk(X)$, including classes of immersed or mildly singular Lagrangians, and a Bridgeland stability condition $(Z,\mathcal{P})$ on $D^{b}Fuk(X)$, whose central charge is

  $$Z(L)=\int_{L}\Omega.$$

  (5)

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  Given a Lagrangian brane $L$ (with grading, orientation, relative spin structure, local system and bounding cochain data) such that the Floer cohomology is unobstructed, and suppose $L$ is generic in its Hamiltonian isotopy class. Then the LMCF $(L_{t})_{t>0}$ (with brane structure) starting from $L$ exists for all time with surgeries at a finite series of singular times. The nature of these surgeries have conjectural descriptions. The Lagrangian can change its Hamiltonian class at these surgeries, but maintains its derived category class. At $t\to\infty$, the Lagrangians $L_{t}$ converges in the geometric measure theory sense to a union of graded special Lagrangian currents $L_{1},\ldots L_{N}$ of phase angle $\hat{\theta}_{1}>\hat{\theta}_{2}>\ldots>\hat{\theta}_{N}$ with multiplicities counted:

  $$\lim_{t\to\infty}L_{t}=L_{1}+\ldots L_{N}.$$

  (6)

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  If there is only one constant phase angle $\hat{\theta}$ appearing in the infinite time limit, then $L$ defines a semistable object of phase $\hat{\theta}$ in $D^{b}Fuk(X)$. Otherwise $L$ is not semistable with respect to any phase angle, and the infinite time limit supposedly give rise to the Harder-Narasimhan decomposition.

Joyce’s program is much more ambitious in the sense that it attempts to capture the entire triangulated category $D^{b}Fuk(X)$. The original Thomas-Yau proposal, which is concerned only with almost calibrated Lagrangians, fits into the Joyce picture as an abelian subcategory $\mathcal{P}(-\pi/2<\theta\leq\pi/2)\subset D^{b}Fuk(X)$,¹⁰¹⁰ 10 Here we ignore the ‘small’ difference between $<$ and $\leq$. This would be justified if $\Omega$ is suitably generic so that there is no special Lagrangian of phase $\pi/2$, namely that the countably many numbers $\text{Arg}\int_{L}\Omega$ for $[L]\in H_{n}(X,\mathbb{Z})$ miss the number $\pi/2$. the heart of a certain $t$-structure. The main appeal of Joyce’s mean curvature flow perspective, is that if the infinite time limiting currents $L_{1},\ldots L_{N}$ can be given suitable brane structures to be admitted as objects of $D^{b}Fuk(X)$, then the program gives a conjectural dynamical mechanism to explain the Harder-Narasimhan decomposition, which is the most nontrivial aspect of the Bridgeland stability. Its principal drawback is that Joyce offers no a priori information on the Bridgeland stability beyond the central charge, other than letting off the mean curvature flow to find its own destiny. This limits its practical applicability, for instance to existence questions of special Lagrangians in prescribed classes in $D^{b}Fuk(X)$. See section 3.6 for more discussions.

At this moment an objection may arise: since the Thomas-Yau-Joyce picture is beset by some vagueness and plenty of technical difficulties, why is it useful as a guiding principle at all? Besides the supportive evidence that we shall soon discuss, the main answer is that the Thomas-Yau philosophy transforms a PDE problem (the existence of special Lagrangians) into a categorical framework, which if better understood is in principle checkable by algebraic means. A Bridgeland stability is the interplay between a category and a numerical property. In the analogous problem of Hermitian Yang-Mills connections, the existence criterion is formulated by $\mu$-stability, which compresses the information of the Kähler form into only certain intersection numbers/cohomological integrals (cf. section 2.5). Likewise, the stability condition responsible for the existence of special Lagrangians, even though it is not adequately specified, is in principle a compression of the analytic data of a holomorphic volume form, and quite plausibly enters only through cohomological integrals of $\Omega$, as will be discussed more fully in Chapter 3. As an indication of the possible predicative power of the Thomas-Yau picture, here is a sample question as food for thought: