The visionary proposal of Thomas and Yau [65][66] is the philosophy that existence and uniqueness questions of certain special Lagrangian branes inside an almost Calabi-Yau manifold $(X,\omega,\Omega)$ should be governed by stability conditions in the derived Fukaya category (cf. section 2.1). This proposal lies at the intersection of two major mathematical disciplines:

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  From the viewpoint of geometric measure theory, the Thomas-Yau proposal promises a systematic method to prove existence theorems for special Lagrangians, which are certain absolute volume minimizers within prescribed homology classes. Currently, the known construction methods are based on perturbative techniques, symmetry reductions, some special ansatzs, and integrable system techniques specific to low dimensions [43]. In contrast, Thomas and Yau suggests the difficult PDE questions may be reducible to largely topological and algebraic questions in Floer theory.

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  From the viewpoint of mirror symmetry, the Thomas-Yau proposal is a window beyond homological mirror symmetry. Currently, most of the mathematical works on mirror symmetry are concerned with the duality between pure symplectic topology and pure complex geometry, and as such have a largely topological and algebraic flavour. As soon as one simultaneously consider the full Kähler geometric data on each side of the mirror, then questions of analytic nature become inevitable, and the Thomas-Yau proposal is a central component of this larger picture (cf. section 2.4).

Two decades have lapsed since their proposal, but we feel only a small part of the mystery has been unveiled. Part of the problem is that the relevance of holomorphic curves to the Thomas-Yau proposal is insufficiently understood, despite their central role in defining the Fukaya category. One of the main goals of this paper is to propose the following picture, in the more specialized setting of exact Lagrangian branes inside (almost) Calabi-Yau Stein manifolds:

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  Using $(n+1)$-dimensional universal families of holomorphic curves associated to certain $(n-1)$-dimensional moduli spaces, one can construct bordism currents between two $n$-dimensional Lagrangians in the same derived category class. More generally, one can also sometimes associate bordism currents between several Lagrangians, the notable example coming from distinguished triangles. This is a special case of the open-closed map well known to symplectic geometers, and some lower brow expositions are given in section 3.1.

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  Solomon [75] defined a functional on the universal cover of the space of Lagrangians within the same Hamiltonian deformation class, whose critical points are precisely special Lagrangians. In the exact setting, we will give a more homological formula for the Solomon functional, and propose that the bordism current produced from holomorphic curves allows one to extend the functional to Lagrangian objects in the same derived category class, which is well defined without the universal cover problem (cf. section 3.2). A different perspective involving integration over moduli spaces is presented in section 3.7, and we suggest further how the Solomon functional may generalize to compact Calabi-Yau manifolds.

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  We further specialize to almost calibrated Lagrangians. We propose that given a distinguished triangle $L_{1}\to L\to L_{2}\to L_{1}[1]$, then there are Floer theoretic necessary conditions for $L$ to be a special Lagrangian:

  $$\arg\int_{L_{1}}\Omega\leq\arg\int_{L}\Omega\leq\arg\int_{L_{2}}\Omega.$$

  Here $L_{1},L_{2}$ are not required to be special Lagrangians, and the cohomological integral is the only quantitative way they enter into obstruction criterions. In other words, obstructions are of numerical nature. We will prove this assuming further that the bordism current from holomorphic curves satisfies automatic transversality and a positivity condition (cf. section 3.3, 3.4, 3.5). The main technique is integration over the $(n-1)$-dimensional moduli spaces of holomorphic curves. We also give generalizations to several Lagrangians, to make possible contact with Harder-Narasimhan decomposition. Compared with the proposal of Joyce [41], the Floer theoretic obstructions capture some features of the Bridgeland stability condition, but instead of tackling the entire derived Fukaya category $D^{b}Fuk(X)$, we restrict to almost calibrated Lagrangian objects, which morally correspond to the heart of a $t$-structure.

  Remark 1.1.

  While Joyce’s prediction of a Bridgeland stability condition is a very important heuristic motivation for this paper, in our proposal we will carefully avoid assuming that the Bridgeland stability condition exists on $D^{b}Fuk(X)$, or that $D^{b}Fuk(X)$ is idempotent closed, or that the almost calibrated Lagrangians form an abelian subcategory of $D^{b}Fuk(X)$, except when we make comparisons with Joyce’s program.

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  When there are no destabilizing distinguished triangles, we say the derived Fukaya category class of $L$ is Thomas-Yau semistable (cf. Definition 3.32). This is similar to the original viewpoint of Thomas and Yau [65][66]. Let $X$ be a Calabi-Yau Stein manifold, such that $(\omega,\Omega)$ satisfies the complex Monge-Ampère equation, and the regularity scale of the Calabi-Yau manifold tends to infinity asymptotically. Let $\mathcal{L}$ be the geometric measure theoretic closure of the class of exact, quantitatively almost calibrated, unobstructed Lagrangians in the $D^{b}Fuk(X)$ class. (Beware that we include immersed and singular Lagrangians as in Joyce [41], so the definition of $\mathcal{L}$ is partially conjectural.) According to our interpretation of the Thomas-Yau conjecture, if $L_{0}\in\mathcal{L}$ is Thomas-Yau semistable, then there exists a special Lagrangian current in $\mathcal{L}$ (cf. section 5.5.1 for the full version). Morever, it is desirable that the special Lagrangian current carries unobstructed brane structure in some formal sense, to represent the $D^{b}Fuk(X)$ class or some weaker equivalence class.

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  We will make a start on the variational approach to tackle the Thomas-Yau conjecture in the exact and quantitative almost calibrated setting. A summary of the variational strategy is in section 5.3. We prove a number of uniform estimates independent of the Lagrangians, notably the potential clustering property, which implies a uniform bound on the energy of holomorphic curves, and that the Solomon functional deviates from an elementary functional by a uniformly bounded amount (cf. section 5.1, 5.2). We are not able to complete the proof of the Thomas-Yau conjecture, but we try to identify the main technical difficulties to be overcome. To make contact with the a priori compactness theorems in geometric measure theory, it is natural to set up the variational program in terms of Lagrangian currents and varifolds. One then encounters both geometric measure theoretic difficulties, to do with the existence of enough Lagrangian competitors, and the difficulty to make sense of Floer theory for Lagrangian currents. We expect that many aspects of Floer theory will not be robust under passage to such weak limits of Lagrangians, but as a heuristic principle, we hope that the bordism currents relevant to our proposal are robust (cf. section 5.4).

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  Assume there is a suitable compactness theory for Lagrangian objects, then we expect the existence of special Lagrangians to be essentially equivalent to the properness of the Solomon functional. The following heuristic principle would then explain why Thomas-Yau conjecture could be true: for a sequence of Lagrangians whose Solomon functional diverges to infinity, the underlying Lagrangian objects should break up into a bounded number of connected components $L_{i}$, such that the Lagrangian potential on each component has uniformly bounded oscillation, and the asymptotic behaviour of the Solomon functional is controlled by Floer theoretic data related to the cohomological integrals $\int_{L_{i}}\Omega$. We will give evidence for this picture (cf. section 5.5), and explain a similar picture for Hermitian Yang-Mills connections (cf. section 2.5).

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  Although we restrict to the exact setting in the above, we will leave the room open for more ambitious speculations in the setting of compact almost Calabi-Yau manifolds. The almost calibrated condition, on the other hand, is essential in our proposal, and cannot be dropped within this framework even with substantial efforts.

Aside from the main goal of making the proposal and presenting evidence, this paper also contains a large amount of expository content, which aims to present both a (biased) overview and a critique of the current literature on the Thomas-Yau conjecture, and to explain how our picture fits into this body of works. The most relevant works are the original papers of Thomas and Yau [65][66], and the major update by Joyce [41]. We will devote substantial attention to their main considerations (cf. section 2.1, 2.2, 2.3, 2.4, 2.5, 2.7, 4.1), and the significance of the Thomas-Yau-Joyce proposal to mirror symmetry (cf. section 2.4), and we will constantly draw comparisons between their pictures and ours. We also touch on a number of other relevant works:

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  (cf. section 2.3) Joyce-Imagi-Santos [40] gave a characterisation of Lawlor necks, in which holomorphic curves made an interesting, albeit somewhat technical appearance.

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  (cf. section 2.5, 2.6) The Hermitian Yang-Mills equation [27][26][79] was a major inspiration to Thomas and Yau. The deformed Hermitian Yang-Mills equation [18][22][23][16] has been advocated as a mirror version of special Lagrangians.

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  (cf. section 2.8) Solomon [75][76][77] introduced a functional and a formal Riemannian metric on the infinite dimensional space of almost calibrated Lagrangians in a given Hamiltonian isotopy class. This space is too small for our variational purpose, but his functional is fundamental to our proposal.

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  (cf. section 2.9) Lotay and Pacini [57][58] introduced a formal GIT picture for totally real submanifolds. Their notion of ‘geodesic’ was our main inspiration for the bordism currents between Lagrangians.

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  (cf. section 4.1, 5.2) Neves [61][62][63] pointed out the inevitability of finite time singularity in the Lagrangian mean curvature flow, which is fundamental to Joyce’s proposal [41]. Some of his technical arguments are adapted to support our variational program in section 5.2.

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  (cf. section 4.2) Joyce’s earlier work on wall crossing and special Lagrangian counting problems [42] is in some sense the elliptic cousin to his Lagrangian mean curvature flow proposal [41].