We now return to some analytic aspects of Joyce’s proposal [41] related to the Lagrangian mean curvature flow (LMCF) inside a Calabi-Yau manifold. Recall a smooth mean curvature flow means a family of immersions $\iota_{t}:L\to M$ parametrised by time $t\in[0,T)$, such that the velocity is equal to the mean curvature:

$$\partial_{t}\vec{x}=\vec{H}.$$

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The starting point of LMCF is an early observation of Smoczyk, which justifies the name:

Assuming $(L_{t})$ is a smooth and compact LMCF inside a Calabi-Yau ambient manifold, then if the initial Lagrangian is graded, i.e. the Lagrangian angle is well defined as a real valued function on the Lagrangian, then so is $L_{t}$. The grading is highly desirable, because of the foundational fact that the Lagrangian angle function $L_{t}\to\mathbb{R}$ satisfies the heat equation

$$(\partial_{t}-\Delta_{L_{t}})\theta_{t}=0,$$

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which among many other things, implies that $\sup_{L_{t}}\theta_{t}$ can only decrease in time, and $\inf_{L_{t}}\theta_{t}$ can only increase in time, and in particular the almost calibrated condition would be preserved by the flow. Inside a Calabi-Yau manifold, the mean curvature of $L_{t}$ is related to the Lagrangian angle $\theta_{t}$ by an appealing formula:

$$\vec{H}=J\nabla\theta_{t},$$

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where $\nabla\theta$ stands for the gradient of $\theta$ along $L_{t}$. Along the LMCF

$$\omega(\partial_{t}\vec{x},\cdot)=\omega(\vec{H},\cdot)=\omega(J\nabla\theta_{t},\cdot)=-d\theta_{t},$$

so the Lagrangians evolve by the local Hamiltonian function $-\theta_{t}$ up to an additive constant.

Mean curvature flow in codimension greater than one does not satisfy the avoidance principle. As such embedded Lagrangians can become immersed during the flow, so the program should at least include immersed Lagrangians. Joyce further suggests that certain ‘stable Lagrangian singularities’ should be admitted. For instance, inside Calabi-Yau 3-folds one should allow Lagrangians with local conical singularity modelled on the Harvey-Lawson $T^{2}$-cone [41, Example 2.7]. The adjective ‘stable’ here means that the flow should preserve this class of singularities at least for a short amount of time, even if one makes a generic perturbation of the initial data.

The central difficulty of the subject is that finite time singularities are in general inevitable, starting from complex dimension two. Indeed, a theorem of Neves [61, Thm. 6.1] says that for any embedded Lagrangian submanifold inside a Calabi-Yau surface, there exists a Lagrangian within the same Hamiltonian isotopy class, such that the LMCF with this initial data forms finite time singularity. ⁴⁵⁴⁵ 45 Whether the same holds for almost calibrated initial data is an interesting open problem. There is also a good geometric reason why singularities must occur in Joyce’s program: the Thomas-Yau uniqueness theorem applies to Lagrangians within the same derived Fukaya category class, which may include several Hamiltonian isotopy classes, at most one of which can have special Lagrangian representatives. In order for an initial Lagrangian in the wrong Hamiltonian isotopy class to find its way back to the right class along the LMCF, it must undergo a sequence of surgeries.

Now there is a substantial theory of weak solutions of mean curvature flows in the context of varifolds and currents, known as ‘Brakke flows’ [12], which exist under very general conditions. The problem is that such solutions are too weak to guarantee uniqueness of the flow, and the total mass of the varifold may jump down at discrete time. Even more fatally for our purpose, once the smoothness of the flow is dropped, the Lagrangian condition may not be preserved any more. It is instructive to look at the prototypical bad behaviours:

With this background in mind, one may better appreciate the upshot of Joyce’s perspective: LMCF should be better behaved if the Lagrangians support unobstructed brane structures, namely the bad singularities that would spell ruin in a more general context do not actually occur in his program. The moral reasons are:

• •

  The Lagrangian branes are always assumed to be graded.

• •

  Unobstructed Lagrangian branes cannot bound holomorphic curves with very small areas unless their Floer theoretic contributions exactly balance out, for otherwise certain positivity requirements in the Novikov ring will be violated.

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  Assume the Lagrangian decomposes into two pieces, one of which is contained inside a small coordinate ball. Since this piece only contributes a zero object in $D^{b}Fuk(X)$, discarding this piece does not affect the $D^{b}Fuk(X)$ class of the Lagrangian.

This gain comes at the burdensome cost of carrying the brane structure along the flow, which leads to somewhat counterintuitive prescriptions such as surgeries before the Lagrangian itself reaches a singularity, so that the unobstructed brane structure may not be lost prematurely. Joyce describes a number of singularities and surgeries that are expected to occur generically in his program:

• •

  (Openning up the neck) The Lagrangian brane can develop new self intersection points, and may flow from unobstructed to obstructed at $t=t_{0}$ through the shrinking of certain holomorphic curves with boundary on $L_{t}$, even through the underlying Lagrangian remains smooth. The Floer theoretic mechanism causing the obstruction (to do with positivity conditions in the Novikov ring) precisely ensures an angle condition at certain self intersection points of $L_{t_{0}}$, so that a Joyce-Lee-Tsui Lagrangian expander [45] can be glued into $L_{t_{0}}$ to continue the LMCF. Woodward and Palmer [81][82] have performed substantial checks that suitable brane structures can be assigned to such surgeries so that $L_{t}$ remains unobstructed, and the Floer cohomology remains continuous throughout the surgery.

• •

  (Neck pinching) In some sense converse to the above process, the LMCF $L_{t}$ may contain a local region modelled on a Lawlor neck with small length scale parameters $\epsilon(t)$, which shrinks ‘slowly’ in time, and $\epsilon(t)\to 0$ at time $t\to t_{0}$. ⁴⁹⁴⁹ 49 A gluing construction of neck pinching examples is in working progress with T. Collins. There is however a crucial sign difference concerning Lagrangian angles, between our work and the Joyce prediction, which may have rather disconcerting consequences for Joyce’s program. In some cases this may cause the domain of the immersed Lagrangian to become disconnected.

• •

  (Collapsing zero objects) After a number of surgeries, the Lagrangian $L_{t}$ may be decomposed into several disconnected pieces, some of which are zero objects in $D^{b}Fuk(X)$, so in particular have homology class zero. A typical situation is that the zero objects are contained in small coordinate balls.⁵⁰⁵⁰ 50 Joyce suggests plausibly that Neves’s example [61] exhibits this behaviour by splitting off a small Whitney sphere, although this is not proven. We simply discard these pieces and continue the flow for the remaining pieces.

• •

  (Stable singularities) As mentioned above, one may need to include Lagrangians with certain local singularities, since generic perturbations cannot remove such singularities. How such singularities can form dynamically starting with smooth initial data is less clear, but Joyce offers some analogy with the setting of $U(1)$-invariant special Lagrangians in $\mathbb{C}^{3}$ [41, Example 2.8], where Harvey-Lawson $T^{2}$-cone singularities can appear and disappear in pairs within a 1-parameter family of deformations, and in particular smooth objects can be continuously deformed to such singular objects.

The Joyce program of LMCF with surgery contains a number of potentially counterintuitive phenomenon.

In summary, the main difficulty of Joyce’s LMCF program is that there is a huge gap between the general Brakke flow framework, and the kind of regularity control required for the long time existence of the LMCF. It would represent very substantial progress ⁵¹⁵¹ 51 Joyce [41] assesses the difficulty of his program in the Calabi-Yau 3-fold case to be comparable to Perelman’s breakthrough on the Poincaré conjecture. Indeed, ruling out the cigar solution in the context of the Ricci flow is in itself already a major achievement of Perelman. if one can classify possible singularity types under suitable genericity assumptions, say for almost calibrated Lagrangians inside Calabi-Yau 3-folds. A large list of problems, from routine level up to the impossible, can be found in Joyce’s excellent original paper [41].

Provided one can prove long time existence of LMCF, the total mass of $L_{t}$ will be uniformly bounded since it decreases during the flow. Under mild conditions to ensure $L_{t}$ does not escape to spatial infinity (e.g. if the ambient Calabi-Yau manifold is compact), one can extract the infinite time subsequential limits of $L_{t}$ as currents. From the heat equation (51) on the Lagrangian angle $\theta$,

$$(\partial_{t}-\Delta_{L_{t}})|\theta|^{2}=-2|\nabla\theta|^{2}=-2|\vec{H}|^{2}.$$

If the Lagrangians remain sufficiently smooth, then an integration by part calculation shows

$$\int_{0}^{T}\int_{L_{t}}|\vec{H}|^{2}dvol_{L_{t}}dt=\frac{1}{2}\left(\int_{L_{0}}|\theta|^{2}dvol_{L_{0}}-\int_{L_{T}}|\theta|^{2}dvol_{L_{T}}\right).$$

Even if the volume mass can jump down at discrete time, such as during the collapsing of zero objects, we still expect

$$\int_{0}^{T}\int_{L_{t}}|\vec{H}|^{2}dvol_{L_{t}}dt\leq\frac{1}{2}\int_{L_{0}}|\theta|^{2}dvol_{L_{0}}<\infty,\quad\forall T>0.$$

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In particular we can find a sequence of time $t_{i}\to\infty$, with

$$\int_{L_{t_{i}}}|\nabla\theta|^{2}dvol\to 0.$$

This strongly suggests that the subsequential limit is a union of special Lagrangian currents with multiplicities.

In the heursitic logic of Thomas-Yau-Joyce prgogram, the infinite time limit supposedly provides the Harder-Narasimhan decomposition. In general one cannot expect the special Lagrangian currents to be smooth, so this raises the question how to make sense of singular Lagrangians as representatives of $D^{b}Fuk(X)$ classes, or whether we should use some weaker equivalence class. Another interesting open problem is whether the limiting current is unique. In order to run the Thomas-Yau argument, one presumably also needs Floer theory for singular Lagrangians.

Joyce [41] already observed that it is not obvious how singular Lagrangians can carry brane structures, and it is logically possible for some Floer theoretic information to be lost in the infinite time limit. The suggestion is that hopefully the Lagrangian $L_{t}$ at large but finite time $t\gg 1$, can serve as a substitute for the infinite time limit, which presumably has better smoothness properties [41]. There is however no known justification (and probably false) that the surgeries terminate after some finite time, and $L_{t}$ decomposes into the union of several Lagrangian objects, in order to provide a Harder-Narasimhan decomposition.

We think Floer theory for singular Lagrangians is one of the foundational open questions necessary for an adequate solution of the Thomas-Yau conjecture. See section 5.4 for further discussions.

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.

• •

  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].

• •

  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.

• •

  (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.