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:

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

(50)

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

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

(51)

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:

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

(52)

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.

• •

  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

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

(53)

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.