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:

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  The Lagrangian branes are always assumed to be graded.

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  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:

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  (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.

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  (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.

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  (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.

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  (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].