Thomas [69] and Thomas and Yau [70] proposed some interesting conjectures on graded Lagrangians $L$ in Calabi–Yau manifolds $(M,J,g,\Omega)$: they defined a notion of ‘stability’ [69, Def. 5.1] for Hamiltonian isotopy classes $[L]$ of (almost calibrated) graded Lagrangians $L$, and conjectured [69, Conj. 5.2] that $[L]$ contains a (unique) special Lagrangian $L^{\prime}$ if and only if $[L]$ is stable. Furthermore, they conjectured [70, Conj. 7.3] that if $[L]$ is stable and $L$ satisfies an extra condition [70, (7.1) or (7.2)] then Lagrangian mean curvature flow $\{L^{t}:t\in[0,\infty)\}$ with $L^{0}=L$ exists for all time, and $\lim_{t\rightarrow\infty}L^{t}=L^{\prime}$.

Thomas and Yau’s papers [69, 70] are remarkably prescient, as they predate (and motivated) much important mathematics relevant to their picture, including the invention of Bridgeland stability on triangulated categories [10], the publication of Fukaya, Oh, Ohta and Ono’s [18, 19, 20] and Seidel’s work [64] on Lagrangian Floer cohomology and Fukaya categories, and progress on singularities of Lagrangian MCF such as Neves [55, 56, 57]. I believe their big picture is correct, although I think they are too optimistic in expecting Lagrangian MCF to exist for all time without singularities even in the stable case, and want to substitute Lagrangian MCF with surgeries instead (see (iv) below).

The aim of this paper is to update the Thomas–Yau conjectures in the light of subsequent discoveries, to add more detail to the picture, and to extend their scope. Thomas and Yau’s papers are clearly intended as a programme for future research rather than as precise conjectures; they have many caveats on points they are uncertain about, and the conjectures they actually state are fairly cautious (for instance, the inclusion of the strong condition [70, (7.1) or (7.2)] on stable Lagrangians for Lagrangian MCF to converge to a special Lagrangian).

I am going to be a lot less cautious, and will make conjectures on unique long-time existence of Lagrangian MCF with surgeries starting from any compact graded Lagrangian with $HF^{*}$ unobstructed. Nonetheless, I ask readers to take the conjectures in the spirit they are intended: as provisional, quite probably false in their current form, to be refined (or discarded) as our understanding improves, but in the mean time, as (hopefully) a useful guide and motivation for research in the area. I will say more on this in the introduction to §3.

In reading Thomas and Yau [69, 70], I think it is helpful to impose the standing assumption that all graded Lagrangians $L$ considered are almost calibrated, that is, have phase variation less than $\pi$. This is not clearly articulated in [69, 70], although bounds on the phase variation are assumed in several places, with the almost calibrated condition used in [70, §5.3]. We need $L$ to be almost calibrated since otherwise the ‘global phase’ $\phi(L)$ in [69, §3] is not well-defined, and so ‘stability’ in [69, Def. 5.1] does not make sense.

Including the almost calibrated assumption, I am not aware of any counterexamples to the precise conjectures stated in [69, 70] (although I do expect such counterexamples to exist, see (iv) below). In particular, Neves’ examples [57] of finite time singularities to Lagrangian MCF discussed in Example 3.28 below are not almost calibrated, and so not counterexamples to [70, Conj. 7.3].

Here are the main differences between our programme and that of [69, 70]:

• (i)

  We work in the ‘derived Fukaya category’ $D^{b}{\mathbin{\mathscr{F}}}(M)$ of $M$, as in Fukaya, Oh, Ohta and Ono [18, 19, 20] (see also Seidel [64]). Objects of $D^{b}{\mathbin{\mathscr{F}}}(M)$ include triples $(L,E,b)$, where $L$ is a compact, graded Lagrangian in $M$ and $E\rightarrow L$ a rank one local system such that $(L,E)$ has ‘$HF^{*}$ unobstructed’, and $b$ is a ‘bounding cochain’ for $(L,E)$, as in [20].

  Rather than working in a Hamiltonian isotopy class $[L]$ as in [69, 70], we work in an isomorphism class $[(L,E,b)]$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$.

• (ii)

  The derived Fukaya category $D^{b}{\mathbin{\mathscr{F}}}(M)$ must be enlarged to include immersed Lagrangians as in [2] in dimension $m\geqslant 2$, and certain classes of singular Lagrangians in dimension $m\geqslant 3$, for the programme to work.

• (iii)

  Our notion of ‘stability’ of Lagrangians is a ‘Bridgeland stability condition’ $(Z,{\mathbin{\cal P}})$ on the triangulated category $D^{b}{\mathbin{\mathscr{F}}}(M)$, as in Bridgeland [10].

• (iv)

  Even for a ‘stable’ object $(L,E,b)$ with small phase variation, I do not expect Lagrangian MCF $\{L^{t}:t\in[0,\infty)\}$ with $L^{0}=L$ to exist without singularities, as hoped in [70]. Instead, in a similar way to the proof of the Poincaré Conjecture by Perelman and others using Ricci flow (see [54]), I expect there to exist a unique family $\{(L^{t},E^{t},b^{t}):t\in[0,\infty)\}$ of objects in the isomorphism class of $(L,E,b)$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$ with $(L^{0},E^{0},b^{0})=(L,E,b)$, where $\{L^{t}:t\in[0,\infty)\}$ satisfies Lagrangian MCF with surgeries.

  That is, at a discrete series of ‘singular times’ $t=T_{1},T_{2},\ldots$ the flow develops a singularity, but one can continue the flow uniquely for $t>T_{i}$ in a way which is continuous at $t=T_{i}$ in a weak sense. The $L^{t}$ for $T_{i}-\epsilon<t<T_{i}$ and for $T_{i}<t<T_{i}+\epsilon$ may have different topologies.

• (v)

  Lagrangians $L$ or pairs $(L,E)$ in $M$ ‘with $HF^{*}$ obstructed’ do not give objects of $D^{b}{\mathbin{\mathscr{F}}}(M)$, and ‘stability’ does not make sense for them.

  For $L$ with $HF^{*}$ obstructed, the author expects that Lagrangian MCF $\{L^{t}:t\in[0,T)\}$ with $L^{0}=L$ may develop finite time singularities at $t=T$ after which it is not possible to continue the flow, even with a surgery. So, the long time existence of Lagrangian MCF with surgeries in (iv) should apply only for Lagrangians with $HF^{*}$ unobstructed.

Part (iv), our insistence on including finite time singularities of Lagrangian MCF and surgeries, is the greatest divergence between our picture and that of [69, 70]. As some justification, note that Neves [57] proves that every Hamiltonian isotopy class $[L]$ of compact Lagrangians $L$ in a Calabi–Yau $m$-fold $(M,J,g,\Omega)$ for $m\geqslant 2$ contains (not almost calibrated) representatives $\tilde{L}$ such that Lagrangian MCF $\{L^{t}:t\in[0,T)\}$ with $L^{0}=\tilde{L}$ develops a finite time singularity at $t=T$, so without (strong) extra assumptions, finite time singularities of Lagrangian MCF are unavoidable.

One of the goals of this paper is to persuade mathematicians working on Lagrangian MCF that obstructions to $HF^{*}$ are important in understanding finite time singularities of Lagrangian MCF, that the flow should be better behaved if $HF^{*}$ is unobstructed, and that tools from symplectic topology such as $J$-holomorphic curves, Lagrangian Floer cohomology, and Fukaya categories, should be used to make the next generation of advances in the field.

Some evidence for this is provided by Imagi, Oliveira dos Santos and the author [31], in which, motivated by this paper, we use Lagrangian Floer cohomology and Fukaya categories to prove that the unique special Lagrangians in ${\mathbin{\mathbb{C}}}^{m}$ asymptotic at infinity to the union $\Pi_{0}\cup\Pi_{\boldsymbol{\phi}}$ of two transverse Lagrangian planes $\Pi_{0},\Pi_{\boldsymbol{\phi}}$ are the ‘Lawlor necks’ of [45], and the unique Lagrangian MCF expanders in ${\mathbin{\mathbb{C}}}^{m}$ asymptotic at infinity to $\Pi_{0}\cup\Pi_{\boldsymbol{\phi}}$ are the examples in Lee, Tsui and the author [43, Th.s C & D], as in Theorems 2.6 and 2.14 below.

Section 2 explains some background material, and §3 states the conjectures.