ScalingStacks

Remark 3.35 . [03PV]

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Remark 3.35.

(i) The most feasible case of the conjecture is that of Lagrangian MCF in dimension m=2m=2, starting from an almost calibrated Lagrangian LL generic in its Hamiltonian isotopy class.

(ii) Assuming the initial object (L,E,b)(L,E,b) is semistable or stable in the sense of Conjectures 3.2 and 3.5 means in part (e) that the limit limt→∞Lt=L1\lim_{t\rightarrow\infty}L^{t}=L_{1} is only one (singular) special Lagrangian, rather than a finite union L1∪⋯∪LnL_{1}\cup\cdots\cup L_{n} of special Lagrangians with different phases, but otherwise it does not simplify things: we still expect nontrivial finite time singularities, and surgeries.

(iii) It is an interesting question whether there are useful extra assumptions on LL which limit the kinds of singularities occurring at the singular times T1,T2,….T_{1},T_{2},\ldots. For example, if LL is generic in its Hamiltonian isotopy class then only singularities of ‘index zero’ appear, as in Remark 3.8(iii), and if LL is almost calibrated, then as in §3.7 ‘collapsing zero objects’ cannot happen. Thomas and Yau give conditions [70, (7.1) or (7.2)] preventing ‘neck pinches’ in §3.5 dividing LL into two pieces L1∐L2L_{1}\amalg L_{2} from happening, although I expect other singularities can.

There are some very special situations in which Lagrangian MCF is known to exist for all time without singularities, such as the Lagrangian TmT^{m}-graphs in T2​mT^{2m} studied by Smoczyk and Wang [68], or Lagrangian MCF starting from a small perturbation of a smooth special Lagrangian. But apart from these, I do not know of any useful, nontrivial conditions on LL under which I expect the flow to exist for all time without singularities, as hoped for in [70, Conj. 7.3].

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