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 , starting from an almost calibrated Lagrangian generic in its Hamiltonian isotopy class.
(ii) Assuming the initial object is semistable or stable in the sense of Conjectures 3.2 and 3.5 means in part (e) that the limit is only one (singular) special Lagrangian, rather than a finite union 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 which limit the kinds of singularities occurring at the singular times For example, if is generic in its Hamiltonian isotopy class then only singularities of ‘index zero’ appear, as in Remark 3.8(iii), and if 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 into two pieces 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 -graphs in 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 under which I expect the flow to exist for all time without singularities, as hoped for in [70, Conj. 7.3].