Let be a Calabi–Yau -fold, and consider compact, graded Lagrangians in . Thomas and Yau conjectured that there should be a notion of 'stability' for such , and that if is stable then Lagrangian mean curvature flow with should exist for all time, and should be the unique special Lagrangian in the Hamiltonian isotopy class of . This paper is an attempt to update the Thomas–Yau conjectures, and discuss related issues.
It is a folklore conjecture, extending , that there exists a Bridgeland stability condition on the derived Fukaya category , such that an isomorphism class in is -semistable if (and possibly only if) it contains a special Lagrangian, which must then be unique.
In brief, we conjecture that if is an object in an enlarged version of , then there is a unique family such that , and in for all , and satisfies Lagrangian MCF with surgeries at singular times and in graded Lagrangian integral currents we have , where is a special Lagrangian integral current of phase for , and correspond to the decomposition of into -semistable objects.
We also give detailed conjectures on the nature of the singularities of Lagrangian MCF that occur at the finite singular times
Cite this article
Dominic Joyce, Conjectures on Bridgeland stability for Fukaya categories of Calabi–Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow. EMS Surv. Math. Sci. 2 (2015), no. 1, pp. 1–62DOI 10.4171/EMSS/8