# Conjectures on Bridgeland stability for Fukaya categories of Calabi–Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow

### Dominic Joyce

Oxford University, UK

## Abstract

Let $(M,J,g,Ω)$ be a Calabi–Yau $m$-fold, and consider compact, graded Lagrangians $L$ in $M$. Thomas and Yau conjectured that there should be a notion of 'stability' for such $L$, and that if $L$ is stable then Lagrangian mean curvature flow ${L_{t}:t∈[0,∞)}$ with $L_{0}=L$ should exist for all time, and $L_{∞}=lim_{t→∞}L_{t}$ should be the unique special Lagrangian in the Hamiltonian isotopy class of $L$. This paper is an attempt to update the Thomas–Yau conjectures, and discuss related issues.

It is a folklore conjecture, extending [81], that there exists a Bridgeland stability condition $(Z,P)$ on the derived Fukaya category $D_{b}F(M)$, such that an isomorphism class in $D_{b}F(M)$ is $(Z,P)$-semistable if (and possibly only if) it contains a special Lagrangian, which must then be unique.

In brief, we conjecture that if $(L,E,b)$ is an object in an enlarged version of $D_{b}F(M)$, then there is a unique family ${(L_{t},E_{t},b_{t}):t∈[0,∞)}$ such that $(L_{0},E_{0},b_{0})=(L,E,b)$, and $(L_{t},E_{t},b_{t})≅(L,E,b)$ in $D_{b}F(M)$ for all $t$, and ${L_{t}:t∈[0,∞)}$ satisfies Lagrangian MCF *with surgeries* at singular times $T_{1},T_{2},…,$ and in graded Lagrangian integral currents we have $lim_{t→∞}L_{t}=L_{1}+⋯+L_{n}$, where $L_{j}$ is a special Lagrangian integral current of phase $e_{iπϕ_{j}}$ for $ϕ_{1}>⋯>ϕ_{n}$, and $(L_{1},ϕ_{1}),…,(L_{n},ϕ_{n})$ correspond to the decomposition of $(L,E,b)$ into $(Z,P)$-semistable objects.

We also give detailed conjectures on the nature of the singularities of Lagrangian MCF that occur at the finite singular times $T_{1},T_{2},….$

## 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. 1–62

DOI 10.4171/EMSS/8