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

Abstract

Let $M$ be a Calabi-Yau $m$-fold, and consider compact, graded Lagrangians $L$ in $M$. Thomas and Yau math.DG/0104196, math.DG/0104197 conjectured that there should be a notion of "stability" for such $L$, and that if $L$ is stable then Lagrangian mean curvature flow $\{L^t:t\in[0,\infty)\}$ with $L^0=L$ should exist for all time, and $L^\infty=\lim_{t\to\infty}L^t$ 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 that there exists a Bridgeland stability condition $(Z,\mathcal P)$ on the derived Fukaya category $D^b\mathcal F(M)$ of $M$, such that an isomorphism class in $D^b\mathcal F(M)$ is $(Z,\mathcal P)$-semistable if (and possibly only if) it contains a special Lagrangian, which must then be unique. We conjecture that if $(L,E,b)$ is an object in an enlarged version of $D^b\mathcal F(M)$, where $L$ is a compact, graded Lagrangian in $M$ (possibly immersed, or with "stable singularities"), $E\to M$ a rank one local system, and $b$ a bounding cochain for $(L,E)$ in Lagrangian Floer cohomology, then there is a unique family $\{(L^t,E^t,b^t):t\in[0,\infty)\}$ such that $(L^0,E^0,b^0)=(L,E,b)$, and $(L^t,E^t,b^t)\cong(L,E,b)$ in $D^b\mathcal F(M)$ for all $t$, and $\{L^t:t\in[0,\infty)\}$ satisfies Lagrangian MCF with surgeries at singular times $T_1,T_2,\dots,$ and in graded Lagrangian integral currents we have $\lim_{t\to\infty}L^t=L_1+\cdots+L_n$, where $L_j$ is a special Lagrangian integral current of phase $e^{i\pi\phi_j}$ for $\phi_1>\cdots>\phi_n$, and $(L_1,\phi_1),\ldots,(L_n,\phi_n)$ correspond to the decomposition of $(L,E,b)$ into $(Z,\mathcal P)$-semistable objects. We also give detailed conjectures on the nature of the singularities of Lagrangian MCF that occur at the finite singular times $T_1,T_2,\ldots.$