Fix a Calabi-Yau manifold $X$ with a holomorphic $(n,0)$-form $\Omega$. In [Th] a stability condition for Lagrangians was described conjectured to be equivalent to the existence of a special Lagrangian (SLag) in the hamiltonian deformation class of a fixed Lagrangian. This was motivated by an infinite dimensional set-up in which $U(1)$ gauge transformations act on the (infinite dimensional) space of Lagrangians (in a Calabi-Yau $n$-fold $X$) with flat $U(1)$ connections on them. There is a natural complex structure and symplectic form on this space and, ignoring issues of integrability of these structures (discussed in [Th]), the formal complexification of the $U(1)$ gauge transformations gives hamiltonian deformations of the Lagrangian, with moment map the $n$-form $\,\mathrm{Im}\,\Omega\arrowvert_{L}$. The stability condition was motivated by an example of Joyce and the ‘angle criterion’, in terms of splittings of the Lagrangian into Seidel’s graded Lagrangian connect sums (as defined in Section 3 below) and family versions thereof, with a certain phase inequality. This led to a conjecture, a sort of globalised version of the angle criterion [L], [N], that the hamiltonian deformation class of a Lagrangian should contain a SLag if and only if the Lagrangian is stable; this SLag representative should then be unique. Here we expand on the conjecture and relate it to mean curvature flow. It was verified for the simplest case $T^{2}$ in [Th]; here we prove it in a series of $n$-dimensional examples with symmetry (Theorem 7.6), and prove uniqueness of SLags in hamiltonian deformation classes whose Floer cohomology [FO3] is defined (Theorem 4.3).

Some notation: we write $\approx$ for “in the same hamiltonian deformation class as”, and use $\,\widetilde{\ }\,$ for the isomorphism $T^{*}L\to TL$ induced by the metric on a Riemannian manifold $L$. Restricting the Ricci-flat metric on a Calabi-Yau manifold $(X,\Omega)$ to a Lagrangian submanifold $L$ we get an induced volume form $\vol$ on $L$, and

$$\Omega\arrowvert_{L}=e^{i\theta}\vol$$

(1.1)

defines an $S^{1}$-valued function $\theta$ on $L$, the phase function of $L$. A grading of $L$ is a lift of $\theta$ to a real valued function. By Lagrangian we will always mean graded Lagrangian (thus the Maslov class of the Lagrangian, which is the class of $d\theta$ in $H^{1}(L;\mathbb{R})$, is assumed to vanish, and we have chosen a lift of $\theta$). $L$ is special Lagrangian (SLag) if $\theta$ is a constant; equivalently, with respect to a suitable phase rotation of $\Omega$, $\,\mathrm{Im}\,\Omega\arrowvert_{L}\equiv 0$. An average, cohomological, measure of the phase of a homology class $[L]$ is given by taking the phase of the complex number $\int_{L}\Omega$; since $L$ is graded this lifts naturally to give a real number $\phi(L)$, which is the phase of any SLag in the same homology class.

We should point out that as in [Th], we do not fully understand the role of holomorphic discs in the theory. These are of course crucial in the definition and hamiltonian deformation invariance of Floer cohomology; until this is fully set up [FO3] and all of its expected properties (such as the spectral sequences of [Oh2] and [P]) are proved and extended to the Calabi-Yau case, some of the arguments below are necessarily conjectural. We also deal exclusively with smooth (S)Lags; how to modify our constructions to include singularities is an important question. Using only (family) Lagrangian connect sums as the degenerations necessary to describe stability of Lagrangians is also probably too restrictive, studying other singularities and splittings may also be necessary.