Aspherical Lagrangian submanifolds, Audin's conjecture and cyclic dilations
Given a closed, oriented Lagrangian submanifold $L$ in a Liouville domain $\overline{M}$, one can define a Maurer-Cartan element with respect to a certain $L_\infty$-structure on the string homology $\widehat{H}_\ast^{S^1}(\mathcal{L}L;\mathbb{R})$, completed with respect to the action filtration. When the first Gutt-Hutchings capacity of $\overline{M}$ is finite, and $L$ is a $K(\pi,1)$ space, it leads to interesting geometric implications. In particular, we show that $L$ bounds a non-constant pseudoholomorphic disc of Maslov index 2. This confirms a general form of Audin's conjecture and generalizes the works of Fukaya and Irie in the case of $\mathbb{C}^n$ to a wide class of Liouville manifolds. In particular, when $\dim_\mathbb{R}(\overline{M})=6$, every closed, orientable, prime Lagrangian 3-manifold $L\subset\overline{M}$ is diffeomorphic to a spherical space form, or $S^1\times\Sigma_g$, where $\Sigma_g$ is a closed oriented surface.