Aspherical Lagrangian submanifolds, Audin's conjecture and cyclic dilations

Abstract

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 ${\hat{H}}_{\ast }^{S^1}(\mathcal{L}L;\mathbb{R})$, 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, we show that $L$ bounds a 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, which includes low degree smooth affine hypersurfaces in ${\mathbb{C}}^{n+1}$. In particular, when ${\dim }_{\mathbb{R}}(\overline{M})=6$, every closed, orientable, prime Lagrangian 3-manifold $L\subset \overline{M}$ is diffeomorphic either to a spherical space form, or $S^1\times {\Sigma }_g$, where ${\Sigma }_g$ is a closed oriented surface.

80 pages, 5 figures. v6: minor correction. To appear in Selecta Mathematica