Aspherical Lagrangian submanifolds, Audin’s conjecture and cyclic dilations

Abstract Given a closed, oriented Lagrangian submanifold L in a Liouville domain $$\overline{M}$$ M ¯ , one can define a Maurer-Cartan element with respect to a certain $$L_\infty $$ L ∞ -structure on the string homology $$\widehat{H}_*^{S^1}(\mathcal {L}L;\mathbb {R})$$ H ^ ∗ S 1 ( L L ; R ) , completed with respect to the action filtration. When the first Gutt-Hutchings capacity [33] of $$\overline{M}$$ M ¯ is finite, and L is a $$K(\pi ,1)$$ K ( π , 1 ) space, we show that L bounds a pseudoholomorphic disc of Maslov index 2. This confirms a general form of Audin’s conjecture [5] and generalizes the works of Fukaya [20] and Irie [36] in the case of $$\mathbb {C}^n$$ C n to a wide class of Liouville manifolds, which includes low degree smooth affine hypersurfaces in $$\mathbb {C}^{n+1}$$ C n + 1 . In particular, when $$\dim _\mathbb {R}(\overline{M})=6$$ dim R ( M ¯ ) = 6 , every closed, orientable, prime Lagrangian 3-manifold $$L\subset \overline{M}$$ L ⊂ M ¯ is diffeomorphic either to a spherical space form, or $$S^1\times \Sigma _g$$ S 1 × Σ g , where $$\Sigma _g$$ Σ g is a closed oriented surface.