Aspherical Lagrangian submanifolds, Audin’s conjecture and cyclic dilations

Given a closed, oriented Lagrangian submanifold L in a Liouville domain M̄ M ¯ , one can define a Maurer-Cartan element with respect to a certain L_∞ L ∞ -structure on the string homology H_*S¹(LL;R) H ^ ∗ S 1 ( L L ; R ) , completed with respect to the action filtration. When the first Gutt-Hutchings capacity [33] of M̄ M ¯ is finite, and L is a K(π ,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 Cⁿ C n to a wide class of Liouville manifolds, which includes low degree smooth affine hypersurfaces in Cⁿ⁺¹ C n + 1 . In particular, when R(M̄)=6 dim R ( M ¯ ) = 6 , every closed, orientable, prime Lagrangian 3-manifold L⊂ M̄ L ⊂ M ¯ is diffeomorphic either to a spherical space form, or S¹× Σ g S 1 × Σ g , where Σ g Σ g is a closed oriented surface.