Aspherical Lagrangian submanifolds, Audin's conjecture and cyclic dilations

2023/08/09 by Li, Yin · 1 citation
#FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2308.05086

Abstract

Given a closed, oriented Lagrangian submanifold L in a Liouville domain M, one can define a Maurer-Cartan element with respect to a certain L_∞-structure on the string homology \widehatH_∗^S1(LL;ℝ), completed with respect to the action filtration. When the first Gutt-Hutchings capacity of M is finite, and L is a K(π,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 ℂⁿ to a wide class of Liouville manifolds. In particular, when dim_ℝ(M)=6, every closed, orientable, prime Lagrangian 3-manifold L⊂M is diffeomorphic to a spherical space form, or S¹×Σ_g, where Σ_g is a closed oriented surface.

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