Aspherical Lagrangian submanifolds, Audin's conjecture and cyclic dilations

TL;DR

The paper proves a broad Audin-type constraint for closed Lagrangian submanifolds in Liouville manifolds with cyclic dilations by constructing an S^1-equivariant L-infinity structure on free loop space homology and identifying a Maurer–Cartan element whose consequences force a Maslov index 2 disc on L when L is a K(π,1). It develops a chain-level framework via de Rham models and a chain-level string bracket, then realizes this through a detailed moduli-space analysis using Kuranishi structures and Cohen–Ganatra interpolations to produce the required elements and identities. The results yield a generalization of Fukaya–Irie’s approach beyond C^n, apply to Milnor fibers, and yield corollaries including a 6-dimensional classification of prime Lagrangian 3-manifolds and obstructions to exact Lagrangian K(π,1) embeddings, tying in Gutt-Hutchings capacities and Viterbo-like functoriality in non-exact settings. The work highlights deep connections between cyclic dilations, symplectic cohomology, and chain-level algebra on loop spaces, with implications for non-formality and the topology of Lagrangian embeddings in a wide class of Liouville manifolds.

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 $\widehat{H}_\ast^{S^1}(\mathcal{L}L;\mathbb{R})$, completed with respect to the action filtration. When the first Gutt-Hutchings capacity of $\overline{M}$ is finite, and $L$ is a $K(π,1)$ 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 $\mathbb{C}^n$ 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 $S^1\timesΣ_g$, where $Σ_g$ is a closed oriented surface.

Figures (6)

• Figure 1: An element of the moduli space $\mathcal{R}_{3,\vartheta}$ (left) and its images under the map $\pi_{\vartheta,1}$ (upper right) and $\pi_{\vartheta,2}$ (lower right)
• Figure 2: An element of the moduli space $\overline{\mathcal{R}}_{8,\vartheta}$, which satisfies $\varphi_2=\mathrm{const}$, and its image under $\pi_{\vartheta,1}$, which gives an element in $\overline{\mathcal{R}}_7$.
• Figure 3: An element in the moduli space ${}_3\mathcal{R}_4^1$
• Figure 4: A cylinder with marked points $p_1$ and $p_2$ bubbles off at $\zeta$, which belongs to the codimension $1$ boundary stratum ${}_2\overline{\mathcal{M}}\times{}_1\overline{\mathcal{R}}_{4}^1$ of ${}_3\overline{\mathcal{R}}_{4}^1$.
• Figure 5: The definition of the auxiliary-rescaling map $\pi_f^0:{}_2\overline{\mathcal{R}}_{4,\tau_0}^1\rightarrow{}_2\overline{\mathcal{R}}_4^{S_{0,1}^1}$

Theorems & Definitions (106)

• Theorem 1: Fukaya kf, Irie ki2
• Definition 2
• Remark 3
• Proposition 4: $yl2$, Corollary 5.1
• Proposition 5
• Theorem 6
• Remark 7
• Corollary 8
• Remark 9
• Remark 10