[Paper Review] Structure of $J$-holomorphic disks with immersed Lagrangian boundary conditions

Perrier Alexandre|arXiv (Cornell University)|Aug 6, 2018

Geometric and Algebraic Topology3 references4 citations

TL;DR

This paper generalizes Lazzarini's structural theorem for $J$-holomorphic disks to the case of immersed Lagrangian boundary conditions, proving that any finite-energy $J$-holomorphic disk with corners and boundary on an immersed Lagrangian factors through finitely many simple or multiply covered disks. The key contribution is a decomposition theorem enabling the use of time-independent almost complex structures in Floer homology computations for monotone Lagrangians with $N_L \geq 3$. This resolves transversality issues in Lagrangian intersection theory without requiring time-dependent almost complex structures.

ABSTRACT

We explain how to generalize Lazzarini's structural Theorem from [Laz11] to the case of curves with boundary on a given Lagrangian immersion. As a consequence of this result, we show that we can compute Floer homology with time-independent almost complex structures. We also give some applications as well as topics for future work.

Motivation & Objective

• To extend Lazzarini’s structural theorem for $J$-holomorphic disks to the case of immersed Lagrangian boundary conditions.
• To establish a decomposition of finite-energy $J$-holomorphic disks with corners and boundary on an immersed Lagrangian into finitely many simple or multiply covered disks.
• To show that Floer homology can be computed using time-independent almost complex structures under monotonicity and Maslov number conditions.
• To provide a foundation for regularity and transversality in Lagrangian intersection theory with immersed Lagrangians.

Proposed method

• Define $J$-holomorphic curves with corners and boundary on an immersed Lagrangian, incorporating corner points at double points of the immersion.
• Construct a $\mathcal{C}^1$-embedded graph (the 'frame') in the domain disk where the curve overlaps with its boundary, using asymptotic expansions near corners.
• Prove that the image of the curve decomposes as a union of images of finitely many simple or multiply covered $J$-holomorphic disks.
• Use a perturbation argument and Gromov compactness to analyze the limit of sequences of teardrop-like curves, showing convergence to broken strips.
• Apply combinatorial index arguments and regularity conditions to rule out complex broken configurations, proving that only simple or multiply covered curves can appear.
• Adapt Lazzarini’s proof strategy to the immersed case by carefully tracking branch points and corner structures.

Experimental results

Research questions

• RQ1 Can Lazzarini’s structural theorem for $J$-holomorphic disks be extended to the case of immersed Lagrangian boundary conditions?
• RQ2 Under what conditions can Floer homology be defined using time-independent almost complex structures for immersed Lagrangians?
• RQ3 What is the decomposition structure of finite-energy $J$-holomorphic disks with corners and boundary on an immersed Lagrangian?
• RQ4 How does the presence of transverse double points in the Lagrangian immersion affect the moduli space of $J$-holomorphic disks?
• RQ5 What topological and analytical constraints ensure that such disks are either simple or multiply covered?

Key findings

• Any non-constant finite-energy $J$-holomorphic disk with corners and boundary on an immersed Lagrangian factors through finitely many simple or multiply covered $J$-holomorphic disks.
• In complex dimension $n \geq 3$, for a generic time-independent almost complex structure $J$, every such disk is either simple or a branched cover of a simple disk.
• The image of the disk is the union of the images of the simple components, and in homology, the class of the original disk is the sum of the classes of the components with multiplicity.
• For two transverse monotone Lagrangians with $N_{L_1}, N_{L_2} \geq 3$, Floer homology is well-defined using time-independent $J$, bypassing the need for time-dependent perturbations.
• The moduli space of teardrop-like curves converges to a broken strip under Gromov convergence, and such broken strips must consist of a single simple curve due to index constraints.
• The tree of broken curves must be trivial (single vertex) due to index inequalities and the condition $n \geq 3$, ruling out nontrivial broken configurations.