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Uniqueness of holomorphic quilts lifted from holomorphic bigons on surfaces

Zuyi Zhang

TL;DR

The paper proves the uniqueness of holomorphic quilts lifted from holomorphic bigons in a quilted Lagrangian Floer framework where figure-eight bubbling occurs during strip-shrinking. The main technique replaces the bounded-derivative argument of Wehrheim–Woodward with a strategy that excises bubbling neighborhoods and applies the Riemann mapping theorem to force identity of competing quilt maps, thereby enabling a combinatorial computation of boundary maps via immersed Lagrangian data on closed surfaces. It connects to broader themes in the Atiyah–Floer program and confirms a Bottman–Wehrheim/Cazassus–Herald–Kirk–Kotelskiy conjecture about figure-eight bubblings, providing multiple explicit examples. The results sharpen the link between disc-level and quilt-level pictures in immersed Floer theory and yield new instances of trivial quilted Floer homology in concrete geometric configurations.

Abstract

In the author's previous paper, the author constructed holomorphic quilts from the bigons of the Lagrangian Floer chain group after performing Lagrangian composition. This paper proves the uniqueness of such holomorphic quilts. As a consequence, it provides a combinatorial method for computing the boundary map of immersed Lagrangian Floer chain groups when the symplectic manifolds are closed surfaces. One outcome is the construction of many examples exhibiting figure eight bubbling, which also confirms a conjecture of Cazassus Herald Kirk Kotelskiy.

Uniqueness of holomorphic quilts lifted from holomorphic bigons on surfaces

TL;DR

The paper proves the uniqueness of holomorphic quilts lifted from holomorphic bigons in a quilted Lagrangian Floer framework where figure-eight bubbling occurs during strip-shrinking. The main technique replaces the bounded-derivative argument of Wehrheim–Woodward with a strategy that excises bubbling neighborhoods and applies the Riemann mapping theorem to force identity of competing quilt maps, thereby enabling a combinatorial computation of boundary maps via immersed Lagrangian data on closed surfaces. It connects to broader themes in the Atiyah–Floer program and confirms a Bottman–Wehrheim/Cazassus–Herald–Kirk–Kotelskiy conjecture about figure-eight bubblings, providing multiple explicit examples. The results sharpen the link between disc-level and quilt-level pictures in immersed Floer theory and yield new instances of trivial quilted Floer homology in concrete geometric configurations.

Abstract

In the author's previous paper, the author constructed holomorphic quilts from the bigons of the Lagrangian Floer chain group after performing Lagrangian composition. This paper proves the uniqueness of such holomorphic quilts. As a consequence, it provides a combinatorial method for computing the boundary map of immersed Lagrangian Floer chain groups when the symplectic manifolds are closed surfaces. One outcome is the construction of many examples exhibiting figure eight bubbling, which also confirms a conjecture of Cazassus Herald Kirk Kotelskiy.

Paper Structure

This paper contains 5 sections, 4 theorems, 24 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Figure-eight Bubblings and Strip-shrinking
  4. Uniqueness Of Holomorphic Quilts
  5. Examples of the Quilted Lagrangian Floer Theory

Key Result

Theorem 1.1

Let $(F_1.\omega_1)$ and $(F_2,\omega_2)$ be two symplectic closed surfaces equipped with compatible almost complex structures. Suppose $L_i\looparrowright F_i$$\mathrm{(}i=1,2\mathrm{)}$ and $F\looparrowright (F_1\times F_2,\omega_1\times(-\omega_2))$ are Lagrangian immersions such that $L_1\times is a holomorphic map such that Then there is a unique holomorphic map $\tilde{u}(x,y):\mathbb{R}\t

Figures (6)

  • Figure 1: Figur-eight bubbling.
  • Figure 2: Shrinking the lower layer.
  • Figure 3: The bottom-left and bottom-right genus two surfaces are denoted as $F_1$ and $F_2$, respectively. The red (blue) curve on the bottom-right (left) surface is the result of performing the Lagrangian composition of the red (blue) curve on the bottom-left (right) surface with the genus two surface $F$.
  • Figure 4: As $t$ tends to 1, the keyholes become smaller and smaller as shown from left to the right in the above picture.
  • Figure 5: The bottom-left and bottom-right genus-two surfaces are $F_1$ and $F_2$ respectively. The red (blue) curve on the bottom-right (left) surface is obtained by performing the Lagrangian composition of the red (blue) curve on the bottom-left (right) surface with the genus-two surface $F$.
  • ...and 1 more figures

Theorems & Definitions (29)

  • Theorem 1.1
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 19 more