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A JSJ-type decomposition theorem for symplectic fillings

Austin Christian, Michael Menke

TL;DR

The paper proves a JSJ-type decomposition for exact and weak symplectic fillings by splitting along a mixed torus, showing that any filling arises from a filling of the split boundary via round symplectic 1-handle attachment. The core method uses a holomorphic cylinder analysis in the symplectic completion to produce a canonical solid-torus decomposition, paired with Avdek’s Liouville-hypersurface gluing to reconstruct the original filling. A key application shows that Legendrian surgery along a knot stabilized both positively and negatively yields a contact manifold with a unique exact (thus weak) filling, by reducing to a unique filling of $(S^3,\xi_{std})$ and translating through the round-handle operation. This provides new structural insight into how toroidal decompositions govern the filling landscape and extends classical results on unique fillings for standard contact 3-manifolds.

Abstract

We establish a JSJ-type decomposition theorem for splitting exact symplectic fillings of contact 3-manifolds along \emph{mixed tori} -- these are convex tori satisfying a particular geometric condition. As an application, we show that if $(M,ξ)$ is obtained from $(S^3,ξ_{\mathrm{std}})$ via Legendrian surgery along a knot which has been stabilized both positively and negatively, then $(M,ξ)$ has a unique exact filling.

A JSJ-type decomposition theorem for symplectic fillings

TL;DR

The paper proves a JSJ-type decomposition for exact and weak symplectic fillings by splitting along a mixed torus, showing that any filling arises from a filling of the split boundary via round symplectic 1-handle attachment. The core method uses a holomorphic cylinder analysis in the symplectic completion to produce a canonical solid-torus decomposition, paired with Avdek’s Liouville-hypersurface gluing to reconstruct the original filling. A key application shows that Legendrian surgery along a knot stabilized both positively and negatively yields a contact manifold with a unique exact (thus weak) filling, by reducing to a unique filling of and translating through the round-handle operation. This provides new structural insight into how toroidal decompositions govern the filling landscape and extends classical results on unique fillings for standard contact 3-manifolds.

Abstract

We establish a JSJ-type decomposition theorem for splitting exact symplectic fillings of contact 3-manifolds along \emph{mixed tori} -- these are convex tori satisfying a particular geometric condition. As an application, we show that if is obtained from via Legendrian surgery along a knot which has been stabilized both positively and negatively, then has a unique exact filling.

Paper Structure

This paper contains 13 sections, 15 theorems, 79 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Contact geometry preliminaries
  4. Bypasses
  5. Basic slices
  6. Contact handles
  7. Sutured contact manifolds
  8. Legendrian surgery
  9. Symplectization
  10. Liouville hypersurfaces and convex gluing
  11. The role of slope in splitting along a torus
  12. Proof of Theorem \ref{['thm:main-thm']}
  13. Proof of Theorem \ref{['thm:twice-stabilized']}

Key Result

Theorem 1.1

Let $(M,\xi)$ be a closed, cooriented 3-dimensional contact manifold and let $(W,\omega)$ be an exact (respectively, weak) symplectic filling of $(M,\xi)$. If there exists a mixed torus $T \subset (M,\xi)$, witnessed by a neighborhood $T^2\times[0,2]$ with slopes $s_0=-1$, $s_1=\infty$, and $s_2$, t

Figures (6)

  • Figure 1: Stabilizations
  • Figure 2: On the left is the dividing set of $\Sigma$ with solid attaching arc $a$. On the right is the result of bypass attachment.
  • Figure 3: The sutured neighborhood constructed in Lemma \ref{['lemma:convex-torus-nbhd']}. The keyhole-shaped regions near $e_2$ result from applying a concave-to-convex modification on a Reeb flowbox, each containing a hyperbolic orbit parallel to $e_2$.
  • Figure 4: A schematic for a standard neighborhood of a mixed torus $T^2\times \{0\}$. The surfaces identified in Lemma \ref{['lemma:bypasschange']} are labeled on the left, while their dividing sets are labeled (left-to-right) on the right. A concave-to-convex modification is carried out in the blue keyhole-shaped regions, producing the blue hyperbolic Reeb orbits.
  • Figure 5: Contact 1-handle attachment may be carried out via the gluing of (convex) sutured contact manifolds, as described in colin2011sutures. Each of the surfaces above is assumed to be sutured along its dividing set; while the components of the dividing set are labeled according to parallel Reeb orbits, they are not taken to be Reeb themselves.
  • ...and 1 more figures

Theorems & Definitions (34)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Remark 2.1
  • Theorem 2.2: honda2000classification
  • Remark 2.3
  • Remark 2.4
  • ...and 24 more