Torus bundle Liouville domains are stably Weinstein

TL;DR

This work develops explicit local Liouville homotopies—notably box folds, chimney folds, and a blocking apparatus—that tame chaotic Liouville vector fields by creating Morse-like dynamics and tightly controlling holonomy. By embedding these operations into standard stabilized regions and combining them with box-hole perturbations, the authors prove that Huang’s torus-bundle Liouville domains become Weinstein after a single stabilization, i.e., are stably Weinstein. The approach provides a constructive, hands-on toolkit for transforming Liouville domains with simple topology into Weinstein domains, and it suggests broader applicability to other globally transversal settings. The results advance understanding of the gap between Liouville and Weinstein dynamics and offer a concrete path toward stabilizing wild dynamics in symplectic topology.

Abstract

We develop explicit local operations that may be applied to Liouville domains, with the goal of simplifying the dynamics of the Liouville vector field. These local operations, which are Liouville homotopies, are inspired by the techniques used by Honda and Huang in [HH19] to show that convex hypersurfaces are $C^0$-generic in contact manifolds. As an application, we use our operations to show that certain Liouville-but-not-Weinstein domains constructed by Huang in [Hua20] are stably Weinstein.

Figures (23)

• Figure 1: A box fold is installed in 2D by identifying a region as on the left and replacing it (via a Liouville homotopy) with a region as on the right.
• Figure 2: A Liouville structure on $S^1\times[-1,1]$, before and after the installation of a box fold. In both figures, the vertical edges are identified, and the skeleton is $S^1\times\{0\}$.
• Figure 3: A schematic for blocking apparatus installation. On the left, the set $K$ represents a cross section of flowlines we wish to trap; the $s$- and $\mathbb{D}^2$-directions are projected out of the picture. On the right, two projections of blocking apparatus profiles are given by the colored boxes. The dashed red path indicates a hypothetical problematic flowline that does not occur thanks to \ref{['prop:mainprop:t-holo-part']}.
• Figure 4: A box fold with $z_0 < t_0$ and the three qualitative types of flowlines entering the fold according to Lemma \ref{['lemma:PL_box_fold_holonomy_low']}. The flowline in green is trapped in backward time, because it spirals around $\underline{t=t_0}\cap \underline{z=z_0}$ via the faces $\underline{s=s_0} \to \underline{z=z_0}\to \underline{s=0}\to \underline{t=t_0} \to \underline{s=s_0}$. The flowline in red passes through the fold, and its holonomy is given by a shift in the $t$-direction. The flowline in blue also passes through the fold, but its holonomy is given by a scaling in the $t$-direction.
• Figure 5: A head-on view of two different box folds, one with $z_0 < t_0$ (left) and $z_0 > t_0$ (right). This is a visual depiction of Lemma \ref{['lemma:PL_box_fold_holonomy_low']}. All of the dashed lines on $\underline{s=s_0}$ have $(t,z)$ slope $-e^{s_0}$ and all of the solid lines on $\underline{s=0}$ have $(t,z)$ slope $-1$. The pink flowlines represent the lower threshold of the trapping region. Observe on the right picture that increasing $z_0$ beyond $t_0$ (with $s_0$ fixed) does not increase the size of the trapping region.

Theorems & Definitions (113)

• Proposition 1.1
• Definition 1.4
• Remark 1.5
• Definition 1.6
• Remark 1.7
• Definition 1.8
• Theorem 1.9
• Theorem 1.10
• Definition 2.1: Symplectization and contactization
• Remark 2.2