Quantitative Results on Symplectic Barriers

TL;DR

This paper investigates quantitative symplectic barriers arising from removing codimension-two subspaces from the $2n$-dimensional ball and studies how different symplectic capacities respond. It shows that removing a single subspace with K\ähler angle $t$ reduces the complement’s size to $\frac{1+t}{2}$ across all normalized capacities, and that removing many such subspaces (with spacing $\varepsilon$) drives the size down toward $t$ as $\varepsilon\to0$, with precise asymptotics for $c_{HZ}$ and the cylindrical capacity, and near-optimal lower bounds for the Gromov width. The methods combine explicit four-dimensional embeddings, Hamiltonian displacement arguments, and capacity inequalities, with a key reduction to 4D geometry for the single-subspace case and product constructions for higher dimensions. The results illuminate how the presence and arrangement of symplectic barriers impose rigidity on embeddings and clarify the role of the K\ähler angle in controlling the symplectic size of the complements, having implications for non-squeezing phenomena and capacity comparisons across configurations.

Abstract

In this paper we present some quantitative results concerning symplectic barriers. In particular, we answer a question raised by Sackel, Song, Varolgunes, and Zhu regarding the symplectic size of the $2n$-dimensional Euclidean ball with a codimension-two linear subspace removed.

Figures (11)

• Figure 1: The projections of $P_t \cap B^4(1)$.
• Figure 2: The domain $S_h$ in blue, and the domain $R$ in purple. The domain in green is used to bound the area of $S_h$ from below.
• Figure 3: The map $\varphi$ from $D(\frac{1+t}{2})$ into $R$.
• Figure 4: An illustration of the area preserving map $\psi_2$ which "pushes" ${\bf K}$ into the set $\{ \pi|z_1|^2 > t \} \cup \{|x_1| > \delta \ {\rm and} \ |y_1| < \delta \}$.
• Figure 5: Pushing the hyperplanes arbitrarily close to the boundary.

Theorems & Definitions (39)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof