On the Existence of Symplectic Barriers

Abstract

In this note we establish the existence of a new type of rigidity of symplectic embeddings coming from obligatory intersections with symplectic planes. More precisely, we prove that if a Euclidean ball is symplectically embedded in the Euclidean unit ball, then it must intersect a sufficiently fine grid of two-codimensional pairwise disjoint symplectic planes. Inspired by analogous terminology for Lagrangian submanifolds, we refer to these obstructions as symplectic barriers.

Figures (3)

• Figure 1: Illustration of the map $\varphi$.
• Figure 2: An illustration of the vector field constructed in the proof of Lemma \ref{['lemma_extension']}.
• Figure 3: A 2-dimensional example for the approximated set $D_{\varepsilon}$.

Theorems & Definitions (11)

• Definition 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Proposition 1.6
• Remark 1.7
• Lemma 2.1
• proof : Proof of Theorem \ref{['embed_thm']}
• Example 3.1
• proof : Proof of Proposition \ref{['prop-complex-hyperplane']}