On the Existence of Symplectic Barriers

TL;DR

This paper proves the existence of symplectic barriers, which are obstructions preventing certain symplectic embeddings, specifically showing that embedded Euclidean balls must intersect a grid of symplectic planes.

Contribution

It introduces the concept of symplectic barriers, demonstrating a new rigidity phenomenon in symplectic embeddings related to obligatory intersections with symplectic planes.

Findings

Embedded Euclidean balls must intersect a grid of symplectic planes.

Symplectic barriers act as obstructions to certain embeddings.

The results reveal a new form of rigidity in symplectic geometry.

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.