Symplectic embeddings of products

TL;DR

This paper extends the understanding of symplectic embeddings from four-dimensional cases to higher dimensions by demonstrating that the 'infinite staircase' phenomenon persists under stabilization, revealing deeper geometric structures.

Contribution

It generalizes the known four-dimensional embedding results to higher dimensions, showing the stability of the Fibonacci-based 'infinite staircase' pattern.

Findings

The 'infinite staircase' phenomenon persists in higher dimensions under stabilization.

Symplectic embedding criteria are extended beyond four dimensions.

Fibonacci numbers continue to play a role in embedding obstructions.

Abstract

McDuff and Schlenk determined when a four-dimensional ellipsoid can be symplectically embedded into a four-dimensional ball, and found that when the ellipsoid is close to round, the answer is given by an "infinite staircase" determined by the odd-index Fibonacci numbers. We show that this result still holds in higher dimensions when we "stabilize" the embedding problem.