The Hofer question on intermediate symplectic capacities

TL;DR

This paper resolves Hofer's question by showing certain symplectic embeddings are possible for R above a specific threshold, and concludes that no intermediate symplectic capacities exist between 1- and n-capacities.

Contribution

It proves the existence of symplectic embeddings for a class of cylinders and demonstrates the non-existence of intermediate capacities, answering longstanding questions in symplectic geometry.

Findings

Established explicit bounds for symplectic embeddings.

Proved the non-existence of intermediate capacities.

Connected results to previous work by Gromov and Guth.

Abstract

Roughly twenty five years ago Hofer asked: can the cylinder B^2(1) \times \mathbb{R}^{2(n-1)} be symplectically embedded into B^{2(n-1)}(R) \times \mathbb{R}^2 for some R>0? We show that this is the case if R \geq \sqrt{2^{n-1}+2^{n-2}-2}. We deduce that there are no intermediate capacities, between 1-capacities, first constructed by Gromov in 1985, and n-capacities, answering another question of Hofer. In 2008, Guth reached the same conclusion under the additional hypothesis that the intermediate capacities should satisfy the exhaustion property.