Higher Symplectic Capacities and the Stabilized Embedding Problem for Integral Ellipsoids

TL;DR

This paper applies higher symplectic capacities to solve the stabilized embedding problem for integral ellipsoids, revealing new embedding conditions based on eccentricity parity and recovering known results in special cases.

Contribution

It introduces the use of higher symplectic capacities to address the stabilized embedding problem for integral ellipsoids, extending previous results and confirming conjectures in specific scenarios.

Findings

Solved the stabilized embedding problem for integral ellipsoids with certain eccentricity conditions

Constructed explicit embeddings that are not always optimal in some cases

Reproduced known results for the ball case and polydiscs

Abstract

The third named author has been developing a theory of "higher" symplectic capacities. These capacities are invariant under taking products, and so are well-suited for studying the stabilized embedding problem. The aim of this note is to apply this theory, assuming its expected properties, to solve the stabilized embedding problem for integral ellipsoids, when the eccentricity of the domain has the opposite parity of the eccentricity of the target and the target is not a ball. For the other parity, the embedding we construct is definitely not always optimal; also, in the ball case, our methods recover previous results of McDuff, and of the second named author and Kerman. There is a similar story, with no condition on the eccentricity of the target, when the target is a polydisc: a special case of this implies a conjecture of the first named author, Frenkel, and Schlenk concerning the rescaled polydisc limit function. Some related aspects of the stabilized embedding problem and some open questions are also discussed.