Symplectic embeddings of ellipsoids in dimension greater than four

TL;DR

This paper explores symplectic embeddings of ellipsoids into higher-dimensional balls, demonstrating flexibility under certain conditions and generalizing known results about ball packings in symplectic geometry.

Contribution

It introduces a method to suspend embeddings to higher dimensions and proves new flexibility results for ellipsoids in dimensions greater than four.

Findings

Ellipsoids with large axis ratios are flexible in dimension 6.

Sufficiently thin ellipsoids can be embedded in higher dimensions.

Any large enough number of small identical balls can fill a higher-dimensional ball.

Abstract

We study symplectic embeddings of ellipsoids into balls. In the main construction, we show that a given embedding of 2m-dimensional ellipsoids can be suspended to embeddings of ellipsoids in any higher dimension. In dimension 6,s if the ratio of the areas of any two axes is sufficiently large then the ellipsoid is flexible in the sense that it fully fills a ball. We also show that the same property holds in all dimensions for sufficiently thin ellipsoids E(1,..., a). A consequence of our study is that in arbitrary dimension a ball can be fully filled by any sufficiently large number of identical smaller balls, thus generalizing a result of Biran valid in dimension 4.