Symplectic embeddings of ellipsoids in dimension greater than four

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 $6$, 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 $E\left(1,\dots ,a\right)$. 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 $4$.