Symplectic embeddings of 4-dimensional ellipsoids

TL;DR

This paper translates the problem of symplectically embedding 4D ellipsoids into balls into a problem of embedding disjoint balls into blow-ups of  P^2, providing new solutions for specific cases and answering a longstanding question.

Contribution

It reduces the symplectic embedding problem of 4D ellipsoids to a ball packing problem in blow-ups of  P^2, enabling solutions for certain cases and addressing a question by Hofer.

Findings

Ellipsoid E(1,k) can be fully embedded into a ball for k=1,4 and all k.

The embedding problem reduces to known ball packing problems.

Provides new solutions and answers to longstanding questions.

Abstract

We show how to reduce the problem of symplectically embedding one 4-dimensional rational ellipsoid into another to a problem of embedding disjoint unions of balls into appropriate blow ups of \C P^2. For example, the problem of embedding the ellipsoid E(1,k) into a ball B is equivalent to that of embedding k disjoint equal balls into \C P^2, and so can be solved by the work of Gromov, McDuff--Polterovich and Biran. (Here k is the ratio of the area of the major axis to that of the minor axis.) As a consequence we show that the ball may be fully filled by the ellipsoid E(1,k) for k=1,4 and all k\ge 9, thus answering a question raised by Hofer.