On symplectic folding

TL;DR

This paper investigates the conditions under which symplectic shapes can be embedded into balls and cubes, establishing sharp criteria and constructing nearly optimal embeddings using symplectic folding, with implications for filling symplectic manifolds.

Contribution

It proves a sharp condition for embedding symplectic ellipsoids into balls and introduces symplectic folding techniques for near-optimal embeddings.

Findings

The condition $r_n^2 \u2264 2 r_1^2$ is necessary for embedding ellipsoids into smaller balls.

Symplectic folding enables construction of nearly optimal embeddings of ellipsoids and polydiscs.

Any finite-volume symplectic manifold can be asymptotically filled with skinny ellipsoids or polydiscs.

Abstract

We study the rigidity and flexibility of symplectic embeddings of simple shapes. It is first proved that under the condition $r_n^2 \le 2 r_1^2$ the symplectic ellipsoid $E(r_1, ..., r_n)$ with radii $r_1 \le ... \le r_n$ does not embed in a ball of radius strictly smaller than $r_n$. We then use symplectic folding to see that this condition is sharp and to construct some nearly optimal embeddings of ellipsoids and polydiscs into balls and cubes. It is finally shown that any connected symplectic manifold of finite volume may be asymptotically filled with skinny ellipsoids or polydiscs.