We study the rigidity and flexibility of symplectic embeddings in the model case in which the domain is a symplectic ellipsoid. It is first proved that under the conditionr n 2 ≤2r 1 2 the symplectic ellipsoidE(r 1,…,r n)with radiir 1≤…≤r ndoes not symplectically embed into a ball of radius strictly smaller thanr n.We then use symplectic folding to see that this condition is sharp. We finally sketch a proof of the fact that any connected symplectic 4-manifold of finite volume can be asymptotically filled with skinny ellipoids.
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