Symplectic embeddings of four-dimensional polydisks into balls

Abstract

In this paper we obtain new obstructions to symplectic embeddings of the four-dimensional polydisk $P(a,1)$ into the ball $B(c)$ for $2\le a<\frac{\sqrt{7}-1}{\sqrt{7}-2}\approx 2.549$, extending work done by Hind-Lisi and Hutchings. Schlenk's folding construction permits us to conclude our bound on $c$ is optimal. Our proof makes use of the combinatorial criterion necessary for one "convex toric domain" to symplectically embed into another introduced by Hutchings in \cite{Beyond}. Additionally, we prove that if certain symplectic embeddings of four dimensional convex toric domains exist then a modified version of this criterion from \cite{Beyond} must hold, thereby reducing the computational complexity of the original criterion from $O(2^n)$ to $O(n^2)$.

31 pages, revised as suggested by referee and M. Hutchings

References in corpus (3)

• Symplectic embeddings into four-dimensional concave toric domains
• Beyond ECH capacities
• Combinatorial embedded contact homology for toric contact manifolds

Cited by in corpus (1)

• An elementary alternative to ECH capacities