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arXiv · 1610.00566

Symplectic embeddings of four-dimensional polydisks into balls

Katherine Christianson·Jo Nelson
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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 $2\leq a<\frac{\sqrt{7}-1} {\sqrt{7}-2} \approx 2.549$, 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 $O(n^2)$.

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Katherine Christianson, Jo Nelson. 2016-10-03. Symplectic embeddings of four-dimensional polydisks into balls. https://doi.org/10.2140/agt.2018.18.2151

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