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Symplectic embeddings of 4-dimensional ellipsoids. (English) Zbl 1166.53051

J. Topol. 2, No. 1, 1-22 (2009); corrigendum ibid. 8, No. 4, 1119-1122 (2015).
For \(m\leq n\) consider the 4-dimensional symplectic ellipsoid
\[ E(m,n)= \bigg\{(x_1,y_1,x_2,y_2)\in{\mathbb R}^4\;\bigg|\;\frac{x_1^2+y_1^2}{m}+\frac{x_2^2+y_2^2}{n}\leq1\bigg\} \]
and denote by \(\mathring {E}(m,n)\) the corresponding open ellipsoid. One of the main theorems of the paper under review is that \(E(1,k)\) symplectically embeds in the open ball \(\mathring {E}(\mu,\mu)\) if and only if \(k\) disjoint balls \(E(1,1)\) embed in \(\mathring {E}(\mu,\mu)\). By combining this with results of M. Gromov [Invent. Math. 82, 307–347 (1985; Zbl 0592.53025)], D. McDuff and L. Polterovich [Invent. Math. 115, No. 3, 405–429 (1994; Zbl 0833.53028)], and P. Biran [Geom. Funct. Anal. 7, No. 3, 420–437 (1997; Zbl 0892.53022)], one obtains that \(E(1,k)\) fully fills the ball \(E(\sqrt{k},\sqrt{k})\) if and only if \(k=1,4\) or \(k\geq 9\).
Another theorem proved in the paper provides for any positive integers \(m,n,m',n'\) and \(\lambda>0\), an integer \(k\) and weights \(\underline{w}_\lambda=(w_1,\dots,w_k)\) such that the problem of embedding of \(E(\lambda m,\lambda n)\) into \(\mathring {E}(m',n')\) is equivalent to the question of whether the \(k\) disjoint balls \(E(w_1,w_1),\dots,E(w_k,w_k)\) embed into the open ball \(\mathring {E}(1,1)\).

MSC:

53D05 Symplectic manifolds (general theory)
57R17 Symplectic and contact topology in high or arbitrary dimension
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References:

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