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Symplectic embeddings of ellipsoids in dimension greater than four. (English) Zbl 1239.53107

Let \(E(a_1,\ldots,a_m)\) denote a symplectic ellipsoid in the standard symplectic \(\mathbb R^{2m}\). The authors provide a construction of an embedding between symplectic ellipsoids in higher dimension from a symplectic embedding in lower dimensions. More precisely, they prove that if there exists a symplectic embedding \(E(a_1,\dots,a_m)\hookrightarrow E(b_1,\dots,b_m)\) then there is a symplectic embedding \[ E(a_1,\dots,a_m,a_{m+1},\dots, a_n)\hookrightarrow E(b_1,\dots,b_m,a_{m+1},\dots,a_n). \]
In dimension six they study the function defined by \(f(a,b) = \inf \{c\,|\, E(1,a,b)\hookrightarrow B^6(c)\}\), where \(B^6(c)\) denote the standard symplectic ball of capacity \(c\). They obtain a set of results concerning the values of this function.
Finally, they prove that in all even dimensions bigger than four a symplectic ball can be fully filled by a discjoint collection of symplectically embedded small enough balls. Since a ball is the affine part of the complex projective space the corresponding result for \(\mathbb{CP}^n\), for \(n>2\) follows immediately.

MSC:

53D35 Global theory of symplectic and contact manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
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References:

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