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Symplectic embeddings of 4-dimensional ellipsoids into polydiscs
Auteur(s)
Stylianou, Dorothee Cosima
Editeur(s)
Date de parution
2012
Résumé
Recently, McDuff and Schlenk determined in [19] the function $c_{EB}(a)$, whose value at $a$ is the infimum of the size of a $4$-ball, into which the ellipsoid $E(1,a)$ symplectically embeds (here, $a > 1$ is the ratio of the area of the large axis to that of the smaller axis of the ellipsoid). This work is focused on the study of embeddings of $4$-dimensional ellipsoids $E(1,a)$ into four-dimensional polydiscs $P(\mu,\mu) = D^2(\mu) \times D^2(\mu)$, where $D^2(\mu)$ denotes the disc in $\mathbb{R}^2$ of area $\mu$, for $a\in [1,\sigma^2]$, where $\sigma^{2}=3+2\sqrt{2}\approx5.83$ is the square of the silver ratio $\sigma:=1+\sqrt{2}$. The embedding capacity function given by $c_{EC}(a):=\inf\left\{\mu \ \big\vert \ E(1,a) \ \rightarrow \ P(\mu,\mu)\right\}$ is defined for $a\in [1,\infty)$. As in the case of embeddings into balls, the structure of the graph of $c_{EC}(a)$ is very rich: since symplectic embeddings are volume preserving, we always have $c_{EC}(a) \geq \sqrt{\frac{a}{2}}$ and it is not hard to see that this lower bound is sharp for $a\ge 8$, that is the function $c_{EC}(a)$ is equal to the volume constraint given by $\sqrt{\frac{a}{2}}$ for $a\geq 8$. For $a$ less than the square of the silver ratio $\sigma:= 1 + \sqrt{2}$, the function $c_{EC}(a)$ turns out to be piecewise linear, having the form of a stair with an infinite number of steps, converging to $\sqrt{\frac{\sigma^2}{2}}$ for $a \rightarrow \sigma^2$. These ``stairs'' will be determined by the Pell numbers and we thus refer to them as the ``Pell stairs''. For the proof, we first translate the embedding problem $E(1, a) \rightarrow P(\mu,\mu)$ to a certain ball packing problem of the ball $B(2\mu)$. This embedding problem is then solved by adapting the method from McDuff and Schlenk in [19], which finds all exceptional spheres in blow-ups of the complex projective plane that provide an embedding obstruction. Furthermore we also prove the equivalence of symplectic embeddings of an ellipsoid into a polydisc and the embedding of its decomposition into disjoint balls into a ball, that is $E(a,b)\rightarrow P(c,d) \Leftrightarrow B(a,b)\sqcup B(c)\sqcup B(d)\rightarrow B(c+d)$.
Notes
Thèse de doctorat : Université de Neuchâtel, 2012
Identifiants
Type de publication
doctoral thesis
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