A Symplectically Non-Squeezable Small Set and the Regular Coisotropic Capacity
A Symplectically Non-Squeezable Small Set and the Regular Coisotropic Capacity
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辛不可压缩小集与正则各向同性容量
DOI：
10.4310/jsg.2013.v11.n4.a1
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
F. Ziltener
中科院分区：
文献类型：
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作者：
J. Swoboda;F. Ziltener
我们证明，对于$n\geq2$，在半径为$\sqrt{2}$的$R^{2n}$中存在闭球的紧子集$X$，使得$X$具有豪斯多夫维数$n$并且不会辛嵌入到标准开辛圆柱中。第二个主要结果是第 $d$ 个正则各向同性容量的下界，该容量急剧上升至 3 倍。对于几何有界非球面辛流形的开子集，此容量是其位移能量的下界。结果的证明涉及线性空间的某个拉格朗日子流形，这是由M. Audin和L. Polterovich考虑的。
We prove that for $n\geq2$ there exists a compact subset $X$ of the closed ball in $R^{2n}$ of radius $\sqrt{2}$, such that $X$ has Hausdorff dimension $n$ and does not symplectically embed into the standard open symplectic cylinder. The second main result is a lower bound on the $d$-th regular coisotropic capacity, which is sharp up to a factor of 3. For an open subset of a geometrically bounded, aspherical symplectic manifold, this capacity is a lower bound on its displacement energy. The proofs of the results involve a certain Lagrangian submanifold of linear space, which was considered by M. Audin and L. Polterovich.