论文标题
生成集合和符号能力的表示
Generating sets and representability for symplectic capacities
论文作者
论文摘要
K. Cieliebak，H。Hofer，J。Latschev和F. Schlenk（CHLS）提出了在给定的符号类别上找到（符号）能力的最小生成集的问题。我们表明，如果该类别包含一定的一个参数的对象家族，那么每一个伯爵生成的（归一化）容量集的基础性具有比连续体更大的（严格）。这似乎是CHL问题的第一个结果，除了D. McDuff的两个结果，关于维度4中的椭圆形类别。 我们还证明，只要该类别包含一个单参数的符号歧管族，在给定的符合性类别上的每一个有限的产生能力集都是不可数的，而``严格的数量增强''因此，Ekeland-Hofer容量和体积能力并没有有限地分化，从而在椭圆形类别上产生所有广义能力。这回答了CHLS问题的变体。 此外，我们证明，如果给定的符合性类别包含一定的单参数对象家族，那么几乎没有归一化的能力是域或目标代表。这为CHL的两个主要问题提供了一些解决方案。
K. Cieliebak, H. Hofer, J. Latschev, and F. Schlenk (CHLS) posed the problem of finding a minimal generating set for the (symplectic) capacities on a given symplectic category. We show that if the category contains a certain one-parameter family of objects, then every countably Borel-generating set of (normalized) capacities has cardinality (strictly) bigger than the continuum. This appears to be the first result regarding the problem of CHLS, except for two results of D. McDuff about the category of ellipsoids in dimension 4. We also prove that every finitely differentiably generating set of capacities on a given symplectic category is uncountable, provided that the category contains a one-parameter family of symplectic manifolds that is ``strictly volume-increasing'' and ``embedding-capacity-wise constant''. It follows that the Ekeland-Hofer capacities and the volume capacity do not finitely differentiably generate all generalized capacities on the category of ellipsoids. This answers a variant of a question of CHLS. In addition, we prove that if a given symplectic category contains a certain one-parameter family of objects, then almost no normalized capacity is domain- or target-representable. This provides some solutions to two central problems of CHLS.
