Generating sets and representability for symplectic capacities

DOI10.4310/JSG.2022.V20.N4.A3zbMath1530.53084arXiv2003.06442MaRDI QIDQ6336725

Publication date: 13 March 2020

Abstract: 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.

Mathematics Subject Classification ID

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

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