Generating systems and representability for symplectic capacities

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.

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D. Joksimović’s work on this article was funded by NWO (The Netherlands Organisation for Scientific Research) via the TOP grant 614.001.508.

F. Ziltener worked on this article during a visit at FIM—the Institute for Mathematical Research at ETH Zürich. He gratefully acknowledges the hospitality of ETH.

Received 20 August 2021

Accepted 13 November 2021

Published 16 March 2023