Generating systems and representability for symplectic capacities

Abstract

We consider the problem by K. Cieliebak, H. Hofer, J. Latschev, and F. Schlenk (CHLS) that is concerned with finding a minimal generating system for (symplectic) capacities on a given symplectic category. We show that under some mild hypotheses every countably Borel-generating set of (normalized) capacities has cardinality bigger than the continuum. This appears to be the first result regarding the problem of CHLS, except for a result by D. McDuff, stating that the ECH-capacities are monotonely generating for the category of ellipsoids in dimension 4. Under the same mild hypotheses we also prove that almost no normalized capacity is domain- or target-representable. This provides some solutions to two central problems by CHLS. In addition, we prove that every finitely differentiably generating system of symplectic 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 by CHLS.