Discontinuous symplectic capacities

Zehmisch, Kai; Ziltener, Fabian

doi:10.1007/s11784-013-0148-x

Cited by 5 publications

(7 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…• The proof of Theorem 17(ii) shows that the cardinality of the set of discontinuous normalized capacities is ℶ 2 . This improves the result of K. Zehmisch and the second author that discontinuous capacities exist, see [17]. 36…”

Section: Remarks 14 (Set Of Capacities and Isomorphism-closedness) (I)supporting

confidence: 81%

See 1 more Smart Citation

Generating systems and representability for symplectic capacities

Joksimović

Ziltener

2022

Journal of Symplectic Geometry

View full text Add to dashboard Cite

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 "embeddingcapacity-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. 837 838 D. Joksimović and F. Ziltener 4 Proof of Theorem 43 (cardinality of the set of capacities, more general setting) 877 5 Proof of Proposition 44 (sufficient conditions for being an I-collection) 890 6 Proof of Theorem 17(iii) (cardinality of a generating set) 895 7 Proof of Theorem 25 (uncountability of every generating set under a mild hypothesis) 900 Appendix A Cardinality of the set of equivalence classes of pairs of manifolds and forms 901 Appendix B Proof of Theorem 29 (monotone generation for ellipsoids) 905 References 908 "The introduction of the cipher 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps . . . " -Alexander Grothendieck 840 D. Joksimović and F. Ziltener Assume now that B, Z ∈ O. Let c be a generalized capacity on C. We call c a capacity iff it satisfies:(iii) (non-triviality) c(B) > 0 and c(Z) < ∞.We call it normalized iff it satisfies:We denote Cap(C) := generalized capacity on C .Remark. There is a set-theoretic issue with this definition, which we will resolve in Definition 12 in the next section. Compare to Remark 11.Example 2 (embedding capacities). Let C = (O, M) be a symplectic category in dimension 2n and (M, ω) an object of Symp 2n . We define the domain-embedding capacity for (M, ω) on C to be the functionWe define the target-embedding capacity for (M, ω) on C to be the functionThese are generalized capacities. 8 We define the Gromov width on C to be (1)w := πc C B,ωst .If B, Z ∈ O, then by Gromov's nonsqueezing theorem the Gromov width is a normalized capacity.25 i.e., cardinalities of some sets 26 ℶ (bet) is the second letter of the Hebrew alphabet.

show abstract

Section: Remarks 14 (Set Of Capacities and Isomorphism-closedness) (I)supporting

confidence: 81%

“…We equip each ellipsoid E with the restriction of ω to E. We define the restriction category of ellipsoids to be the full subcategory Ell V of Op V consisting of ellipsoids. 17 The objects of Ell V are uniquely determined by the Ekeland-Hofer capacities, up to isomorphism, see [3,FACT 10,p. 27].…”

mentioning

confidence: 99%

Generating systems and representability for symplectic capacities

Joksimović

Ziltener

2022

Journal of Symplectic Geometry

View full text Add to dashboard Cite

show abstract

“…Let the dimension of P × Q × C be 2n. If the (n − 1)-st power of the symplectic form ω = ω P + ω Q + dx ∧ dy has a primitive µ, as it is the case if ω P is exact, the helicity (see [32]) can be used as in [41] to show that Σ is the concave boundary of (C, ω C ). Indeed, by Stokes theorem the symplectic volume of (D Σ , ω) equals Σ µ ∧ ω, where Σ is equipped with the boundary orientation induced by the symplectic orientation of (D Σ , ω).…”

Section: Examples and Applicationsmentioning

confidence: 99%

Polyfolds, cobordisms, and the strong Weinstein conjecture

Suhr

Zehmisch

2017

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…The continuity of symplectic capacities is discussed in [2,3,6,27]. The semitoric and toric packing capacities are each defined on categories of integrable systems which have a natural topology [15,18], but we can only discuss the continuity of the (m, k)-equivariant Gromov radius on a subcategory of its domain which has a topology, so we restrict to the case of (m, k) = (n, n).…”

Section: Introductionmentioning

confidence: 99%