The Sharp Energy-Capacity Inequality

Usher, Michael

doi:10.1142/s0219199710003889

Cited by 47 publications

(62 citation statements)

References 33 publications

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“…In fact, we deduce Theorem 1.2 from part (ii) of the following Theorem 1.3 about the deformed spectral invariants, which is new even in the undeformed case Á D 0 (though an important special case appears in Usher [62] We obtain Theorem 1.2 from Theorem 1.3 by using the assumed nonzero GromovWitten invariant to find a value (indeed an open dense set of values) Á2 L n 1 iD0 H 2i .M I C/ so that OEpt Á a 0 has a nontrivial component in H 2n .M I ƒ ! /; the triangle inequality and a standard duality property (Proposition 3.13(vii)) of the spectral invariants allow one to use this to obtain an estimate as in (ii) of Theorem 1.3.…”

Section: Hofer-zehnder Capacitymentioning

confidence: 62%

Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms

Usher

2011

Geom. Topol.

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We develop a family of deformations of the differential and of the pair-of-pants product on the Hamiltonian Floer complex of a symplectic manifold .M; !/ which upon passing to homology yields ring isomorphisms with the big quantum homology of M . By studying the properties of the resulting deformed version of the OhSchwarz spectral invariants, we obtain a Floer-theoretic interpretation of a result of Lu which bounds the Hofer-Zehnder capacity of M when M has a nonzero GromovWitten invariant with two point constraints, and we produce a new algebraic criterion for .M; !/ to admit a Calabi quasimorphism and a symplectic quasistate. This latter criterion is found to hold whenever M has generically semisimple quantum homology in the sense considered by Dubrovin and Manin (this includes all compact toric M ), and also whenever M is a point blowup of an arbitrary closed symplectic manifold. 53D40, 53D45

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Section: Hofer-zehnder Capacitymentioning

confidence: 62%

Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms

Usher

2011

Geom. Topol.

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“…(2) H ∈ H(A, M ) is called HZ • -admissible if the flow φ t H has no non-constant contractible periodic orbit whose period is less than 1. Usher also asked the following question in [13].…”

Section: Resultsmentioning

confidence: 99%

“…Question 1 (Usher [13]) Does the (π 1 -sensitive) sharp energy-capacity inequality hold also on non-compact symplectic manifolds?…”

Section: Resultsmentioning

confidence: 99%

“…The factor 2 in (3) was then removed for symplectically aspherical manifolds (closed or open convex) in Frauenfelder-Ginzburg-Schlenk ( [3]). This is the π 1 -sensitive sharp energy-capacity inequality: π 1 -sensitive Hofer-Zehnder capacity ≤ displacement energy (4) From here, Usher went on to remove the assumption of symplectic asphericity in the closed case [13]. He used Floer homology and its spectral invariants to study the (π 1 -sensitive) Hofer-Zehnder capacity and the displacement energy.…”

Section: Introductionmentioning

confidence: 99%

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The sharp energy–capacity inequality on convex symplectic manifolds

Sugimoto

2018

J. Fixed Point Theory Appl.

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In symplectic geometry, symplectic invariants are useful tools in studying symplectic phenomena. One such invariant is the Hofer-Zehnder capacity, which is defined for any subset of a symplectic manifold. On the other hand, we can associate the so-called displacement energy to any subset. Many symplectic geometers tried to relate the Hofer-Zehnder capacity and the displacement energy to study the behaviour of closed orbits of Hamiltonian diffeomorphisms. Usher proved the so-called (π1-sensitive) sharp energy-capacity inequality between the Hofer-Zehnder capacity and the displacement energy for closed symplectic manifolds. In this paper, we consider a certain Floer homology on symplectic manifolds with boundaries (not symplectic homology) and its spectral invariants. Then we extend the π1-sensitive sharp energy-capacity inequality to convex symplectic manifolds. As a corollary, we also prove the almost existence theorem of closed characteristics near displaceable hypersurfaces in convex symplectic manifolds. In particular, we prove the existence of closed characteristics on displaceable contact type hypersurfaces in convex symplectic manifolds (the Weinstein conjecture).

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“…These applications are related, among others, to the notions of energy and displacement energy of a subset (see for instance [11,10,18,21,6]) and to Hofer-Zehnder [9] capacities.…”

Section: Introductionmentioning

confidence: 99%

Capacities and Displacement Energy in Contact Hofer Geometry

Pitiş

2011

Int. J. Geom. Methods Mod. Phys.

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A contact version of a Laudenbach's engulfing theorem is proved. Some properties of the notions of contact displacement energy and contact Hofer-Zehnder capacities are presented and, under the condition of existence of a modified action selector on a contact manifold, we can prove some inequalities involving these invariants. These inequalities are similar to the ones obtained by Frauenfelder, Ginzburg and Schlenk, in the symplectic case.

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The Sharp Energy-Capacity Inequality

Cited by 47 publications

References 33 publications

Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms

Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms

The sharp energy–capacity inequality on convex symplectic manifolds

Capacities and Displacement Energy in Contact Hofer Geometry

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