The weinstein conjecture in the presence of holomorphic spheres

Hofer, Helmut; Viterbo, Claude

doi:10.1002/cpa.3160450504

Cited by 89 publications

(106 citation statements)

References 18 publications

(30 reference statements)

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“…Because its limit at ∞ is a point, it also extends to a continuous map S 2 → V that represents the class A. It is shown in [8] that the algebraic number of solutions to this equation is still 1 for small λ.…”

Section: Definition 32mentioning

confidence: 99%

“…We begin by sketching the proof of this inequality for semi-positive M using the set up in Hofer-Viterbo [8]. Section 3.2 contains more technical details, and Section 3.3 discusses the case of general M .…”

Section: The Area-capacity Inequalitymentioning

confidence: 99%

“…8 The resulting trajectories u have domain C and in terms of the coordinates (s, t) of (−∞, ∞) × S 1 satisfy the following equation for some λ:…”

Section: Definition 32mentioning

confidence: 99%

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Hofer–Zehnder capacity and length minimizing Hamiltonian paths

McDuff

Slimowitz

2001

Geom. Topol.

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We use the criteria of Lalonde and McDuff to show that a path that is generated by a generic autonomous Hamiltonian is length minimizing with respect to the Hofer norm among all homotopic paths provided that it induces no non-constant closed trajectories in M . This generalizes a result of Hofer for symplectomorphisms of Euclidean space. The proof for general M uses LiuTian's construction of S 1 -invariant virtual moduli cycles. As a corollary, we find that any semifree action of S 1 on M gives rise to a nontrivial element in the fundamental group of the symplectomorphism group of M . We also establish a version of the area-capacity inequality for quasicylinders.

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Section: Definition 32mentioning

confidence: 99%

Section: The Area-capacity Inequalitymentioning

confidence: 99%

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Hofer–Zehnder capacity and length minimizing Hamiltonian paths

McDuff

Slimowitz

2001

Geom. Topol.

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“…Using the deformed Hamiltonian Floer complexes, we give a new proof of the following theorem of Lu (see also Hofer and Viterbo [30] and Liu and Tian [38] for earlier related results). We prove this Theorem below as Corollary 4.3.…”

Section: Hofer-zehnder Capacitymentioning

confidence: 99%

“…Meanwhile if D dˇE 0 then [32, Lemma 1.1] shows that the invariant h i ; j ; k i 0;3;ˇi s zero. Thus all of the terms in (30) arising from invariants h i ; j ; k i 0;3;ˇw ithˇ… H 2 .M / contribute terms belonging to the submodule Z n 1 N Ä N , while all of the terms arising from h i ; j ; k i 0;3;ˇw ithˇ2 H 2 .M / contribute terms in H 2 .M / Ä N . Now we consider the invariants h i ; j ; E k i 0;3;ˇa ppearing in (30).…”

Section: 34mentioning

confidence: 99%

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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The Weinstein conjecture and theorems of nearby and almost existence

Ginzburg

Progress in Mathematics

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The Weinstein conjecture, as the general existence problem for periodic orbits of Hamiltonian or Reeb flows, has been among the central questions in symplectic topology for over two decades and its investigation has led to understanding of some fundamental properties of Hamiltonian flows.In this paper we survey some recently developed and well-known methods of proving various particular cases of this conjecture and the closely related almost existence theorem. We also examine differentiability and continuity properties of the Hofer-Zehnder capacity function and relate these properties to the features of the underlying Hamiltonian dynamics, e.g., to the period growth. 1 2. The nearby existence theorems Nearby existence theorems are usually proved by Floer homological methods. Below we discuss some of these methods, focusing on the key ideas and omitting many (often quite non-trivial) technical details.

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The weinstein conjecture in the presence of holomorphic spheres

Cited by 89 publications

References 18 publications

Hofer–Zehnder capacity and length minimizing Hamiltonian paths

Hofer–Zehnder capacity and length minimizing Hamiltonian paths

Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms

The Weinstein conjecture and theorems of nearby and almost existence

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