101:. In several cases, the existence of a periodic orbit was known. For instance, Rabinowitz showed that on star-shaped level sets of a Hamiltonian function on a symplectic manifold, there were always periodic orbits (Weinstein independently proved the special case of convex level sets). Weinstein observed that the hypotheses of several such existence theorems could be subsumed in the condition that the level set be of contact type. (Weinstein's original conjecture included the condition that the first
151:, then extended to cotangent bundles by Hofer–Viterbo and to wider classes of aspherical manifolds by Floer–Hofer–Viterbo. The presence of holomorphic spheres was used by Hofer–Viterbo. All these cases dealt with the situation where the contact manifold is a contact submanifold of a symplectic manifold. A new approach without this assumption was discovered in dimension 3 by
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is a contact type level set (of a canonically defined
Hamiltonian) and the Reeb vector field is a Hamiltonian flow. That is, any contact manifold can be made to satisfy the requirements of the Weinstein conjecture. Since, as is trivial to show, any orbit of a Hamiltonian flow is contained in a level
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and pursues a strategy analogous to Taubes' proof that the
Seiberg-Witten and Gromov invariants are equivalent on a symplectic four-manifold. In particular, the proof provides a shortcut to the closely related program of proving the Weinstein conjecture by showing that the
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It has been known that any contact form is isotopic to a form that admits a closed Reeb orbit; for example, for any contact manifold there is a compatible
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86:, whose binding is a closed Reeb orbit. This is not enough to prove the Weinstein conjecture, though, because the Weinstein conjecture states that
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Hofer, H. (1993). "Pseudoholomorphic curves in symplectizations with applications to the
Weinstein conjecture in dimension three".
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contact form admits a closed Reeb orbit, while an open book determines a closed Reeb orbit for a form which is only
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Hofer, H.; Viterbo, C. (1992). "The
Weinstein conjecture in the presence of holomorphic spheres".
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the
Hamiltonian vector field into the symplectic form. In this case, the Hamiltonian flow is a
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Ginzburg (2003). "The
Weinstein conjecture and the theorems of nearby and almost existence".
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Taubes, C. H. (2007). "The
Seiberg-Witten equations and the Weinstein conjecture".
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group of the level set is trivial; this hypothesis turned out to be unnecessary).
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245:"Periodic solutions of a Hamiltonian system on a prescribed energy surface"
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The
Weinstein conjecture was first proved for contact hypersurfaces in
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set, the
Weinstein conjecture is a statement about contact manifolds.
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500:"Taubes's proof of the Weinstein conjecture in dimension three"
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Annales de l'institut Henri Poincaré (C) Analyse non linéaire
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39:. More specifically, the conjecture claims that on a compact
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on that level set. It is a fact that any contact manifold (
202:"On the hypotheses of Rabinowitz' periodic orbit theorems"
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By definition, a level set of contact type admits a
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162:. The proof uses a variant of Seiberg–Witten
508:Bulletin of the American Mathematical Society
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313:{\displaystyle \mathbb {R} ^{2n}}
250:Journal of Differential Equations
207:Journal of Differential Equations
133:{\displaystyle \mathbb {R} ^{2n}}
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589:Unsolved problems in geometry
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272:10.1016/0022-0396(79)90069-X
229:10.1016/0022-0396(79)90070-6
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169:embedded contact homology
390:Inventiones Mathematicae
458:10.2140/gt.2007.11.2117
435:Geometry & Topology
243:Rabinowitz, P. (1979).
84:open book decomposition
498:Hutchings, M. (2010).
374:10.1002/cpa.3160450504
362:Comm. Pure Appl. Math.
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286:Viterbo, C. (1987).
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94:to the given form.
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584:Conjectures
397:: 515–563.
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140:in 1986 by
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30:Hamiltonian
18:mathematics
568:Categories
187:References
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522:0906.2444
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175:See also
92:isotopic
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153:Hofer
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