Weinstein conjecture

In mathematics, the Weinstein conjecture refers to a general existence problem for periodic orbits of Hamiltonian or Reeb vector flows. More specifically, the current understanding is that a regular compact contact type level set of a Hamiltonian on a symplectic manifold should carry at least one periodic orbit of the Hamiltonian flow. The conjecture is stated for any Hamiltonian on any "2n"-dimensional symplectic manifold.

By definition, a level set of contact type admits a contact form obtained by contracting the Hamiltonian vector field into the symplectic form. In this case, the Hamiltonian flow is a Reeb vector field on that level set. It is a fact that any contact manifold ("M",α) can be embedded into a canonical symplectic manifold, called the symplectization of "M", such that "M" 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 it is known that any orbit of a Hamiltonian flow is contained in a level set, the Weinstein conjecture is a statement about contact manifolds.

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 open book decomposition, whose binding is a closed Reeb orbit. This is not enough to prove the Weinstein conjecture, though, because the Weinstein conjecture states that "every" contact form admits a closed Reeb orbit, while an open book determines a closed Reeb orbit for a form which is only isotopic to the given form.

The conjecture was formulated in 1978 by Alan Weinstein. 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 de Rham cohomology group of the level set is trivial; this hypothesis turned out to be unnecessary).

In October 2006, Clifford Taubes posted an [http://arxiv.org/abs/math.SG/0611007 article] to the mathematics arXiv which contains a proposed proof of the conjecture for 3-dimensional manifolds. His proof uses a variant of Seiberg-Witten Floer homology and pursues a strategy analogous to his proof that the Seiberg-Witten and Gromov invariants are equivalent on a symplectic four-manifold. Taubes's proof provides a shortcut to a the closely related program of proving the Weinstein conjecture by showing that the Embedded contact homology of any contact three-manifold is nontrivial.

References

* [http://arxiv.org/abs/math.DG/0310330 Ginzburg, Viktor. "The Weinstein conjecture and the theorems of nearby and almost existence"]

* [http://arxiv.org/math.SG/0611007 Taubes, C. H., "The Seiberg-Witten equations and the Weinstein conjecture".]

* Weinstein Alan, "On the hypotheses of Rabinowitz' periodic orbit theorem". Journal of Diff. Eq., Vol. 33 , 1979 , pp. 353 - 358.