Weinstein conjecture
Encyclopedia
In mathematics
, the Weinstein conjecture refers to a general existence problem for periodic orbits of Hamiltonian or Reeb vector flow
s. 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, as is trivial to show, 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
(Weinstein 1979). 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).
The Weinstein conjecture has now been proven for all closed 3-dimensional manifolds by Clifford Taubes
(Taubes 2007). The proof uses a variant of Seiberg-Witten Floer homology
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 a the closely related program of proving the Weinstein conjecture by showing that the embedded contact homology
of any contact three-manifold is nontrivial.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Weinstein conjecture refers to a general existence problem for periodic orbits of Hamiltonian or Reeb vector flow
Vector flow
In mathematics, the vector flow refers to a set of closely related concepts of the flow determined by a vector field. These appear in a number of different contexts, including differential topology, Riemannian geometry and Lie group theory...
s. More specifically, the current understanding is that a regular compact contact type level set
Level set
In mathematics, a level set of a real-valued function f of n variables is a set of the formthat is, a set where the function takes on a given constant value c....
of a Hamiltonian
Hamiltonian vector field
In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field, defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations...
on a symplectic manifold
Symplectic manifold
In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...
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
Reeb vector field
In mathematics, the Reeb vector field, named after the French mathematician Georges Reeb, is a notion that appears in various domains of contact geometry including:...
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
Symplectization
In mathematics, the symplectization of a contact manifold is a symplectic manifold which naturally corresponds to it.- Definition :Let be a contact manifold, and let x \in V. Consider the setof all nonzero 1-forms at x, which have the contact plane \xi_x as their kernel...
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, as is trivial to show, 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
Open book decomposition
In mathematics, an open book decomposition is a decomposition of a closed oriented 3-manifold M into a union of surfaces and solid tori...
, 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
Alan Weinstein
Alan David Weinstein is a professor of mathematics at the University of California, Berkeley who works in symplectic geometry, Poisson Geometry and Mathematical Physics....
(Weinstein 1979). 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
De Rham cohomology
In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes...
group of the level set is trivial; this hypothesis turned out to be unnecessary).
The Weinstein conjecture has now been proven for all closed 3-dimensional manifolds by Clifford Taubes
Clifford Taubes
Clifford Henry Taubes is the William Petschek Professor of Mathematics at Harvard University and works in gauge field theory, differential geometry, and low-dimensional topology.-Early career:Taubes received his Ph.D...
(Taubes 2007). The proof uses a variant of Seiberg-Witten Floer homology
Floer homology
Floer homology is a mathematical tool used in the study of symplectic geometry and low-dimensional topology. First introduced by Andreas Floer in his proof of the Arnold conjecture in symplectic geometry, Floer homology is a novel homology theory arising as an infinite dimensional analog of finite...
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 a the closely related program of proving the Weinstein conjecture by showing that the embedded contact homology
Floer homology
Floer homology is a mathematical tool used in the study of symplectic geometry and low-dimensional topology. First introduced by Andreas Floer in his proof of the Arnold conjecture in symplectic geometry, Floer homology is a novel homology theory arising as an infinite dimensional analog of finite...
of any contact three-manifold is nontrivial.