mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Weinstein conjecture refers to a general existence problem for

periodic orbit In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. Iterated functions Given ...

s of

Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...

or Reeb

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. In d ...

s. More specifically, the conjecture claims that on a compact

contact manifold In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution (differential geometry), distribution in the tangent bundle satisfying a condition called 'complete non-integrability' ...

, its Reeb vector field should carry at least one periodic orbit. By definition, a level set of contact type admits a contact form obtained by

contracting A contract is an agreement that specifies certain legally enforceable rights and obligations pertaining to two or more parties. A contract typically involves consent to transfer of goods, services, money, or promise to transfer any of those a ...

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 Alan David Weinstein (born 17 June 1943) is a professor of mathematics at the University of California, Berkeley, working in the field of differential geometry, and especially in Poisson manifold, Poisson geometry. Early life and education ...

. 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 In mathematics, de Rham cohomology (named after Georges de Rham) 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 adapte ...

group of the level set is trivial; this hypothesis turned out to be unnecessary). The Weinstein conjecture was first proved for contact hypersurfaces in

\mathbb R^

in 1986 by , 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 Hofer and is at the origin of contact homology. The Weinstein conjecture has now been proven for all closed 3-dimensional manifolds by

Clifford Taubes Clifford Henry Taubes (born February 21, 1954) is the William Petschek Professor of Mathematics at Harvard University and works in gauge field theory, differential geometry, and low-dimensional topology. His brother is the journalist Gary Taub ...

. The proof uses a variant of Seiberg–Witten

Floer homology In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is an invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer intro ...

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

embedded contact homology In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is an invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer intro ...

of any contact three-manifold is nontrivial.

See also

References

Further reading