Math 2.0

Differential geometry Contact geometry Conjecture

Weinstein conjecture

Builds on Reeb vector fieldContact manifoldPeriodic orbit

Statement

Let MM be a closed smooth manifold of odd dimension. Let α\alpha be a contact form on MM, so that (M,ker⁡α)(M, \ker \alpha) is a contact manifold. Then the Reeb vector field RαR_\alpha of α\alpha has at least one periodic orbit. That is, there are a point x∈Mx \in M and a time T>0T > 0 such that the flow of RαR_\alpha maps xx to itself at time TT.

Remarks

Weinstein stated the conjecture for hypersurfaces of contact type in symplectic manifolds. He was motivated by Rabinowitz's theorem on periodic orbits on star-shaped energy surfaces in ℝ2n\R^{2n}. The form above is the standard intrinsic version. Viterbo proved it for hypersurfaces of contact type in ℝ2n\R^{2n} in 1987. Hofer proved it for S3S^3 and for closed contact 33-manifolds that are overtwisted or have π2≠0\pi_2 \ne 0, using pseudoholomorphic curves in the symplectization. Taubes proved it for all closed 33-manifolds in 2007, using Seiberg–Witten Floer homology. In dimension 55 and higher it is open in general, although it is known in many cases.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Contact manifold, concept, no evidence levelContact manifoldconceptPeriodic orbit, concept, no evidence levelPeriodic orbitconceptReeb vector field, concept, no evidence levelReeb vector fieldconceptWeinstein conjecture, conjecture, no evidence levelWeinstein conjectureconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Contact manifold concept
  2. Periodic orbit concept
  3. Reeb vector field concept
  4. Weinstein conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Weinstein_conjecture
About this page
TagMP6B64DS31
URIm2://MP6B64DS31
Kindconjecture
WritingW0 stub
MSC53D10, 37J55, 53D42
StatusThis copy of the site has no build data. Run m2 status MP6B64DS31.
AboutReeb vector field
UsesReeb vector field, Contact manifold, Periodic orbit
Same as
Changed2 November 2026

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