---
tag: MP6B64DS31
kind: conjecture
name: Weinstein conjecture
msc: [53D10, 37J55, 53D42]
uses: [GGM63KXCF2, HBQ8BZ7XB5, 0AA97XA70K]
about: [GGM63KXCF2]
sources:
  - url: https://en.wikipedia.org/wiki/Weinstein_conjecture
same_as:
  wikidata: Q966711
  wikipedia: Weinstein conjecture
history:
  posed_by: "Alan Weinstein"
  first_stated: "1979"
---

## Statement

Let $M$ be a closed smooth manifold of odd dimension.
Let $\alpha$ be a contact form on $M$, so that $(M, \ker \alpha)$ is a [[contact manifold|HBQ8BZ7XB5]].
Then the [[Reeb vector field|GGM63KXCF2]] $R_\alpha$ of $\alpha$ has at least one [[periodic orbit|0AA97XA70K]].
That is, there are a point $x \in M$ and a time $T > 0$ such that the flow of $R_\alpha$ maps $x$ to itself at time $T$.

## 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 $\R^{2n}$.
The form above is the standard intrinsic version.
Viterbo proved it for hypersurfaces of contact type in $\R^{2n}$ in 1987.
Hofer proved it for $S^3$ and for closed contact $3$-manifolds that are overtwisted or have $\pi_2 \ne 0$, using pseudoholomorphic curves in the symplectization.
Taubes proved it for all closed $3$-manifolds in 2007, using Seiberg–Witten Floer homology.
In dimension $5$ and higher it is open in general, although it is known in many cases.

## Explanations

### L1 human

On a closed contact manifold, a contact form determines a flow, the Reeb flow.
The conjecture says that this flow always has a closed orbit, a trajectory that comes back to where it started.
For energy surfaces of a mechanical system, this means a periodic motion with that energy.