The Weinstein conjecture is a general existence problem for periodic orbits of Hamiltonian or Reeb vector flows, claiming that on a compact contact manifold, its Reeb vector field should carry at least one periodic orbit. It was formulated in 1978 by Alan Weinstein, generalizing several earlier existence theorems that had been proved for star-shaped and convex level sets. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementOn a compact contact manifold, its Reeb vector field should carry at least one periodic orbit. 1 Proposed Year Progress Toward ResolutionFirst proved for contact hypersurfaces in R^2n by Viterbo in 1986, then extended by Hofer and Viterbo and by Floer, Hofer and Viterbo; proven for all closed 3-dimensional manifolds by Clifford Taubes. 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Weinstein Conjecture (Wikipedia)
Sources
1. Weinstein Conjecture (Wikipedia)
Wikimedia FoundationLead section
The conjecture was formulated in 1978 by Alan Weinstein.
Lead section, first paragraph
More specifically, the conjecture claims that on a compact contact manifold, its Reeb vector field should carry at least one periodic orbit.
Final paragraph of the lead section
The Weinstein conjecture has now been proven for all closed 3-dimensional manifolds by Clifford Taubes.
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