The Seiberg–Witten equations and the Weinstein conjecture

By Clifford H. Taubes. 2006

309 citations recorded by the source corpus.

Let M denote a compact, oriented 3–dimensional manifold and let a denote a contact 1–form on M; thus a∧da is nowhere zero. This article proves that the vector field that generates the kernel of da has a closed integral curve.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/54ecd72217a0faf272eb98959221e892b406a7ac

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

Citations in the corpus

Torsion contact forms in three dimensions have two or infinitely many Reeb orbits (2017, unknown): “This was proved by Taubes [25] in 2006, and various special cases had been proved earlier; see eg the survey by Hutchings [13].”
Sharp systolic inequalities for Reeb flows on the three-sphere (2015, result): “is ﬂow always has periodic orbits, and we denote by Tmin(α) the minimum over all their periods. For arbitrary closed contact three-manifolds, the existence of a periodic orbit was proved by Taubes in [Tau07], but in the case of S3 we are considering here this follows from previous results of Rabinow”
Lecture Notes on Embedded Contact Homology (2013, methodology): “The proof of the Volume property is beyond the scope of these notes; it is given in [10], using ingredients from Taubes’s proof of the Weinstein conjecture [57].”
The Restricted Three-Body Problem and Holomorphic Curves (2018, methodology): “Later on Taubes [105] proved the Weinstein conjecture in dimension three completely using quite different methods than finite energy planes, namely Seiberg-Witten invariants.”
Symplectic Floer homology of area-preserving surface diffeomorphisms (2008, background): “SYMPLECTIC FLOER HOMOLOGY OF SURFACE DIFFEOMORPHISMS 5 conjectured to agree with SWFH and HF+. Taubes has recently announced a result relating ECH to SWFH [Ta2, Ta3] which may additionally work for PFH.”
On the existence of almost contact structure and the contact magnetic field (2009, unknown): “In a recent paper Taubes [15] proved that the conjeture is true but the second hypothesis is superfluos, that is, on any compact oriented 3- dimensional contact manifold the vector field ξ has a closed orbit.”
Lectures on Holomorphic Curves in Symplectic and Contact Geometry (2010, methodology): “ closed odd-dimensional manifold Mwith a contact form α, Xα has a closed orbit. The Weinstein conjecture is still open in general, though a proof in dimension three was produced recently by C. Taubes [Tau07], using Seiberg-Witten theory. Before this, there was a long history of partial results using”
Reeb vector fields and open book decompositions (2008, unknown): “conjecture). Let (M,ξ) be a contact 3-manifold. Then for any contact form αwith kerα= ξ, the corresponding Reeb vector ﬁeld R= R αadmits a periodic orbit. During the preparation of this paper, Taubes [Ta] gave a complete proof of the Weinstein conjecture in dimension three. Our methods are completel”
Embedded contact homology and Seiberg–Witten Floer cohomology II (2008, methodology): “Theorem 1 can also be viewed as a generalization of the author’s papers [24; 25], which use the existence of certain nontrivial Seiberg–Witten Floer homology classes to find closed orbits of the vector field that generates the kernel of da.”
Quantitative embedded contact homology (2010, background): “12 is related to the question of whether the Weinstein conjecture in three dimensions [21] can be refined to show that a closed contact 3-manifold has a Reeb orbit with an explicit upper bound on the length, see Remark 8.”
Sutures and contact homology I (2010, methodology): “[T3] C. Taubes,The structure of pseudoholomorphic subvarieties for degenrate almost complex structure and symplectic form onS1 ×B3, Geom.”
Proof of the Arnold chord conjecture in three dimensions, II (2010, background): “ric one has |λ| = 1, dλ= 2∗λ. (2.10) Remark 2.2. The factor of 1/2 in (2.9) and the factor of 2 in (2.10) could probably be dropped, but we have included these factors for consistency with the papers [21, 23] and their sequels. With these choices made, if s= (S,cl) is any spin-c structure, then ther”