Timeline for Reeb flows on $S^3$ versus volume preserving flows
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jan 13, 2017 at 3:42 | comment | added | Ian Agol | @BenoîtKloeckner: I don't think that their example is proved to be volume-preserving. | |
| Jan 13, 2017 at 1:46 | comment | added | alvarezpaiva | Actually, my answer below does not really use the Rabinowitz-Hofer theorem: if the contact form $\alpha$ is the lift of a contact form on the unit tangent of an ergodic Finslerian or Riemannian metric on the sphere, all we need is Birkoff's theorem on the existence of a closed geodesic. I'll edit it. | |
| Jan 12, 2017 at 23:29 | comment | added | alvarezpaiva | It would be fun to give examples that do not rely on the solutions to two famous conjectures. Taking a hint from my answer, perhaps given a Reeb vector field $X$, some (most?) nowhere zero functions $f$ are such that $fX$ cannot be a Reeb vector field. | |
| Jan 12, 2017 at 21:04 | comment | added | Benoît Kloeckner | @IanAgol K. and G. Kuperberg gave real-analytic counter-examples to the Seifert conjecture: arxiv.org/abs/math/9802040, so that gives examples. | |
| Jan 12, 2017 at 18:16 | vote | accept | aglearner | ||
| Jan 12, 2017 at 13:12 | comment | added | alvarezpaiva | I took the liberty to change the differential calculus tag for the differential geometry tag and to add the dynamical systems tag. | |
| Jan 12, 2017 at 12:28 | history | edited | alvarezpaiva |
Changed the diffential calculus tag for the differential geometry tag and then added the dynamical systems tag.
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| Jan 12, 2017 at 10:05 | answer | added | alvarezpaiva | timeline score: 11 | |
| Jan 12, 2017 at 1:52 | comment | added | Ian Agol | Kuperberg proved the existence of a $C^1$ volume-preserving flow without closed orbits. ams.org/mathscinet-getitem?mr=1371679 On the other hand, a Reeb flow on $S^3$ has a closed orbit by a result of Hofer. ams.org/mathscinet-getitem?mr=1244912 However, Hofer's result is for smooth Reeb flows, so doesn't necessarily imply that Kuperberg's flow is not Reeb. | |
| Jan 12, 2017 at 1:19 | history | asked | aglearner | CC BY-SA 3.0 |