Seifert conjecture
In mathematics, the Seifert conjecture states that every nonsingular, continuous vector field on the 3-sphere has a closed orbit. It is named after Herbert Seifert. In a 1950 paper, Seifert asked if such a vector field exists, but did not phrase non-existence as a conjecture. He also established the conjecture for perturbations of the Hopf fibration. The conjecture was disproven in 1974 by Paul Schweitzer, who exhibited a [math]\displaystyle{ C^1 }[/math] counterexample. Schweitzer's construction was then modified by Jenny Harrison in 1988 to make a [math]\displaystyle{ C^{2+\delta} }[/math] counterexample for some [math]\displaystyle{ \delta \gt 0 }[/math]. The existence of smoother counterexamples remained an open question until 1993 when Krystyna Kuperberg constructed a very different [math]\displaystyle{ C^\infty }[/math] counterexample. Later this construction was shown to have real analytic and piecewise linear versions. In 1997 for the particular case of incompressible fluids it was shown that all [math]\displaystyle{ C^\omega }[/math] steady state flows on [math]\displaystyle{ S^3 }[/math] possess closed flowlines[1] based on similar results for Beltrami flows on the Weinstein conjecture.[2]
References
- ↑ Etnyre, J.; Ghrist, R. (1997). "Contact Topology and Hydrodynamics". arXiv:dg-ga/9708011.
- ↑ Hofer, H. (1993). "Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three.". Inventiones Mathematicae 114 (3): 515–564. doi:10.1007/BF01232679. ISSN 0020-9910. Bibcode: 1993InMat.114..515H. https://eudml.org/doc/144157.
- Ginzburg, Viktor L.; Gurel, Basak Z. (2001). "A C2-smooth counterexample to the Hamiltonian Seifert conjecture in R4". arXiv:math/0110047.
- Harrison, Jenny (1988). "[math]\displaystyle{ C^2 }[/math] counterexamples to the Seifert conjecture". Topology 27 (3): 249–278. doi:10.1016/0040-9383(88)90009-2.
- Kuperberg, Greg (1996). "A volume-preserving counterexample to the Seifert conjecture". Commentarii Mathematici Helvetici 71 (1): 70–97. doi:10.1007/BF02566410.
- Kuperberg, Greg; Kuperberg, Krystyna (1996). "Generalized counterexamples to the Seifert conjecture". Annals of Mathematics. Second series 143 (3): 547–576. doi:10.2307/2118536.
- Kuperberg, Krystyna (1994). "A smooth counterexample to the Seifert conjecture". Annals of Mathematics. Second series 140 (3): 723–732. doi:10.2307/2118623.
- Schweitzer, Paul A. (1974). "Counterexamples to the Seifert Conjecture and Opening Closed Leaves of Foliations". Annals of Mathematics 100 (2): 386–400. doi:10.2307/1971077.
- Seifert, Herbert (1950). "Closed Integral Curves in 3-Space and Isotopic Two-Dimensional Deformations". Proceedings of the American Mathematical Society 1 (3): 287–302. doi:10.2307/2032372.
Further reading
- Kuperberg, Krystyna (1999). "Aperiodic dynamical systems". Notices of the AMS 46 (9): 1035–1040. http://www.ams.org/notices/199909/fea-kuperberg.pdf.
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