Unknotted periodic orbits for Reeb flows on the three-sphere
Keywords
Reeb flows, unknotted periodic orbits, three-sphere, theory for PDE of Cauchy-Riemann typeAbstract
It is well known that a Reeb vector field on $S^3$ has a periodic solution. Sharpening this result we shall show in this note that every Reeb vector field $X$ on $S^3$ has a periodic orbit which is unknotted and has self-linking number equal to $-1$. If the contact form $\lambda$ is non-degenerate, then there is even a periodic orbit $P$ which, in addition, has an index $\mu (P) \in \{2,3\}$, and which spans an embedded disc whose interior is transversal to $X$. The proofs are based on a theory for partial differential equations of Cauchy-Riemann type for maps from punctured Riemann surfaces into ${\mathbb R} \times S^3$, equipped with special almost complex structures related to the contact form $\lambda$ on $S^3$.Downloads
Published
1996-06-01
How to Cite
1.
HOFER, H., WYSOCKI, K. and ZEHNDER, E. Unknotted periodic orbits for Reeb flows on the three-sphere. Topological Methods in Nonlinear Analysis. Online. 1 June 1996. Vol. 7, no. 2, pp. 219 - 244. [Accessed 24 October 2024].
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