Topological Methods in Nonlinear Analysis

Unknotted periodic orbits for Reeb flows on the three-sphere

H. Hofer, K. Wysocki, and E. Zehnder

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Abstract

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 $S^3$.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 7, Number 2 (1996), 219-244.

Dates
First available in Project Euclid: 16 November 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1479265300

Mathematical Reviews number (MathSciNet)
MR1481697

Zentralblatt MATH identifier
0898.58018

Citation

Hofer, H.; Wysocki, K.; Zehnder, E. Unknotted periodic orbits for Reeb flows on the three-sphere. Topol. Methods Nonlinear Anal. 7 (1996), no. 2, 219--244. https://projecteuclid.org/euclid.tmna/1479265300


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