Unknotted periodic orbits for Reeb flows on the three-sphere

H. Hofer, K. Wysocki, E. Zehnder


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 $S^3$.


Reeb flows; unknotted periodic orbits; three-sphere; theory for PDE of Cauchy-Riemann type

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