### Unknotted periodic orbits for Reeb flows on the three-sphere

DOI: http://dx.doi.org/10.12775/TMNA.1996.010

#### 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 $S^3$.

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 $S^3$.

#### Keywords

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

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