Unknotted periodic orbits for Reeb flows on the three-sphere

It is well known that a Reeb vector field on $S^3$has a periodic solution. Sharpening thisresult 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-linkingnumber equal to $-1$. If the contact form $\lambda$ is non-degenerate,then there is even a periodic orbit $P$ which, in addition, hasan index $\mu (P) \in \{2,3\}$, and which spans an embedded disc whoseinterior is transversal to $X$. The proofs are based on a theory forpartial differential equations of Cauchy-Riemann type for maps frompunctured Riemann surfaces into ${\mathbb R} \times S^3$, equipped withspecial almost complex structures related to the contact form $\lambda$on $S^3$.