Unknotted periodic orbits for Reeb flows on the three-sphere
Considering the three-sphere \(S^3\), equipped with a contact 1-form \(\lambda\) (i.e., \(\lambda\wedge d\lambda\) is a volume form), it was shown by Hofer before that the corresponding Reeb vector field \(X_\lambda\) has a periodic orbit. The main theorem of the present paper states more precisely that there is an unknotted periodic orbit with self-linking number equal to \(-1\). The contact form is said to be non-degenerate if all periodic orbits of the associated Reeb field are non-degenerate, meaning that the linearizations of a Poincaré section map and its iterates do not have an eigenvalue 1. A Maslov-type index \(\mu(P)\) can be defined for such non-degenerate orbits \(P\). Another main theorem states that such an orbit \(P\) spans an embedded disc whose interior is transversal to \(X_\lambda\) and that \(\mu(P)\) is either 2 or 3. Following a classification introduced by Eliashberg, contact structures are either `overtwisted' or `tight'. The above result is proved for both cases separately and it is shown that in the overtwisted case, there is an unknotted periodic orbit with self-linking number \(-1\) and index 2. The techniques of proof rely heavily on previous work by the authors concerning pseudoholomorphic curves in symplectizations of contact manifolds.
scientific article; zbMATH DE number 1127671