Topological properties of Reeb orbits on boundaries of star-shaped domains in R4

Abstract

Let c be a periodic Reeb orbit on the boundary S of a compact star-shaped domain C in R4. We show that if there is an immersed symplectic disc f in C with boundary c then the self-linking number lk(c) of c equals 2 tan(f)-1 where tan(f) is the tangential self-intersection number of f. We also show that if C is convex and if the principal curvatures of S are suitably pointwise pinched then the self-linking number of a periodic Reeb orbit of Maslov index 3 equals -1.