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

TL;DR

This paper explores the topological properties of Reeb orbits on boundaries of star-shaped domains in R4, establishing relationships between self-linking numbers, symplectic discs, and curvature conditions.

Contribution

It introduces new formulas linking self-linking numbers of Reeb orbits to symplectic disc properties and curvature conditions in star-shaped domains.

Findings

Self-linking number equals 2 times the tangential self-intersection minus 1 for certain symplectic discs.

In convex domains with pinched curvatures, Reeb orbits of Maslov index 3 have self-linking number -1.

Provides conditions under which topological invariants of Reeb orbits can be explicitly computed.

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.