Quantitative conditions for right-handedness of flows

TL;DR

This paper establishes a numerical criterion for right-handedness of certain flows on the 3-sphere, linking geometric pinching conditions to dynamical properties and global surface structures.

Contribution

It introduces a quantitative condition for right-handedness of dynamically convex Reeb flows and applies it to geodesic flows on 2-spheres with pinched metrics.

Findings

Derived an explicit constant δ* < 0.7225 for geodesic flow right-handedness.

Showed that δ-pinched metrics with δ > δ* have geodesic flows that lift to right-handed flows.

Proved that such flows have open book decompositions with global surfaces of section.

Abstract

We give a numerical condition for right-handedness of a dynamically convex Reeb flow on the $3$-sphere. Our condition is stated in terms of an asymptotic ratio between the amount of rotation of the linearised flow and the linking number of trajectories with a periodic orbit that spans a disk-like global surface of section. As an application, we find an explicit constant $\delta_* < 0.7225$ such that if a Riemannian metric on the $2$-sphere is $\delta$-pinched with $\delta > \delta_*$, then its geodesic flow lifts to a right-handed flow on the $3$-sphere. In particular, all finite non-empty collections of periodic orbits of such a geodesic flow bind open books whose pages are global surfaces of section.