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Quantitative conditions for right-handedness of flows

Anna Florio, Umberto Hryniewicz

TL;DR

This work provides a nonperturbative numerical criterion for right-handedness of dynamically convex Reeb flows on $S^3$, expressed through the asymptotic invariant $oldsymbol{ extkappa}(oldsymbol{gamma}_0)$ exceeding $2oldsymbol{ extpi}$. By connecting transverse rotation numbers to linking data via disk-like global surfaces of section, the authors show that $oldsymbol{ extkappa}(oldsymbol{gamma}_0)>oldsymbol{2oldsymbol{ extpi}}$ implies right-handedness for the flow, and deduce consequences for the binding of finite periodic orbits by open books. They then specialize to strictly convex energy levels to obtain a curvature–return-time criterion guaranteeing right-handedness. The results culminate in an explicit, nonperturbative condition: if a metric on $S^2$ is $oldsymbol{ extdelta}$-pinched with $oldsymbol{ extdelta}>oldsymbol{ extdelta}_*$ (with $oldsymbol{ extdelta}_* \,=\ x_*^{2}$ and $x_*$ the root of $oldsymbol{P}(x)=4x^{3}-2x^{2}-1$ near $0.84$–$0.85$), then its geodesic flow lifts to a right-handed flow on $S^3$, yielding bound open books for all finite collections of periodic orbits. This provides a concrete geometric regime in which right-handedness can be verified explicitly and has applications to global surface-of-section theory and transverse knot theory.

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 $δ_* < 0.7225$ such that if a Riemannian metric on the $2$-sphere is $δ$-pinched with $δ> δ_*$, 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.

Quantitative conditions for right-handedness of flows

TL;DR

This work provides a nonperturbative numerical criterion for right-handedness of dynamically convex Reeb flows on , expressed through the asymptotic invariant exceeding . By connecting transverse rotation numbers to linking data via disk-like global surfaces of section, the authors show that implies right-handedness for the flow, and deduce consequences for the binding of finite periodic orbits by open books. They then specialize to strictly convex energy levels to obtain a curvature–return-time criterion guaranteeing right-handedness. The results culminate in an explicit, nonperturbative condition: if a metric on is -pinched with (with and the root of near –), then its geodesic flow lifts to a right-handed flow on , yielding bound open books for all finite collections of periodic orbits. This provides a concrete geometric regime in which right-handedness can be verified explicitly and has applications to global surface-of-section theory and transverse knot theory.

Abstract

We give a numerical condition for right-handedness of a dynamically convex Reeb flow on the -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 such that if a Riemannian metric on the -sphere is -pinched with , then its geodesic flow lifts to a right-handed flow on the -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.

Paper Structure

This paper contains 11 sections, 27 theorems, 214 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Global surfaces of section
  3. Reeb flows
  4. Right-handedness from dynamical pinching
  5. Transverse rotation numbers
  6. Right-handedness
  7. Proof of Theorem \ref{['thm_main_0']}
  8. Right-handedness on strictly convex energy levels
  9. Pinched two-spheres
  10. Lifting geodesic flows on $S^2$ and Birkhoff annuli
  11. Asymptotic estimates on linking and intersection numbers

Key Result

Theorem 1.1

Every non-empty finite collection of periodic orbits of a right-handed flow binds an open book whose pages are global surfaces of section.

Figures (6)

  • Figure 1: Examples of geodesic polygons, convex $(a)$ and non convex $(b)$.
  • Figure 2: Examples of paths $\alpha_+$.
  • Figure 3: The chosen loops $e$ and $f$.
  • Figure 4: The path $\ell_{m(T,x)-1}$.
  • Figure 5: An example of the paths $\ell_i,\nu_i$ and $\varepsilon$.
  • ...and 1 more figures

Theorems & Definitions (58)

  • Theorem 1.1: Ghys ghys
  • Theorem 1.2
  • Definition 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 1.6
  • Remark 1.7
  • Definition 1.8: Hofer, Wysocki and Zehnder char2
  • Theorem 1.9: Hofer, Wysocki and Zehnder convex
  • Theorem 1.10: openbook
  • ...and 48 more