Global properties of tight Reeb flows with applications to Finsler geodesic flows on $S^2$

TL;DR

This paper investigates the topological and dynamical properties of Finsler geodesic flows on the 2-sphere, establishing conditions under which certain closed geodesics with self-intersections cannot exist, and exploring related global dynamics.

Contribution

It introduces sharp curvature pinching conditions that prevent specific closed geodesics with self-intersections on $S^2$, using pseudo-holomorphic curve techniques from symplectic geometry.

Findings

No closed geodesics with one transverse self-intersection under given curvature conditions.

Constructed examples of Randers metrics satisfying the pinching condition.

Analyzed global dynamics of energy levels close to $S^3$.

Abstract

We show that if a Finsler metric on $S^2$ with reversibility $r$ has flag curvatures $K$ satisfying $(\frac{r}{r+1})^2 <K \leq 1$, then closed geodesics with specific contact-topological properties cannot exist, in particular there are no closed geodesics with precisely one transverse self-intersection point. This is a special case of a more general phenomenon, and other closed geodesics with many self-intersections are also excluded. We provide examples of Randers type, obtained by suitably modifying the metrics constructed by Katok \cite{katok}, proving that this pinching condition is sharp. Our methods are borrowed from the theory of pseudo-holomorphic curves in symplectizations. Finally, we study global dynamical aspects of 3-dimensional energy levels $C^2$-close to $S^3$.