Existence of a positive hyperbolic Reeb orbit in three spheres with finite free group actions

TL;DR

This paper proves the existence of a positive hyperbolic Reeb orbit in certain three-sphere manifolds with finite free group actions, extending previous results to new classes of manifolds.

Contribution

It establishes the presence of a positive hyperbolic Reeb orbit in lens spaces with odd p, answering a question about manifolds with finite free group actions.

Findings

Existence of positive hyperbolic Reeb orbit in lens spaces with odd p.

Extension of previous results to manifolds with finite free group actions.

Addresses open question for manifolds with b_1(Y)=0.

Abstract

Let $(Y,\lambda)$ be a non-degenerate contact three manifold. D. Cristfaro-Gardiner, M. Hutshings and D. Pomerleano showed that if $c_{1}(\xi=\mathrm{Ker}\lambda)$ is torsion, then the Reeb vector field of $(Y,\lambda)$ has infinity many Reeb orbits otherwise $(Y,\lambda)$ is a lens space or three sphere with exaxtly two simple elliptic orbits. In the same paper, they also showed that if $b_{1}(Y)>0$, $(Y,\lambda)$ has a simple positive hyperbolic orbit directly from the isomorhphism between Seiberg-Witten Floer homology and Embedded contact homology. In addition to this, they asked whether $(Y,\lambda)$ with infinity many simple orbits also has a positive hyperbolic orbit under $b_{1}(Y)=0$. In the present paper, we answer this question under $Y \simeq S^{3}$ with nontrivial finite free group actions, especially lens spaces $(L(p,q),\lambda)$ with odd $p$ as quotient spaces of $S^{3}$.