Existence of a positive hyperbolic orbit in the presence of an elliptic orbit in three-dimensional Reeb flows

TL;DR

This paper proves the existence of a positive hyperbolic orbit in three-dimensional Reeb flows with at least one elliptic orbit, especially in cases where the first Betti number is zero, using ECH spectrum properties and holomorphic curve moduli spaces.

Contribution

It establishes the existence of positive hyperbolic orbits in new cases where previous methods failed, particularly when the first Betti number is zero.

Findings

Existence of positive hyperbolic orbit under certain conditions.

Use of ECH spectrum volume property and moduli space compactification.

Results extend previous work to cases with $b_1=0$.

Abstract

Nondegenerate periodic orbits in three-dimensional Reeb flows can be classified into three types, positive hyperbolic, negative hyperbolic and elliptic. As a problem which involves refining the three-dimensional Weinstein conjecture, D. Cristofaro-Gardiner, M. Hutchings and D. Pomerleano proposed whether every nondegenerate closed contact three manifold has at least one positive hyperbolic orbit except for lens spaces. In the same paper, they showed the existence of at least one simple hyperbolic orbit under $b_{1}>0$ by the isomorphism between ECH and Seiberg-Witten Floer (co)homology, especially, using the result that if $b_{1}>0$, the odd part of ECH which detects the existence of a positive hyperbolic orbit does not vanish. But in the case of $b_{1}=0$, such a way doesn't work. In the present paper, we prove the existence of a positive hyperbolic orbit in the presence of at least one elliptic orbit except for some well-known trivial cases under $b_{1}=0$. The key points in this paper are the volume property with respect to ECH spectrums and the compactification of the moduli spaces of certain $J$-holomorphic curves counted by the $U$-map.