发表机构
Research Institute for Mathematical Sciences, Kyoto University（京都大学数学科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明在闭连通三维流形上，若Reeb流具有至少三条简单周期轨道，则必存在一条正双曲轨道，方法基于ECH谱不变量的Weyl定律和亏格零全纯曲线的紧致性。
AI 中文摘要
三维Reeb流中的非退化周期轨道分为三类：正双曲、负双曲和椭圆。本文考虑具有非退化接触形式的闭连通接触三维流形。我们证明，若其Reeb流至少具有三条简单周期轨道，则存在一条简单正双曲轨道。我们主要研究不存在椭圆轨道的情形。我们证明，在闭连通三维流形上，当$b_1=0$时，不存在非退化接触形式使得所有简单周期轨道均为负双曲。作为推论，结合作者先前在存在椭圆轨道情形下的结果以及$b_1>0$时的已知结果，我们得到主要结论。证明使用了ECH谱不变量的Weyl定律。我们还使用了由$U$-映射计数的亏格零$J$-全纯曲线的紧致性。在相反假设下，每个作用区间$[L,2L]$中简单轨道的数量关于$L$一致有界。我们利用这一性质研究ECH生成元和亏格零$U$-曲线。
英文摘要
Non-degenerate periodic orbits in three-dimensional Reeb flows are classified into three types: positive hyperbolic, negative hyperbolic and elliptic. In the present paper, we consider a closed connected contact three-manifold with a non-degenerate contact form. We show that its Reeb flow has a simple positive hyperbolic orbit if it has at least three simple periodic orbits. We mainly study the case in which no elliptic orbit exists. We prove that there is no non-degenerate contact form on a closed connected three-manifold with $b_1=0$ such that all simple periodic orbits are negative hyperbolic. As a corollary, by combining the author's previous result in the presence of an elliptic orbit and the known result for $b_1>0$, we obtain the main result. The proof uses the Weyl law for ECH spectral invariants. We also use compactness for genus zero $J$-holomorphic curves counted by the $U$-map. Under the contrary assumption, the number of simple orbits in each action interval $[L,2L]$ is uniformly bounded with respect to $L$. We use this property to study ECH generators and genus zero $U$-curves.
Comments22pages
