论文标题
最小轨道的Conley-Zehnder索引和正双曲线轨道的存在
The Conley-Zehnder index of a minimal orbit and existence of a positive hyperbolic orbit
论文作者
论文摘要
作为Weinstein猜想的改进，是否存在特定类型的Reeb轨道是一个自然的问题。 D. Cristofaro-Gardiner，M。Hutchings和D. Pomerleano表明，每个非排效器封闭的接触三歧管，带有$ b_ {1}> 0 $，至少具有一个正阳性双曲线轨道，直接使用ECH和Seiberg-witten Floer（CO）同源性之间的同构。在同一篇论文中，他们还询问了$ b_ {1} = 0 $的情况。假设$（s^{3}，λ）$是无限范围的无限范围的无限范围。在本文中，我们证明在$（s^{3}，λ）$上存在一个简单的积极双曲线轨道，条件是，在最小的周期性周期性圆盘引起的最小周期性轨道的conley-Zehnder索引中，界限圆盘的琐碎磁盘的诱因大于或等于3。 $（s^{3}，λ）$带有许多简单轨道。在PracteRR中，这意味着A $ c^{\ infty} $通用紧凑型严格凸出能量超出表面的$ \ mathbb {r}^{4} $带有一个正质量简单轨道。
As a refinement of the Weinstein conjecture, it is a natural question whether a Reeb orbit of particular types exists. D. Cristofaro-Gardiner, M. Hutchings and D. Pomerleano showed that every nondegenerate closed contact three manifold with $b_{1}>0$ has at least one positive hyperbolic orbit by directly using the isomorphism between ECH and Seiberg-Witten Floer (co)homology. In the same paper, they also asked whether the case of $b_{1}=0$ does. Suppose that $(S^{3},λ)$ is non-degenerate contact three sphere with infinity many orbits. In the present paper, we prove the existence of a simple positive hyperbolic orbit on $(S^{3},λ)$ under the condition that the Conley-Zehnder index of a minimal periodic orbit induced by the trivialization of a bounding disc is larger than or equal to 3. As an immediate corollary, we have the existence of a simple positive hyperbolic orbit on a non-degenerate dynamically convex contact three sphere $(S^{3},λ)$ with infinity many simple orbits. In particulr, this implies that a $C^{\infty}$ generic compact strictly convex energy hypersurface in $\mathbb{R}^{4}$ carries a positive hyperbolic simple orbit.
