论文标题
在三维REEB流中存在椭圆轨道的存在下存在阳性双曲线轨道
Existence of a positive hyperbolic orbit in the presence of an elliptic orbit in three-dimensional Reeb flows
论文作者
论文摘要
三维REEB流中的非等级周期轨道可以分为三种类型，即阳性双曲线，负双曲线和椭圆形。作为一个涉及完善三维的温斯坦猜想的问题，D。Cristofaro-Gardiner，M。Hutchings和D. Pomerleano提出，除了镜头空间外，每个非等级封闭的封闭触点是否至少具有一个阳性荷毛轨道。在同一篇论文中，他们通过ECH和Seiberg-Witten Floer（CO）同源物之间的同构象征性至少在$ b_ {1}> 0 $下存在至少存在一个简单的双曲线轨道，尤其是使用结果，如果$ b_ {1}> 0 $ 0 $，则检测到奇怪的ECH的奇怪部分，它可以检测到阳性的超重或物质均不范围。但是，在$ b_ {1} = 0 $的情况下，这种方式行不通。在本文中，我们证明了至少一个椭圆轨道存在的正双曲线轨道的存在，除了$ b_ {1} = 0 $下的某些众所周知的琐事情况。本文的要点是相对于ECH频谱的卷属性以及$ u $ -map计数的某些$ j $ holomorphic曲线的模量空间的压缩。
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.
