论文标题
在奇异的温斯坦猜想和$ b $ beltrami领域的逃生轨道的存在
On the singular Weinstein conjecture and the existence of escape orbits for $b$-Beltrami fields
论文作者
论文摘要
由Poincaré的轨道在（受限制的）三体（见[26]和[6]）中的无限范围内的动机，我们研究了$ b $ - contact形式的关键集合中的杂斜状轨道的一般存在。这是通过使用Etnyre的奇异对应物[3] - Ghrist的联系/Beltrami对应[9]，并且有关Uhlenbeck [29]建立的laplacian的特征性函数的通用结果。具体来说，我们分析了$ b $ b $ beltrami vector Fields $ b $ - 尺寸$ 3 $的manifolds，并证明了它们表现出的一般渐近性$ b $ metric，它们展示了逃生轨道。我们还表明，在渐近平坦的$ b $ - manifold上，渐近$ B $ b $ beltrami矢量场具有概括性的周期性轨道，至少有$ 4 $的逃生轨道。广义的奇异周期轨道是向量场的轨迹，其$α$ - 和$ω$ -LIMIT设置与关键表面相交。这些结果是证明奇异的温斯坦猜想的第一步。
Motivated by Poincaré's orbits going to infinity in the (restricted) three-body (see [26] and [6]), we investigate the generic existence of heteroclinic-like orbits in a neighbourhood of the critical set of a $b$-contact form. This is done by using the singular counterpart [3] of Etnyre--Ghrist's contact/Beltrami correspondence [9], and genericity results concerning eigenfunctions of the Laplacian established by Uhlenbeck [29]. Specifically, we analyze the $b$-Beltrami vector fields on $b$-manifolds of dimension $3$ and prove that for a generic asymptotically exact $b$-metric they exhibit escape orbits. We also show that a generic asymptotically symmetric $b$-Beltrami vector field on an asymptotically flat $b$-manifold has a generalized singular periodic orbit and at least $4$ escape orbits. Generalized singular periodic orbits are trajectories of the vector field whose $α$- and $ω$-limit sets intersect the critical surface. These results are a first step towards proving the singular Weinstein conjecture.
