Zoll接触形式的扰动的周期轨道
Periodic orbits for perturbations of Zoll contact forms
AI总结：
该研究通过构造与区域相交的周期Reeb轨道，将Zoll接触形式的扰动结果应用于精确磁场，证明了周期磁测地线的存在性，其核心依赖于Rabinowitz作用泛函的梯度流线与模空间紧性估计。
AI中文摘要：
我们考虑在底流形的某个指定区域上，将Zoll接触形式乘以一个在该区域外取值恒为1的正函数，以此对其进行扰动。对于每一个足够C⁰-小的此类扰动，我们找到了一条与该区域相交的、被扰动接触形式的周期Reeb轨道。作为应用，从一个Zoll黎曼流形出发，我们证明对于每一个在某个给定区域外消失的、具有C⁰-小的磁势的精确磁场，都存在一条与该区域相交的周期磁测地线。该定理是一个具有独立意义的结果的相当直接的推论：对于与定义哈密顿量足够C⁰-接近的哈密顿量，我们证明存在对应的Rabinowitz作用泛函的梯度流线，且该流线满足圆柱分量在(0,0)∈ℝ×𝕊¹处的位置约束。这依赖于Rabinowitz作用泛函的同伦延拓论证，在该过程中，我们必须推导一些精细的估计，以确保出现的模空间的紧性。
英文摘要：
We consider the perturbation of a Zoll contact form on some prescribed domain of the underlying manifold by multiplying it with a positive function which is constant of value $1$ outside the domain. For every sufficiently $C^0$-small such perturbation we find a periodic Reeb orbit of the perturbed contact form intersecting the domain. As an application, starting from a Zoll Riemannian manifold, we demonstrate that for every exact magnetic field, with $C^0$-small magnetic potential vanishing outside some given domain, there exists a periodic magnetic geodesic intersecting this domain. The theorem is a rather direct consequence of a result which is of independent interest: For Hamiltonians sufficiently $C^0$-close to a defining Hamiltonian, we show existence of a gradient flow line of the corresponding Rabinowitz action functional with a constraint on the position of the cylinder component at $(0,0) \in \mathbb{R} \times \mathbb{S}^1$. This relies on a homotopy stretching argument for the Rabinowitz action functional, in the course of which we have to derive some delicate estimates to ensure compactness of the appearing moduli spaces.
