无共轭点的磁流平均洛伦兹力的上界——以水平圆的测地曲率表示
An upper bound for the mean Lorentz force of magnetic flows without conjugate points in terms of the geodesic curvature of horocycles
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中文总结 AI 辅助
本文研究亏格至少二的闭黎曼曲面上无共轭点的磁流，证明在 Mañé 临界值之上，洛伦兹力积分受水平圆测地曲率约束，并由此推出基于面积和高斯曲率的显式上界。
中文摘要 AI 辅助
我们研究亏格至少为二的闭黎曼曲面上的磁流。设 $(M,g)$ 为无焦点的曲面，$\Omega$ 为磁场。我们证明，在 Mañé 临界值之上，无共轭点的条件对磁流施加了几何限制。首先，我们以底层黎曼度量的水平圆的测地曲率获得了洛伦兹力积分的界。作为推论，高斯曲率的下界给出了以曲面面积表示的显式界。证明结合了磁测地线的全局几何与 Liouville 公式，该公式用于在由万有覆盖上的 Busemann 函数的水平集和测地线给出的正交坐标中计算测地曲率。
英文摘要
We study magnetic flows on closed Riemannian surfaces of genus at least two. Let $(M,g)$ be a surface without focal points and let $Ω$ be a magnetic field. We prove that, above the Mañé critical value, the absence of conjugate points imposes geometric restrictions on the magnetic flow. First, we obtain a bound for the integral of the Lorentz force in terms of the geodesic curvature of the horocycles of the underlying Riemannian metric. As a consequence, a lower bound on the Gaussian curvature yields an explicit bound in terms of the area of the surface. The proofs combine the global geometry of magnetic geodesics with Liouville's formula for the geodesic curvature in the orthogonal coordinates given by the level sets and geodesics of a Busemann function on the universal cover.
