周期磁测地线在任意低能量下的存在性与局部化
Periodic Magnetic Geodesics with Every Low Energy: Existence and Localization
- IMPA（巴西国家纯数学与应用数学研究所）
- SISSA（的里雅斯特高级研究学院）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文证明在磁场强度于紧集上达到严格局部极大值时，任意低能量水平均存在局部化的可缩周期磁测地线，并给出变分法证明。
中文摘要 AI 辅助
设 $(Q,g)$ 为配备非恒为零磁场（由闭 $2$-形式 $\eta$ 表示）的黎曼流形。我们允许 $Q$ 非紧致，且不假设度量 $g$ 完备。我们证明：若磁场强度（定义为 $\eta$ 的点态范数）在非空紧集 $K$ 上达到严格局部极大值，则每个充分低的能量水平都承载一条位于 $K$ 附近的 $(g,\eta)$ 的可缩周期磁测地线。更精确地，这样的轨道存在于 $K$ 的每个邻域中，且其长度随能量趋于零。特别地，若 $Q$ 紧致，则每个充分小的能量水平都承载一条可缩周期磁测地线。我们还证明，一般而言，$K$ 的非空性与紧致性均不可省略。我们的证明依赖于拉格朗日作用泛函的变分法，并使用了若干新思想。具体而言，我们克服了以下困难：(i) 通过将极小极大限制在短环集合上处理 $\eta$ 的非恰当性；(ii) 通过将 $Q$ 紧化与 Thom 的 Jet 横截性及能量趋于零的磁测地线序列的爆破相结合，处理 $g$ 的非完备性；(iii) 通过第一作者建立的 Ricci 磁曲率在低能量下的正性，处理 Palais--Smale 序列可能的非紧致性。与先前工作不同，Struwe 的单调性论证不适用于我们的目的，我们转而依赖 Abbondandolo 与 Majer 提出的双 Lyapunov 函数论证。
英文摘要
Let $(Q,g)$ be a Riemannian manifold equipped with a non-identically zero magnetic field represented by a closed $2$-form $β$. We allow $Q$ to be non-compact and do not assume that the metric $g$ is complete. We prove that if the magnetic strength, defined as the pointwise norm of $β$, attains a strict local maximum on a non-empty compact set $K$, then every sufficiently low energy level carries a contractible periodic magnetic geodesic of the pair $(g,β)$ localized near $K$. More precisely, such orbits exist in every neighborhood of $K$, and their lengths converge to zero with the energy. In particular, if $Q$ is compact, then every sufficiently small energy level carries a contractible periodic magnetic geodesic. We also show that, in general, neither the non-emptiness nor the compactness of $K$ can be omitted. Our proof relies on the calculus of variations of the Lagrangian action functional, and uses several new ideas. More precisely, we overcome: (i) the non-exactness of $β$ by restricting the minimax to the set of short loops; (ii) the non-completeness of $g$ by combining a compactification of $Q$ with Thom's Jet Transversality and a blow-up for sequences of magnetic geodesics with energy tending to zero; (iii) the possible non-compactness of Palais--Smale sequences by the positivity of the Ricci magnetic curvature for low energy established by the first-named author. Unlike previous work, Struwe's monotonicity argument cannot be used for our purposes, and we rely on a two-Lyapunov-function argument due to Abbondandolo and Majer.
