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arXiv 0906.1192math.DSmath.DG
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弱正合磁场中电荷的闭轨道

Closed orbits of a charge in a weakly exact magnetic field

Will J. Merry

备注：
25 pages. v3 - minor corrections, this version to appear in PJM
发表信息：
Pacific J. Math. 247 (2010) 189-212
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摘要：

我们证明了在闭连通黎曼流形上的弱正合磁系统，几乎所有的能级都包含一个闭轨道。更确切地说，我们证明了以下更强有力的陈述。设$(M,g)$表示闭连通黎曼流形，$σ$表示弱正合2-形式.令$ϕ_{t}$表示由$σ$确定的磁通，并且令$c$表示对$(g,σ)$的Mane临界值。证明了如果$k>c$，则对$M$上的每个非平凡自由同伦环类，存在一个能量为$k$的闭轨道，其到$M$的投影属于该自由同伦环类.我们还证明了对于几乎所有的$k<c$，存在一个能量为$k$的闭轨道，它在$M$上的投影是可缩的.特别地，当$c=\infty$时，这意味着几乎每个能级都有一个可收缩的闭合轨道。作为推论，我们推导出：如果$σ$不是精确的，并且$M$有一个顺从的基本群（这意味着$c=\infty$），那么几乎在每个能级上都存在可收缩的闭轨道.

英文摘要：

We prove that for a weakly exact magnetic system on a closed connected Riemannian manifold, almost all energy levels contain a closed orbit. More precisely, we prove the following stronger statements. Let $(M,g)$ denote a closed connected Riemannian manifold and $σ$ a weakly exact 2-form. Let $ϕ_{t}$ denote the magnetic flow determined by $σ$, and let $c$ denote the Mane critical value of the pair $(g,σ)$. We prove that if $k>c$, then for every non-trivial free homotopy class of loops on $M$ there exists a closed orbit with energy $k$ whose projection to $M$ belongs to that free homotopy class. We also prove that for almost all $k<c$ there exists a closed orbit with energy $k$ whose projection to $M$ is contractible. In particular, when $c=\infty$ this implies that almost every energy level has a contractible closed orbit. As a corollary we deduce that if $σ$ is not exact and $M$ has an amenable fundamental group (which implies $c=\infty$) then there exist contractible closed orbits on almost every energy level.

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