发表机构
Politecnico di Bari; Università degli Studi di Bari Aldo Moro（巴里理工大学; 巴里阿尔多·莫罗大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过局部共形形变构造能量势垒，证明 Finsler 流形上存在任意多条连接两点的测地线，并推广到 Randers 度量、电磁轨迹及光线，得到多重性结果。
AI 中文摘要
给定一个 Finsler 流形 $(M,F)$ 以及合适的嵌套区域 $B_1\subset B_2\subset B_3$，其中 $M\setminus B_1$ 连通且不可收缩，例如在 $\mathbb R^n$（$n\geq2$）中同心球的模型情形下，我们考虑共形形变 $\varphi_\lambda F$，该形变可以选取得使得在 $B_3$ 之外保持 $F$ 不变，并在 $B_2\setminus B_1$ 上增大。这种增大产生一个能量势垒，其随 $\lambda$ 发散，低能量曲线无法穿越。因此，极小极大族可以在 $M\setminus B_1$ 的路径空间中选择，该空间包含任意大范畴的紧子集。由此，对于每个 $m\in\mathbb N$ 和所有足够大的 $\lambda$，至少存在 $m$ 条连接两个给定点的测地线，且所有这些测地线都避开 $B_1$。这将对 Capozzi、Fortunato 和 Greco 的思想推广到 Finsler 情形。同一方案适用于 Randers 度量 $\alpha+\beta$（其中 $\alpha$ 被形变而 $\beta$ 不变），也适用于非相对论电磁系统的固定能量轨迹、相对论洛伦兹力轨迹，以及具有因果 Killing 场的时空中光线。在非相对论情形下，仅扰动标量势，而磁场保持不变。当 Killing 场并非处处类时，Fermat 原理在类空超曲面上产生一个可能奇异的 Randers--Kropina 度量。在自然的可容许范畴条件下，将局部势垒与 Randers 逼近相结合，得到 $m$ 条两两长度不同的 Randers--Kropina 测地线，等价于 $m$ 条到达时间两两不同的未来指向光线。
英文摘要
Given a Finsler manifold $(M,F)$ and suitable nested regions $B_1\subset B_2\subset B_3$, with $M\setminus B_1$ connected and non-contractible, as in the model case of concentric balls in $\mathbb R^n$, $n\geq2$, we consider conformal deformations $φ_λF$ which can be chosen so as to leave $F$ unchanged outside $B_3$ and grow on $B_2\setminus B_1$. This growth creates an energy barrier, diverging with $λ$, that low-energy curves cannot cross. The minimax families can therefore be chosen in the path space of $M\setminus B_1$, which contains compact subsets of arbitrarily large category. Consequently, for every $m\in\mathbb N$ and all sufficiently large $λ$, there exist at least $m$ geodesics joining two prescribed points, all avoiding $B_1$. This extends to the Finsler setting an idea of Capozzi, Fortunato and Greco. The same scheme applies to Randers metrics $α+β$, with $α$ deformed and $β$ unchanged, as well as to fixed-energy trajectories of non-relativistic electromagnetic systems and relativistic Lorentz-force trajectories, and light rays in spacetimes endowed with a causal Killing field. In the non-relativistic case only the scalar potential is perturbed, while the magnetic field is preserved. When the Killing field is not everywhere timelike, the Fermat principle yields a possibly singular Randers--Kropina metric on a spacelike hypersurface. Under a natural admissible category condition, combining localized barriers with Randers approximation gives $m$ Randers--Kropina geodesics with pairwise distinct lengths, equivalently $m$ future-pointing light rays with pairwise distinct arrival times.
Comments27 pages, AMSLaTex
