Multiplicity of Finsler geodesics and electromagnetic trajectories via localized deformations

Abstract

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 ${\mathbb{R}}^n$, $n\ge 2$, we consider conformal deformations ${\phi }_{\lambda }F$ 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 $\lambda$, 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 $\lambda$, 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 $\alpha +\beta$, with $\alpha$ deformed and $\beta$ 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.

27 pages, AMSLaTex