A variational theory for light rays in stably causal Lorentzian manifolds: Regularity and multiplicity results

This paper is dedicated to the study of light rays joining an event p with a timelike curve gamma in a light-convex subset Lambda of a stably causal Lorentzian manifold M. We set up a functional framework, defined intrinsically, consisting of a family of manifolds L-p,gamma,epsilon(+) and a positive functional Q defined on them. The critical points of Q on L-p,gamma,epsilon(+) approach, as epsilon --> 0, the lightlike, future pointing geodesics joining p and gamma. We prove some regularity results, including the C-1-regularity of L-p,gamma,epsilon(+), the C-2-regularity of Q on L-p,gamma,epsilon(+) and the C-2-regularity of its critical points. Using them, we develop a Ljusternik-Schnirelman theory for light rays, obtaining some multiplicity results, depending on the topology of the space of all lightlike curves joining p and gamma.