A Variational Theory for Light Rays in Stably Causal Lorentzian Manifolds: Regularity and Multiplicity Results ts

This paper is dedicated to the study of light rays joining an event p with a timelike curve γ in a light–convex subset &\Lambda; of a stably causal Lorentzian manifold . We set up a functional framework, defined intrinsically, consisting of a family of manifolds and a positive functional Q defined on them. The critical points of Q on approach, as , the lightlike, future pointing geodesics joining p and γ. We prove some regularity results, including the C 1–regularity of , the C 2–regularity of Q on 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 γ