Periodic trajectories with fixed energy on Riemannian and Lorentzian manifolds with boundary (Q1570435)

Let \(M\) be a Riemannian manifold (possibly non-compact and incomplete) which is convex close to its (singular) boundary points. In the paper existence and multiplicity results for periodic solutions with prescribed energy of a Lagrangian system and for closed geodesics on \(M\) are obtained. The proof of the main theorem is intrinsic. An estimate of the Morse index of the critical points is used in the proof to get the existence of the infinitely many orbits. The existence of periodic solutions of a Lagrangian system with fixed energy has been studied by \textit{E. Mirenghi} and \textit{M. Tucci} [J. Math. Anal. Appl. 199, 334-348 (1996; Zbl 0853.58085)]. Static Lorentzian manifolds with boundary, Schwarzschild and Reissner-Nordström (outer) spacetimes are important in relativity. As an application, the existence and the multiplicity of \(t\)-periodic trajectories with fixed energy on these manifolds is studied.