Periodic orbits on Riemannian manifolds with convex boundary

We look for T-periodic solutions on a convex Riemannian manifold M of the differential equation Dsẋ(s)+ ▽Vz(x(s),s) = 0 where Dsẋ(s) is the covariant derivative of ẋ(s), V is a C2 real function on M × R, T-periodic in s. The manifold is allowed to be noncompact and to have boundary, so the action integral associated to the equation does not satisty the Palais-Smale compactness condition. We overcome this problem under a assumption on the sectional curvature of M which allows to control the Morse index of the critical points of f at "infinity". If M has a "rich" topology it is proved tha there exist infinitely many periodic solutions.