The Palais-Smale condition for contact type energy levels for convex lagrangian systems

We prove that for a uniformly convex Lagrangian system L on a compact manifold M, almost all energy levels contain a periodic orbit. We also prove that below Ma ne’s critical value of the lift of the Lagrangian to the universal cover, almost all energy levels have conjugate points. We prove that if the energy level [E=k] is of contact type and M is not the 2-torus then the free time action functional of L+k satisfies the Palais-Smale condition.