Lagrangian Graphs, Minimizing Measures And Mañé's Critical Values

. Let L be a convex superlinear Lagrangian on a closed connected manifold M . We consider critical values of Lagrangians as defined by R. Ma~n'e in [13]. We show that the critical value of the lift of L to a covering of M equals the infimum of the values of k such that the energy level k bounds an exact Lagrangian graph in the cotangent bundle of the covering. As a consequence we show that up to reparametrization, the dynamics of the Euler-Lagrange flow of L on an energy level that contains minimizing measures with nonzero homology can be reduced to Finsler metrics. We also show that if the EulerLagrange flow of L on the energy level k is Anosov, then k must be strictly bigger than the critical value cu(L) of the lift of L to the universal covering of M . It follows that given k ! cu(L), there exists a potential / with arbitrarily small C 2 -norm such that the energy level k of L + / possesses conjugate points. 1. Introduction Let M be a closed connected smooth manifold and let L ..