Lagrangian Graphs, Minimizing Measures and Mañé's Critical Values

Abstract. Let $\Bbb L$ be a convex superlinear Lagrangian on a closed connected manifold N. We consider critical values of Lagrangians as defined by R. Mañé in [M3]. We show that the critical value of the lift of $\Bbb L$ to a covering of N 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 $\Bbb L$ on an energy level that contains supports of minimizing measures with non-zero rotation vector can be reduced to Finsler metrics. We also show that if the Euler-Lagrange flow of $\Bbb L$ on the energy level k is Anosov, then k must be strictly bigger than the critical value cu ( $\Bbb L$) of the lift of L to the universal covering of N. It follows that given k < cu ( $\Bbb L$), there exists a potential $\psi$ with arbitrarily small C2-norm such that the energy level k of $\Bbb L + \psi$ possesses conjugate points. Finally we show the existence of weak KAM solutions for coverings of N and we explain the relationship between Fathi's results in [F1,2] and Mañé's critical values and action potentials.