Critical values of autonomous Lagrangian systems

.   Let M be a closed manifold and a convex superlinear Lagrangian. We consider critical values of Lagrangians as defined by R. Mañé in [5]. Let c u (L) denote the critical value of the lift of L to the universal covering of M and let c a (L) denote the critical value of the lift of L to the abelian covering of M. It is easy to see that in general, . Let c 0 (L) denote the strict critical value of L defined as the smallest critical value of where ranges among all possible closed 1-forms. We show that c a (L) = c 0 (L). We also show that if there exists k such that the Euler-Lagrange flow of L on the energy level k' is Anosov for all , then . Afterwards, we exhibit a Lagrangian on a compact surface of genus two which possesses Anosov energy levels with energy , thus answering in the negative a question raised by Mañé. This example also shows that the inequality could be strict. Moreover, by a result of M.J. Dias Carneiro [4] these Anosov energy levels do not have minimizing measures. Finally, we describe a large class of Lagrangians for which c u (L) is strictly bigger than the maximum of the energy restricted to the zero section of TM.