CRITICAL-VALUES OF AUTONOMOUS LAGRANGIAN SYSTEMS

Let M be a closed manifold and L:TM --> R a convex superlinear Lagrang ian. We consider critical values of Lagrangians as defined by R. Mane 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, c(u)(L) less than or equal to c(a)(L). Let c(0)(L) denote the strict critical Value of L defined as the smallest critical value of L - w where w ranges among all possible closed I-forms. We show that c (a),(L) = c(0)(L). We also show that if there exists Ic such that the Euler-Lagrange flow of L on the energy level k' is Anosov for all k' g reater than or equal to k, then k > c(u)(L). Afterwards, we exhibit a. Lagrangian on a compact surface of genus two which possesses Anosov e nergy levels with energy k < c(a)(L), thus answering in the negative a question raised by Mane. This example also shows that the inequality c(u)(L) less than or equal to c(a)(L) could be strict. Moreover, by a result of M.J. Dias Carneiro [4] these Anosov energy levels do not hav e minimizing measures. Finally, we describe a large class of Lagrangia ns for which c(u) (L) is strictly bigger than the maximum of the energ y restricted to the zero section of TM.