Critical values of autonomous Lagrangian systems

Abstract

Let M be a closed manifold and $L:TM\to \mathbf{R}$ a convex superlinear Lagrangian. We consider critical values of Lagrangians as defined by R. Mañé in [5]. Let cu(L) denote the critical value of the lift of L to the universal covering of M and let ca(L) denote the critical value of the lift of L to the abelian covering of M. It is easy to see that in general, $c_u(L)\le c_a(L)$ . Let c0(L) denote the strict critical value of L defined as the smallest critical value of $L-\omega$ where $\omega$ ranges among all possible closed 1-forms. We show that ca(L) = c0(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 $k^{\prime }\ge k$ , then $k>c_u(L)$ . Afterwards, we exhibit a Lagrangian on a compact surface of genus two which possesses Anosov energy levels with energy $k<c_a(L)$ , thus answering in the negative a question raised by Mañé. This example also shows that the inequality $c_u(L)\le c_a(L)$ 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 cu(L) is strictly bigger than the maximum of the energy restricted to the zero section of TM.