The Palais-Smale Condition and Mañé's Critical Values

Abstract

Let $ {\Bbb L} $ be a convex superlinear autonomous Lagrangian on a closed connected manifold N. We consider critical values of Lagrangians as defined by R. Mane in [23]. We define energy levels satisfying the Palais-Smale condition and we show that the critical value of the lift of $ {\Bbb L} $ to any covering of N equals the infimum of the values of k such that the energy level t satisfies the Palais-Smale condition for every t > k provided that the Peierls barrier is finite. When the static set is not empty, the Peierls barrier is always finite and thus we obtain a characterization of the critical value of $ {\Bbb L} $ in terms of the Palais-Smale condition.¶We also show that if an energy level without conjugate points has energy strictly bigger than c u ($ {\Bbb L} $) (the critical value of the lift of $ {\Bbb L} $ to the universal covering of N), then two different points in the universal covering can be joined by a unique solution of the Euler-Lagrange equation that lives in the given energy level. Conversely, if the latter property holds, then the energy of the energy level is greater than or equal to c u ($ {\Bbb L} $). In this way, we obtain a characterization of the energy levels where an analogue of the Hadamard theorem holds. We conclude the paper showing other applications such as the existence of minimizing periodic orbits in every non-trivial homotopy class with energy greater than c u ($ {\Bbb L} $) and homologically trivial periodic orbits such that the action of $ {\Bbb L} $ + k is negative if c u ($ {\Bbb L} $) < k < c a ($ {\Bbb L} $), where c a ($ {\Bbb L} $) is the critical value of the lift of $ {\Bbb L} $ the abelian covering of N. We also prove that given an Anosov energy level, there exists in each non-trivial free homotopy class a unique closed orbit of the Euler-Lagrange flow in the given energy level.