Nonexistence of invariant graphs in all supercritical energy levels of mechanical Lagrangians in T2

Let (T-2, g) be a smooth Riemannian structure in the torus T-2. We show that given epsilon > 0 and any C-infinity function U : T-2 degrees -> R there exists a C-1 function U-epsilon with Lipschitz derivatives that is epsilon-C-0 close to U for which there are no continuous invariant graphs in any supercritical energy level of the mechanical Lagrangian L-epsilon : TT2 -> R given by L (p, v) = 2/1 g (upsilon, upsilon) - U-epsilon (p). We also show that given n is an element of N, the set of C-infinity potentials U : T-2 -> R for which there are no continuous invariant graphs in any supercritical energy level E <= n of L(p, upsilon) = 2/1g(upsilon, upsilon) - U(p) is C-0 dense in the set of C-infinity functions.