ON C1,β DENSITY OF METRICS WITHOUT INVARIANT GRAPHS

We show that given any C-infinity Riemannian structure (T-2, g) in the two torus, epsilon > 0 and beta is an element of 2 (0, 1/3), there exists a C-infinity Riemannian metric (g) over bar with no continuous Lagrangian invariant graphs that is epsilon-C-1,C-beta close to g. The main idea of the proof is inspired in the work of V. Bangert who introduced caps from smoothed cone type C-1 small perturbations of metrics with non-positive curvature to get conjugate points. Our new contribution to the subject is to show that positive curvature cone type small perturbations are "less singular" than non-positive curvature cone type perturbations. Positive curvature geometry allows us to get better estimates for the variation of the C-1 norm of the singular cone in a neighborhood of its vertex.