Abstract
We show that given any $C^{\infty}$ Riemannian structure $(T^{2},g)$ in the two torus, $\epsilon >0$ and $\beta \in (0,\frac{1}{3})$, there exists a $C^{\infty}$ Riemannian metric $\bar{g}$ with no continuous Lagrangian invariant graphs that is $\epsilon$-$C^{1,\beta}$ 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.
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