On \(C^{1,\beta}\) density of metrics without invariant graphs (Q2409077)

The authors give an answer to the following problem: what is the largest \(\alpha \geq 0\) such that \(C^{1,\alpha}\) arbitrarily small perturbations of an integrable Hamiltonian do not have any continuous Lagrangian invariant graph? Using Bangert's idea to obtain small perturbations of geodesic flows on the \(n\)-dimensional torus \(T^n\) without Lagrangian invariant graphs and considering the gap between \(C^1\) and \(C^2\) perturbations, they obtain better estimates for the variation of the \(C^1\)-norm of the singular cone in a neighborhood of its vertex.