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An introduction to the study of variational and topological aspects of invariant tori of the geodesic flow of surfaces, Rafael Oswaldo Ruggiero
The purpose of this book is two folded: to give a proof of the fact that an invariant Lagrangian torus of the geodesic flow of the torus is a graph if and only if the torus is minimizing; and to discuss the generic non-existence of invariant graphs of the geodesic flow in the torus. The former result was proved by Bialy and Polterovich in [6] for C3 invariant tori, but in fact their arguments granted the proof of the converse of this statement assuming class C0 and the absence of periodic orbits of the torus. Bangert in [3] gave independently a proof of the same fact, that C0 minimizing tori without periodic orbits are graphs. We prove here that continuous invariant graphs of the geodesic flow of a torus are minimizing, combining some basic ideas of calculus of variations and the theory of ordinary differential equations in the torus. The approach presented here is totally different from the classical approach used to deal with the problem, it is closely related with the theory of codimension-one foliations satisfying geometric properties (totally geodesic foliations, foliations by minimal surfaces). This exposition is contained in [49] and generalizes the previous works about the problem where it was needed to assume at least the C1 smoothness of the invariant torus. We show the converse of this statement assuming class C1 for the invariant torus with no other assumption on the dynamics of the geodesic flow in the invariant torus. The ideas presented here to show that minimizing tori are graph are taken from [12], where a more general version of the proof is applied to study invariant Lagrangian tori of the Euler-Lagrange flow of any convex, superlinear Lagrangian in the torus.