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BIRKHOFF FIRST THEOREM FOR LAGRANGIAN, INVARIANT TORI IN DIMENSION 3

We show that C 2 Lagrangian, minimizing torus that is invariant by the geodesic ow of a Riemannian metric g in the three torus T 3 is a graph of the canonical projection. This result known as the rst Birkho Theorem for Lagrangian, invariant tori was proved in dimension two by Bialy and Polterovich in the early 1980's and since then, no further progress in higher dimensions has been made. As a consequence of our arguments, we get the proof of some partial versions of the rst Birkho Theorem in any dimension.