On variational and topological properties of C¹ invariant Lagrangian tori

Abstract

We study C¹ invariant, two-dimensional tori of constant energy without singularities of the Euler–Lagrange flow of a convex, superlinear Lagrangian defined in the tangent space of a closed surface. We show that in regular energy levels, such tori are graphs of the canonical projection if and only if they are minimizing. This result was known for C³ invariant, Lagrangian tori of Finsler Lagrangians, which implied the statement for C³ invariant Lagrangian tori of unimodal Lagrangians with a high-energy level. We also show that graphs have energy $E \geq c_0(L)$, where c₀(L) is Mañé's strict critical value; and that the presence of Reeb components in such tori implies that the energy is c₀(L). Moreover, if we assume that the Lagrangian is unimodal, we show that Lagrangian invariant tori with no singularities have no Reeb components, regardless of the energy level.