A Multidimensional Birkhoff Theorem for some $C^0$ Lagrangians

Abstract

We prove a multidimensional Birkhoff theorem for a new class of"$C^0$ Lagrangian subsets"in cotangent bundles, obtained as Hausdorff limits of compact exact Lagrangian submanifolds with control on their Liouville primitives. If the successive images of such a subset $L$ under the flow of a Tonelli Hamiltonian admit convergent subsequences in both positive and negative time, then $L$ and all its images are Lipschitz graphs over the base. This extends Birkhoff's celebrated theorem for twist maps of the annulus and its known higher-dimensional generalizations, and provides a first version of such results for merely continuous objects. The proof combines Floer-theoretic graph selectors, variational solutions of the Hamilton-Jacobi equation, and weak KAM theory. We also investigate the rigidity and uniqueness of limiting primitives for this new class of singular Lagrangian subsets, which may be of independent interest in $C^0$ symplectic topology.