Length maximizing invariant measures in Lorentzian geometry

Abstract

We introduce a version of Aubry-Mather theory for the length functional of causal curves in a compact Lorentzian manifold. Results include the existence of maximal invariant measures, calibrations and calibrated curves. We prove two versions of Mather’s graph theorem for Lorentzian manifolds. A class of examples (Lorentzian Hedlund examples) shows the optimality of the results.