Aubry-Mather Theory for Lorentzian Manifolds

TL;DR

This paper extends Aubry-Mather theory to Lorentzian manifolds, establishing the existence of invariant measures and calibrated curves for causal geodesics, and demonstrating the optimality of results with specific examples.

Contribution

It introduces a novel Aubry-Mather framework for Lorentzian geometry, including new theorems and examples specific to causal curves in Lorentzian manifolds.

Findings

Existence of maximal invariant measures for causal curves

Development of calibration and calibrated curves in Lorentzian setting

Lorentzian Hedlund examples demonstrating optimality of results

Abstract

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