AN APPLICATION OF AUBRY-MATHER THEORY IN LORENTZIAN GEOMETRY

The results presented in this note have been obtained in the framework of a research in progress with Albert Fathi, see [4]. We give an application of the Aubry-Mather theory for non regular Hamiltonian, recently developed in [6], see also [5], [3] in Lorentzian ge-ometry. Namely, we provide an alternative proof, based on Aubry-Mather theory, of the existence of a smooth time function for Lorentzian manifolds enjoying suitable assumptions. Our approach is related to that of [9], where Aubry-Mather theory is exploited to construct a Lyapunov function for a multivalued dynamics. We make in this way a connection which is, at a first sight, quite sur-prising since the Aubry-Mather theory concerns the qualitative analysis of Hamilton-Jacobi equations at the critical value, i.e. when subsolutions, but not strect subsolutions, do exist, while, in the time function problem, no Hamiltonians are involved. In what follows we try to explain how to bridge this gap. 2. AUBRY SET In terms of Hamilton-Jacobi equations the issue we are interested on can be described as follows: Given an Hamiltonian $H(x,p) $ defined on a cotangent bundle of a (smooth, $bo$undaryless, connected, paracompact, Hausdorff) manifold $M $ , we look for a smooth $(C^{\infty}) $ strict subsolution of the equation $H(x, Du)=0$, (1) or at least for a strict subsolution which is smooth in a open subset of $M $ as large as possible. Here and throughout the paper subsolution means locally Lipschitz-continuous $a.e $. subsolution. The Hamiltonian $H $ is required to be 1542 2007 12-23 12 continuous in both arguments, convex and coercive in $p $. Clearly, to make the problem sensible, we also have to assume that some subsolutions to (1) do exist. We then consider the point$sy $ of $M $ possessing a neighborhood where some subsolution $u $ to (1) satisfies $H(x, Du(x).)\leq-\delta $ a.e. for some $\delta>0 $ , (2) and denote by $M_{0} $ the (possibly empty) open subset made up by such points. We have: Theorem 2.1. There exists a subsolution to (1), which is smooth and stric