On construction of time functions

By Albert Fathi (ENS, Lyon and IUF)

This is a joint work with Antonio Siconolfi. The purpose of this lecture is to introduce a different way to obtain Lipschitz time functions on causal Lorentzian manifolds. We will show that they can be obtained from a kind of degenerate auxiliary Finsler metric. We will also explain how this leads to the existence of smooth time functions. A feature of this approach is that it works more generally for fields of cones on a manifold. Our approach evolved from our work on smooth sub-solutions of the Hamilton-Jacobi Equation and the relevance of the Aubry set to this problem. However no prior knowledge of this last subject will be used in the lecture.