Weak KAM theory and viscosity solutions on metric spaces

Weak KAM theory originally connected Mather theory of Lagrangian Systems with Viscosity Theory of the solutions of the corresponding Hamilton-Jacobi Equation, at least when the Hamiltonian is obtained from a Lagrangian. In such a case the Mañé potential is the minimal action necessary to join two points in arbitrary time. We will show that we can recover just from the Mañé potential concepts like Peierls barrier, Aubry sets, viscosity subsolutions and solutions. This allows the theory to apply in the more general framework of compact metric spaces, opening a way to define solutions of the Hamilton-Jacobi equation on general metric spaces.