Optimal transport and regularity of weak Kantorovich potentials on a globally hyperbolic spacetime

We consider the optimal transportation problem on a globally hyperbolic spacetime for some cost function $c_2$, which corresponds to the optimal transportation problem on a complete Riemannian manifold where the cost function is the Riemannian distance squared. Building on insights from previous studies on the Riemannian and Lorentzian case, our main goal is to investigate the regularity of $\pi$-solutions (weak versions of Kantorovich potentials), from which we can conclude, in a classical way, the existence, uniqueness and structure of an optimal transport map between given Borel probability measures $\mu$ and $\nu$, under suitable assumptions.