Parabolic Optimal Transport Equations on Manifolds

We study a parabolic equation for finding solutions to the optimal transport problem on compact Riemannian manifolds with general cost functions. We show that if the cost satisfies the strong Ma-Trudinger-Wang condition and the stay-away singularity property, then the solution to the parabolic flow with any appropriate initial condition exists for all time and converges exponentially to the solution to the optimal transportation problem. Such results hold in particular on the sphere for the distance squared cost of the round metric and for the far-field reflector antenna cost, among others.