  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 MTW condition and the stay-away
singularity property, then the solution to the parabolic flow with any
appropriate initial condition exists for all time and it 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.
