A quantitative stability result for regularity of optimal transport on compact manifolds

We use a Korevaar-style maximum principle approach to show the following: Fixing a $C^{2}$ bound on the log densities of a set of smooth measures, there is a quantifiably-sized Wasserstein neighborhood over which all pairs of such measures will enjoy smooth optimal transport. \ We do this in spite of unhelpful MTW\ curvature, by showing that when the gradient of the Kantorovich potential is small enough, the Hessian ``bound" places the Hessian in one of two disconnected regions, one bounded and the other unbounded. \ Tracking the estimate along a continuity path which starts in the bounded region, we conclude the Hessian must stay bounded.