Boundary Regularity of Optimal Transport Maps in the Monge Problem on Riemannian Manifolds

We study the boundary regularity of the optimal transport maps in the Monge problem on Riemannian manifolds. For the quadratic cost induced by the Riemannian distance, we show that the optimal map is Hölder continuous up to the boundary when the source domain has a strictly convex boundary and the ambient manifold has bounded sectional curvature. Our argument is based on a combination of c-convex analysis and barrier constructions adapted to the Fermi coordinates in a neighbourhood of the boundary. We also give explicit counterexamples showing that regularity may fail when the convexity of the boundary is violated. These results generalise classical boundary regularity results of the Euclidean theory to curved geometries and clarify the geometric conditions required for continuity of optimal maps. We discuss applications to geometric PDE's on manifolds with boundary, and further open questions.