Gradient rearrangement for diffeomorphisms of a compact manifold

Dealing with smooth diffeomorphisms on a compact riemannian manifold, we recast in differential geometric terms the results of Brenier and McCann on optimal mass transportation via gradient rearrangement, which lack of a regularity theory. We proceed to a pde approach of the gradient rearrangement, proving uniqueness and local existence of classical solutions; we reduce global existence to a priori estimates (left open, except near flat metrics). We discuss the link between factorization of diffeomorphisms and the Helmholtz decomposition of vector fields, including a new result on the Moser–Ebin–Marsden factorization. A nonlinear comparison principle of independent interest is established.