Polar factorization of maps on Riemannian manifolds

Let (M, g) be a connected compact manifold, C-3 smooth and without boundary, equipped with a Riemannian distance d(x, y). If s : M --> M is merely Borel and never maps positive volume into zero volume, we show s = t circle u factors uniquely a.e. into the composition of a map t(x) = exp(x)[-del psi (x)] and a volume-preserving map u : M --> M, where psi : M --> R satisfies the additional property that (psi (c))(c) = psi with psi (c)(y) := inf {c(x,y) - psi (x) \ x is an element of M} and c(x, y) = d(2)(x, y)/2. Like the factorization it generalizes from Euclidean space, this nonlinear decomposition can be linearized around the identity to yield the Hodge decomposition of vector fields. The results are obtained by solving a Riemannian version of the Monge-Kantorovich problem, which means minimizing the expected value of the cost c(x, y) for transporting one distribution f greater than or equal to 0 of mass in L-1(M) onto another. Parallel results for other strictly convex cost functions c(x, y) greater than or equal to 0 of the Riemannian distance on non-compact manifolds are briefly discussed.