Polar factorization of maps on Riemannian manifolds

Abstract. Let (M,g) be a connected compact manifold, C3 smooth and without boundary, equipped with a Riemannian distance d(x,y). If $ s : M \to M $ is merely Borel and never maps positive volume into zero volume, we show $ s = t \circ u $ factors uniquely a.e. into the composition of a map $ t(x) = {\rm exp}_x[-\nabla\psi(x)] $ and a volume-preserving map $ u : M \to M $, where $ \psi : M \to {\bold R} $ satisfies the additional property that $ (\psi^c)^c = \psi $ with $ \psi^c(y) :={\rm inf}\{c(x,y) - \psi(x)\,\vert\,x \in M\} $ and c(x,y) = d2(x,y)/2. Like the factorization it generalizes from Euclidean space, this non-linear 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 \ge 0 $ of mass in L1(M) onto another. Parallel results for other strictly convex cost functions $ c(x,y) \ge 0 $ of the Riemannian distance on non-compact manifolds are briefly discussed.