Polar factorization of maps on Riemannian manifolds

Abstract.

Let (M,g) be a connected compact manifold, C ³ smooth and without boundary, equipped with a Riemannian distance d(x,y). If $s : M \to M$ 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) = d ²(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 L ¹(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.