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MiS Preprint

Polar factorization of maps on Riemannian manifolds

Robert J. McCann


Let (M,g) be a connected compact manifold, C3 smooth and without boundary, equipped with a Riemannian distance d(x,y). If a Borel mapping of M to itself never maps positive volume into zero volume, we show it factors uniquely a.e. into the composition of a map t(x) = expx[ - grad f(x) ] and a volume-preserving map u, where f is a real-valued function on M given by an infimal convolution with c(x,y) = d2 /2. Like Brenier's factorization which 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 L1 distribution g of mass throughout M onto another. A companion article extends this solution to strictly convex or concave cost functions c(x,y) which increase with the Riemannian distance on a non-compact manifold.


Related publications

2001 Repository Open Access
Robert J. McCann

Polar factorization of maps on Riemannian manifolds

In: Geometric and functional analysis, 11 (2001) 3, pp. 589-608