Polar factorization of maps on Riemannian manifolds

Robert J. McCann

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Submission date: 02. Jun. 1999
Pages: 18
published in: Geometric and functional analysis, 11 (2001) 3, p. 589-608 
DOI number (of the published article): 10.1007/PL00001679
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Abstract:
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) = exp_x[ - 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.

Robert J. McCann
Department of Mathematics
University of Toronto
Toronto Ontario Canada M5S 3G3
mccann@math.toronto.edu