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Polar factorization of maps on Riemannian manifolds. (English) Zbl 1011.58009
Summary: 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 $$t(x)= \exp_x [-\nabla \psi(x)]$$ and a volume-preserving map $$u:M\to M$$, where $$\psi: M\to\mathbb{R}$$ satisfies the additional property that $$(\psi^c)^c=\psi$$ with $$\psi^c(y):= \inf\{c(x,y) -\psi(x)\mid x\in M\}$$ 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 $$f\geq 0$$ of mass in $$L^1(M)$$ onto another. Parallel results for other strictly convex cost functions $$c(x,y)\geq 0$$ of the Riemannian distance on non-compact manifolds are briefly discussed.

##### MSC:
 58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals 46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics 49Q20 Variational problems in a geometric measure-theoretic setting 53C20 Global Riemannian geometry, including pinching
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