Polar factorization of maps on Riemannian manifolds
Polar factorization of maps on Riemannian manifolds
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DOI：
10.1007/pl00001679
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发表时间：
2001-01-01
影响因子：
2.2
通讯作者：
McCann, RJ
中科院分区：
文献类型：
--
作者：
McCann, RJ
设（M，g）是连通的紧致流形，C-3光滑且无边界，具有黎曼距离d（x，y）.若s：M --> M仅仅是Borel的，且不将正体积映射到零体积，则证明了s = t圆u因子唯一地a.e.映射t（x）= exp（x）[-del psi（x）]和保体积映射u：M -> M的合成，其中psi：M -> R满足附加性质（psi（c））（c）= psi，其中psi（c）（y）：= inf {c（x，y）- psi（x）\ x是M的元素}且c（x，y）= d（2）（x，y）/2。这个非线性分解可以在单位元周围线性化，得到向量场的霍奇分解。结果是通过求解蒙格-康托洛维奇问题的黎曼版本得到的，这意味着最小化将L-1（M）中一个大于或等于0的质量分布f转移到另一个上的成本c（x，y）的期望值。对非紧流形上黎曼距离大于或等于0的其它严格凸代价函数c（x，y）的平行结果进行了简要的讨论.
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 --> M is merely Borel and never maps positive volume into zero volume, we show s = t circle u factors uniquely a.e. into the composition of a map t(x) = exp(x)[-del psi (x)] and a volume-preserving map u : M --> M, where psi : M --> R satisfies the additional property that (psi (c))(c) = psi with psi (c)(y) := inf {c(x,y) - psi (x) \ x is an element of 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 greater than or equal to 0 of mass in L-1(M) onto another. Parallel results for other strictly convex cost functions c(x, y) greater than or equal to 0 of the Riemannian distance on non-compact manifolds are briefly discussed.