Rectifiability of the singular set of energy minimizing maps
Rectifiability of the singular set of energy minimizing maps
复制标题
奇异能量最小化图集的可修正性
DOI：
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发表时间：
1995
期刊：
影响因子：
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通讯作者：
L. Simon
中科院分区：
文献类型：
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作者：
L. Simon
Schoen和Uhlenbeck [SU]的能量最小化正则性理论将u从一个具有任意光滑黎曼度量的域~ 2crn映射到一个紧的光滑黎曼目标流形n，建立了奇异集singu总是具有Hausdorff维数_ 0，其中~ s表示s维Hausdorff测度。Giaquinta和Giusti [GG]在将图像包含在坐标图中独立得到了类似的结果。以防目标歧管N是真正的分析,本文的主要定理(定理1 - 3下)建立可矫正性这样奇异的属性集的维N 3,和其他尺寸m < N 3如果N是所有能源最小化等地图u N有暗唱< m。回想一下,一个子集C R N据说m-rectifiable如果~ flm (a) < oo,如果有一个近似切线空间a.e. ~ m的感觉。e。有一个m维子空间Lz满足
The regularity theory of Schoen and Uhlenbeck [SU] for energy minimizing maps u from a domain ~2 C R n (equipped with any smooth Riemannian metric) into a compact smooth Riemannian target manifold N , established that the singular set s ingu always has Hausdorff dimension _ 0, where ~ s denotes s-dimensional Hausdorff measure. Similar results were obtained independently by Giaquinta and Giusti [GG] in the case when the image is contained in a coordinate chart. In case the target manifold N is real analytic, the main theorems of this paper (Theorems 1-3 below) establish rectifiability properties for such singular sets in the dimension n 3, and in other dimensions m < n 3 in case N happens to be such that all energy minimizing maps u into N have dim sing u < m. Recall that a subset A C R n is said to be m-rectifiable if ,~flm(A) < oo, and if A has an approximate tangent space a.e. in the sense that for ~ m a . e . z E A there is an m-dimensional subspace Lz such that