Rectifiable and flat G chains in a metric space
Rectifiable and flat G chains in a metric space
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度量空间中的可矫正且平坦的 G 链
DOI：
10.1353/ajm.2012.0004
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发表时间：
2012
影响因子：
1.7
通讯作者：
R. Hardt
中科院分区：
文献类型：
--
作者：
Thierry de Pauw;R. Hardt
完备度量空间中的链，它的系数在赋范空间中
研究了阿贝尔群G$。一个$m$维可改链是
利普希茨推进$^m$中的区域，配备了
可测量的$G$值密度。平面链是通过使用以下工具完成的
关于多面体或Lipschitz链逼近的某种平坦范数。
这些链的几何测度论的许多基本结果是
导出了有限质量扁平链的可纠性
那个$G$不包含非常数的Lipschitz曲线。这里的工作
概括并引用了B.White在1999年发表的论文中的许多观点
以及L.Ambrosio和B.Kirchheim在2000年发表的论文
公制空间中的电流。平链和可调式链条的使用
几何变分问题或定义几何同调理论
可以揭示空间的几何性质。
Chains, in a complete metric space, which have coefficients in a normed
abelian group $G$ are studied. An $m$ dimensional rectifiable chain is the
Lipschitz push-forward of a region in ${\Bbb R}^m$ equipped with a
measurable $G$-valued density. Flat chains are obtained by completion using
a certain flat norm on polyhedral or Lipschitz chain approximations.
Numerous basic results of geometric measure theory for these chains are
derived including the rectifiability of finite mass flat chains provided
that $G$ contains no nonconstant Lipschitz curves. The work here
generalizes, and uses many ideas from, the 1999 paper of B. White on at $G$
chains in ${\Bbb R}^n$ and the 2000 paper of L. Ambrosio and B. Kirchheim on
currents in a metric space. The use of flat and rectifiable chains in
geometric variational problems or in defining geometric homology theories
may reveal geometric properties of spaces.