Regularity of the singular sets of two-dimensional area-minimizing flat chains modulo 3 inR3
Regularity of the singular sets of two-dimensional area-minimizing flat chains modulo 3 inR3
复制标题
二维面积最小化平链模 3 inR3 奇异集的正则性
DOI：
10.1007/bf01392299
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发表时间：
1973
影响因子：
3.1
通讯作者：
Jean E. Taylor
中科院分区：
文献类型：
--
作者：
Jean E. Taylor
确定具有给定边界的最小面积曲面的存在性和结构是一个长期感兴趣的问题。成功的几何测量理论的方法，近年来在显示的存在和规律性几乎无处不在的解决方案，以各种不同的配方问题的最小面积（一类问题经常统称高原的问题）现在已经集中越来越多的注意力问题的结构确定的非正规，或奇异，这些解决方案的点。这篇论文的结果是一个贡献的低维正则性的研究某些这些奇异集。我们在这里考虑的曲面类，即模3可求长的平坦链，其边界也是模3可求长的平坦链，可以以多种方式定义[5，4.2]。26; 7]。它们的特征在于，除了在任意小的二维和一维区域内，它们和它们的边界同时与系数在整数rood 3中的C~奇异链的像一致，特定的链取决于所需的逼近程度。由于模3的抵消，在这些表面中可以出现一维奇异性。例如，考虑由三个具有相同直径且彼此成120 °角的半圆盘组成的表面Y（参见图1;在球坐标中
Determining the existence and structure of surfaces of minimum area having a given boundary is a problem of long-standing interest. The success of geometric measure theoretic methods in recent years in showing the existence and regularity almost everywhere of solutions to a variety of different formulations of the problem of least area (the class of problems frequently collectively called Plateau's Problem) has now focused increased attention on the problem of determining the structure of the non-regular, or singular, points of these solutions. The results of this thesis are a contribution to the study of the lower dimensional regularity of certain of these singular sets. The class of surfaces we consider here, namely rectifiable flat chains modulo 3 whose boundaries are also rectifiable flat chains modulo 3, can be defined in a number of ways [5, 4.2. 26; 7]. They are characterized by the property that both they and their boundaries simultaneously agree, except in a set of arbitrarily small two and one-dimensional area respectively, with the images of C~ singular chains with coefficients in the integers rood 3, the particular chains depending on the degree of approximation desired. One dimensional singularities can arise in these surfaces because of cancellation modulo 3. For instance, consider the surface Y consisting of three half disks with common diameter and at angles of 120 to each other (see Fig. 1; in spherical coordinates