Existence and soap film regularity of solutions to Plateau’s problem
Existence and soap film regularity of solutions to Plateau’s problem
复制标题
Plateau问题解的存在性及皂膜规律
DOI：
10.1515/acv-2015-0023
复制
发表时间：
2013
影响因子：
1.7
通讯作者：
H. Pugh
中科院分区：
文献类型：
--
作者：
J. Harrison;H. Pugh
高原问题是求一个跨越给定边界的面积最小的曲面。本文给出了一个余维1曲面的定理，在这个定理中，用一个新的代数拓扑概念代替了常用的同调定义。特别是，我们的新定义在边界具有多个连接组件的情况下，比现有的同构定义提供了显著的改进。设M是维数为n-2的连通定向紧流形${n-2}$和？利用Hausdorff球测度作为“大小”的概念，我们证明了：在？${\mathfrak{S}}$的最小大小。任何这样的X 0 ${X_{0}}$包含一个“核心”X 0 *∈？${X_{0}^{*}\in\mathfrak{S}}$具有以下性质：它是M的凸包的子集，并且是a.e.（在（n-1） ${(n-1)}$维Hausdorff测度意义上）一个实解析（n-1） ${(n-1)}$维极小子流形。如果n=3 ${n=3}$，则X 0 * ${X_{0}^{*}}$具有肥皂膜的局部结构。进一步，将集合理论解提升到具有丰富连续算子代数的空间中的当前解。
Abstract Plateau’s problem is to find a surface with minimal area spanning a given boundary. Our paper presents a theorem for codimension one surfaces in ℝ n ${\mathbb{R}^{n}}$ in which the usual homological definition of span is replaced with a novel algebraic-topological notion. In particular, our new definition offers a significant improvement over existing homological definitions in the case that the boundary has multiple connected components. Let M be a connected, oriented compact manifold of dimension n - 2 ${n-2}$ and ? ${\mathfrak{S}}$ the collection of compact sets spanning M. Using Hausdorff spherical measure as a notion of “size,” we prove: There exists an X 0 ${X_{0}}$ in ? ${\mathfrak{S}}$ with smallest size. Any such X 0 ${X_{0}}$ contains a “core” X 0 * ∈ ? ${X_{0}^{*}\in\mathfrak{S}}$ with the following properties: It is a subset of the convex hull of M and is a.e. (in the sense of ( n - 1 ) ${(n-1)}$ -dimensional Hausdorff measure) a real analytic ( n - 1 ) ${(n-1)}$ -dimensional minimal submanifold. If n = 3 ${n=3}$ , then X 0 * ${X_{0}^{*}}$ has the local structure of a soap film. Furthermore, set theoretic solutions are elevated to current solutions in a space with a rich continuous operator algebra.