论文标题
球形量和球形高原问题
Spherical volume and spherical Plateau problem
论文作者
论文摘要
给定封闭的歧管或更一般的组同源类别，我们引入了球形高原问题，这是一个变异问题，与称为球形体积的拓扑不变性相对应。原则上，其解决方案应通过球形商的最小表面来实现。我们解释说，在许多几何有趣的情况下，这些解决方案本质上是独一无二的。我们首先综述了对度量电流的Ambrosio-Kirchheim理论，以及由Besson-Courtois-Gallot开发的Barycenter Map方法。然后，我们概述了以下应用：（1）球形高原解决方案的固有独特性，用于负弯曲，局部对称，封闭式的流形，（2）球形高原解决方案的固有唯一性，用于所有3维封闭的封闭式封闭式折线，（（3）较高较高的较高维度的较高型较高量的超级替补。我们还提出了一些公开问题。
Given a closed oriented manifold or more generally a group homology class, we introduce the spherical Plateau problem, which is a variational problem corresponding to a topological invariant called the spherical volume. In principle, its solutions should be realized by minimal surfaces in quotients of spheres. We explain that in many geometrically interesting cases, those solutions are essentially unique. We start with a review of the Ambrosio-Kirchheim theory of metric currents, and the barycenter map method developed by Besson-Courtois-Gallot. Then, we outline the following applications: (1) the intrinsic uniqueness of spherical Plateau solutions for negatively curved, locally symmetric, closed oriented manifolds, (2) the intrinsic uniqueness of spherical Plateau solutions for all 3-dimensional closed oriented manifolds, (3) the construction of higher-dimensional analogues of hyperbolic Dehn fillings. We also propose some open questions.
