论文标题
高原道格拉斯问题，用于单数配置和一般度量空间
The Plateau-Douglas problem for singular configurations and in general metric spaces
论文作者
论文摘要
假设您在$ \ mathbb {r}^n $中给您一个有限的配置$γ$。高原道格拉斯问题询问是否存在所有紧凑型属的面积最小化，最多是$ p $，跨越$γ$。虽然解决此问题的解决方案是众所周知的，但如果人们允许曲线可能是非分散或自我切断的单数配置$γ$，则经典方法会分解。我们的主要结果解决了这种潜在的奇异配置的高原道格拉斯问题。此外，我们的证明不仅可以在$ \ mathbb {r}^n $中起作用，而且可以使用适当的度量空间。因此，我们还能够扩展JürgenJost以及第二作者与Stefan Wenger的常规配置的先前已知的存在结果。特别是，在一般完整的Riemannian歧管中，生存是Jordan曲线的不相交配置的新事物。即使在最常规的设置中，固定属$ p $界面的最小表面也不总是存在。关于这个问题，我们还通过最小序列概括了奇异构型的方法，从而满足了凝聚和粘附条件对度量空间的设置的方法。
Assume you are given a finite configuration $Γ$ of disjoint rectifiable Jordan curves in $\mathbb{R}^n$. The Plateau-Douglas problem asks whether there exists a minimizer of area among all compact surfaces of genus at most $p$ which span $Γ$. While the solution to this problem is well-known, the classical approaches break down if one allows for singular configurations $Γ$ where the curves are potentially non-disjoint or self-intersecting. Our main result solves the Plateau-Douglas problem for such potentially singular configurations. Moreover, our proof works not only in $\mathbb{R}^n$ but in general proper metric spaces. Thus we are also able to extend previously known existence results of Jürgen Jost as well as of the second author together with Stefan Wenger for regular configurations. In particular, existence is new for disjoint configurations of Jordan curves in general complete Riemannian manifolds. A minimal surface of fixed genus $p$ bounding a given configuration $Γ$ need not always exist, even in the most regular settings. Concerning this problem, we also generalize the approach for singular configurations via minimal sequences satisfying conditions of cohesion and adhesion to the setting of metric spaces.
