论文标题
公制空间中最小磁盘的分支集
The branch set of minimal disks in metric spaces
论文作者
论文摘要
我们研究了满足二次等级不平等的度量空间中高原问题的分支解决方案的结构。在我们的第一个结果中，我们给出了具有等速度常数任意接近欧几里得等恒定常数$（4π）^{ - 1} $的示例。这对Lytchak的最新结果进行了补充 - wenger和Stadler分别指出，与欧几里得等级常数的任何空间都是CAT（$ 0 $）的空间（$ 0 $）的空间，以及在CAT（$ 0 $）空间中高原问题的解决方案，只有隔离的分支点。我们还表明，任何类似于平面的单元格的集合都可以作为平台问题解决方案的分支集。这些结果回答了Lytchak和Wenger提出的两个问题。此外，我们研究了有关度量磁盘的能量最小化参数化的几个相关问题：当这样的映射为准对称时，其分支集为空时，并且当它是唯一的，直到共形差异性。
We study the structure of the branch set of solutions to Plateau's problem in metric spaces satisfying a quadratic isoperimetric inequality. In our first result, we give examples of spaces with isoperimetric constant arbitrarily close to the Euclidean isoperimetric constant $(4π)^{-1}$ for which solutions have large branch set. This complements recent results of Lytchak--Wenger and Stadler stating, respectively, that any space with Euclidean isoperimetric constant is a CAT($0$) space and solutions to Plateau's problem in a CAT($0$) space have only isolated branch points. We also show that any planar cell-like set can appear as the branch set of a solution to Plateau's problem. These results answer two questions posed by Lytchak and Wenger. Moreover, we investigate several related questions about energy-minimizing parametrizations of metric disks: when such a map is quasisymmetric, when its branch set is empty, and when it is unique up to a conformal diffeomorphism.
