通过不动点指数实现精简磁盘配置的刚性
Rigidity of thin disk configurations, via fixed-point index
摘要：
本文证明了闭圆盘构形的几个刚性定理。首先，在黎曼球面$\hat{\mathbb{C}}$中固定两个封闭圆盘的集合$\mathcal{C}$和$\tilde{\mathcal{C}}$，共享一个（主要是-）三角化$\hat{\mathbb{C}}$的接触图，使得对于所有相应的相交圆盘$D_i, D_j \in \mathcal{C}$和$\tilde D_i, \tilde D_j \in \tilde{\mathcal{C}}$对，我们有$D_i$和$D_j$之间的重叠角与$\tilde D_i$和$\tilde D_j$之间的重叠角一致。我们需要一个额外的条件，即集合是“瘦”的，这意味着没有一对$\mathcal{C}$的磁盘在第三个磁盘的内部相遇，$\tilde{\mathcal{C}}$也是如此。然后$\mathcal{C}$和$\tilde{\mathcal{C}}$通过莫比乌斯或反莫比乌斯变换而不同。我们还证明了类似的声明，在复杂的平面$\mathbb{C}$，并在双曲平面$\mathbb{H}^2$的封闭磁盘的集合。我们的证明方法是基本的和独立的，只依赖于平面拓扑参数和莫比乌斯变换的操作。特别是，我们推广了一个不动点的论点，这是以前应用的Schramm和他证明我们的定理在圆包装设置类似物，也就是说，在磁盘的问题是成对的adjorwise不相交。以前人们认为，这些证明方法过于依赖于盘的两两相邻不相交性，因此没有希望将它们推广到重叠盘的配置。最后，我们陈述了一些公开的问题和假设，包括我们的主要结果和我们的主要技术定理的有约束的推广。具体地说，我们猜想，我们的薄条件是不必要的陈述，我们的主要定理。
英文摘要：
We prove some rigidity theorems for configurations of closed disks. First, fix two collections $\mathcal{C}$ and $\tilde{\mathcal{C}}$ of closed disks in the Riemann sphere $\hat{\mathbb{C}}$, sharing a contact graph which (mostly-)triangulates $\hat{\mathbb{C}}$, so that for all corresponding pairs of intersecting disks $D_i, D_j \in \mathcal{C}$ and $\tilde D_i, \tilde D_j \in \tilde{\mathcal{C}}$ we have that the overlap angle between $D_i$ and $D_j$ agrees with that between $\tilde D_i$ and $\tilde D_j$. We require the extra condition that the collections are "thin", meaning that no pair of disks of $\mathcal{C}$ meet in the interior of a third, and similarly for $\tilde{\mathcal{C}}$. Then $\mathcal{C}$ and $\tilde{\mathcal{C}}$ differ by a Möbius or anti-Möbius transformation. We also prove the analogous statements for collections of closed disks in the complex plane $\mathbb{C}$, and in the hyperbolic plane $\mathbb{H}^2$. Our method of proof is elementary and self-contained, relying only on plane topology arguments and manipulations by Möbius transformations. In particular, we generalize a fixed-point argument which was previously applied by Schramm and He to prove the analogs of our theorems in the circle-packing setting, that is, where the disks in question are pairwise interiorwise disjoint. It was previously thought that these methods of proof depended too crucially on the pairwise interiorwise disjointness of the disks for there to be a hope for generalizing them to the setting of configurations of overlapping disks. We end by stating some open problems and conjectures, including conjectured generalizations both of our main result and of our main technical theorem. Specifically, we conjecture that our thinness condition is unnecessary in the statements of our main theorems.
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