Rigidity of thin disk configurations, via fixed-point index

Abstract

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 M\"obius or anti-M\"obius 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 M\"obius 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.