On the Uniqueness of Certain Types of Circle Packings on Translation Surfaces

TL;DR

This work addresses the problem of uniqueness for circle packings on translation surfaces, concentrating on genus two in the stratum $\mathcal{H}(1,1)$ and a doubled-slit torus decomposition. The main approach introduces splitting bigons and a pair-of-collections representation $(P,Q)$ to model circle configurations, yielding finite bounds on the number of packings with a fixed contact graph when two double circles are fixed along the slit; the bounds are expressed in terms of $d(T)$, the number of splitting bigons, and extend to higher genus via products of slit-parameters $k_i$, culminating in a general bound of $2 \prod_{i=1}^{g-1} k_i - 1$. The results rely on hyperelliptic involution symmetries and a careful analysis of how bigons partition slitted surfaces into genus-zero and genus-one components, providing a clear path toward understanding existence and uniqueness in the discrete-circle-packing setting on translation surfaces. These findings contribute to discrete conformal geometry on translation surfaces and offer concrete combinatorial bounds that inform future work on higher-genus packings and related discrete analytic theories.

Abstract

Consider a collection of finitely many polygons in $\mathbb C$, such that for each side of each polygon, there exists another side of some polygon in the collection (possibly the same) that is parallel and of equal length. A translation surface is the surface formed by identifying these opposite sides with one another. The $H(1, 1)$ stratum consists of genus two translation surfaces with two singularities of order one. A circle packing corresponding to a graph $G$ is a configuration of disjoint disks such that each vertex of $G$ corresponds to a circle, two disks are externally tangent if and only if their vertices are connected by an edge in $G$, and $G$ is a triangulation of the surface. It is proven that for certain circle packings on $H(1, 1)$ translation surfaces, there are only a finite number of ways the packing can vary without changing the contacts graph, if two disks along the slit are fixed in place. These variations can be explicitly characterized using a new concept known as splitting bigons. Finally, the uniqueness theorem is generalized to a specific type of translation surfaces with arbitrary genus $g \geq 2$.

Figures (12)

• Figure 1: In each of the three diagrams above, opposite and identified edges have been given the same color. The first two surfaces are tori. All three of the above surfaces are translation surfaces.
• Figure 2: The construction involves taking $k + 1$ copies of the upper half plane and $k + 1$ copies of the lower half plane with the usual flat metric, and gluing them along the infinite rays $[0, \infty)$ and $(-\infty, 0]$ in alternating order. The image above is taken from wright and depicts the case for $k = 1$.
• Figure 3: The square-tiled surface shown above has genus $2$ and two singularities, both of order $1$ and cone angle $4\pi$. The image is taken from massart.
• Figure 4: A valid circle packing on a sphere, taken from stephenson.
• Figure 5: The surface depicted above is an arbitrary genus $2$ surface that has been expressed as the connected sum of two slit tori. It belongs in the $\mathcal{H}(1, 1)$ stratum. In order to express an $\mathcal{H}(2)$ surface as per the theorem, one of the slits needs to start and end at the same point of the torus. This image is taken from bouwman.

Theorems & Definitions (86)

• Definition 2.1
• Example 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Remark 2.7
• Definition 2.8
• Example 2.9
• Proposition 2.10