arXiv subjects

Philip L. Bowers

Publications and source records attributed to Philip L. Bowers.

7 recordsLinked to original sources

arXiv2026

Rigidity of Koebe Polyhedra and Inversive Distance Circle Packings

Hyperbolic inversive distance circle packings on the $2$-sphere correspond to Koebe polyhedra in the Beltrami-Klein model $\mathbb{B}^{3}$ of hyperbolic $3$-space. Koebe polyhedra are triangulated convex hyperbolic polyhedra with hyperideal vertices whose faces meet $\mathbb{B}^{3}$. We prove the global rigidity of these circle packings or, equivalently, of these Koebe polyhedra under mild assumptions on the links of their vertices. Previous rigidity results apply only when all edges of the Koebe polyhedron are tangent or, alternatively, when no edge is tangent to the ideal boundary of hyperbolic space. We remove these restrictions. This generalizes the global rigidity results of both Bao-Bonahon and Bowers-Bowers-Pratt (arXiv:1703.09338), as well as the uniqueness part of the celebrated Koebe-Andre'ev-Thurston Theorem to the case where adjacent circles need not touch.

math.MG↗

arXiv2024

Infinite circle packings on surfaces with conical singularities

We show that given an infinite triangulation $K$ of a surface with punctures (i.e., with no vertices at the punctures) and a set of target cone angles smaller than $π$ at the punctures that satisfy a Gauss-Bonnet inequality, there exists a hyperbolic metric that has the prescribed angles and supports a circle packing in the combinatorics of $K$. Moreover, if $K$ is very symmetric, then we can identify the underlying Riemann surface and show that it does not depend on the angles. In particular, this provides examples of a triangulation $K$ and a conformal class $X$ such that there are infinitely many conical hyperbolic structures in the conformal class $X$ with a circle packing in the combinatorics of $K$. This is in sharp contrast with a conjecture of Kojima-Mizushima-Tan in the closed case.

math.GT↗

arXiv2023

Tame and relatively elliptic $\mathbb{CP}^1$-structures on the thrice-punctured sphere

Suppose a relatively elliptic representation $ρ$ of the fundamental group of the thrice-punctured sphere $S$ is given. We prove that all projective structures on $S$ with holonomy $ρ$ and satisfying a tameness condition at the punctures can be obtained by grafting certain circular triangles. The specific collection of triangles is determined by a natural framing of $ρ$. In the process, we show that (on a general surface $Σ$ of negative Euler characteristics) structures satisfying these conditions can be characterized in terms of their Möbius completion, and in terms of certain meromorphic quadratic differentials.

math.GT↗

arXiv2020

Combinatorics encoding geometry: the legacy of Bill Thurston in the story of one theorem

This article presents a whirlwind tour of some results surrounding the Koebe-Andre'ev-Thurston Theorem, Bill Thurston's seminal circle packing theorem that appears in Chapter 13 of The Geometry and Topology of Three-Manifolds. It will appear as a chapter in the volume: In the tradition of Thurston: geometry and topology (ed. K. Ohshika and A. Papadopoulos), Springer, 2020.

math.GT↗

arXiv2018

Almost all circle polyhedra are rigid

We verify the infinitesimal inversive rigidity of almost all triangulated circle polyhedra in the Euclidean plane $\mathbb{E}^{2}$, as well as the infinitesimal inversive rigidity of tangency circle packings on the $2$-sphere $\mathbb{S}^{2}$. From this the rigidity of almost all triangulated circle polyhedra follows. The proof adapts Gluck's proof in~\cite{gluck75} of the rigidity of almost all Euclidean polyhedra to the setting of circle polyhedra, where inversive distances replace Euclidean distances and Möbius transformations replace rigid Euclidean motions.

math.MG↗

arXiv2017

Rigidity of Circle Polyhedra in the 2-Sphere and of Hyperideal Polyhedra in Hyperbolic 3-Space

We generalize Cauchy's celebrated theorem on the global rigidity of convex polyhedra in Euclidean $3$-space $\mathbb{E}^{3}$ to the context of circle polyhedra in the $2$-sphere $\mathbb{S}^{2}$. We prove that any two convex and proper non-unitary c-polyhedra with Möbius-congruent faces that are consistently oriented are Möbius-congruent. Our result implies the global rigidity of convex inversive distance circle packings in the Riemann sphere as well as that of certain hyperideal hyperbolic polyhedra in $\mathbb{H}^{3}$.

math.MG↗

arXiv2016

Ma-Schlenker c-Octahedra in the 2-Sphere

We present constructions inspired by the Ma-Schlenker example of~\cite{Ma:2012hl} that show the non-rigidity of spherical inversive distance circle packings. In contrast to the use in~\cite{Ma:2012hl} of an infinitesimally flexible Euclidean polyhedron, embeddings in de Sitter space, and Pogorelov maps, our elementary constructions use only the inversive geometry of the $2$-sphere.

math.DG↗