Combinatorial scalar curvature and rigidity of ball packings
type: Item
instance of: scholarly article
title: Combinatorial scalar curvature and rigidity of ball packings
zbMATH Open document ID: 0868.51023
author: Daryl Cooper
author: Igor Rivin
published in: Mathematical Research Letters
publication date: 1996-08-05T00:00:00Z
review text: The authors define a notion of conformal simplex in Euclidean 3-space and study its conformal deformations. A conformal simplex is defined as a simplex for which the lengths \(\ell(e_{ij})\) of the edge from the \(i\)th to \(j\)th edge-point satisfy \(\ell(e_{ij}) =r_i+ r_j\) for positive numbers \(r_1, \dots, r_4\). For the study of the space of conformal simplices the authors introduce a functional \(S\) whose critical points correspond to metrics of constant scalar curvature. As a first result they show that the regular simplex with \(r_i= 1/4\) is a critical point and cannot be conformally deformed. The next result says that a conformal simplex cannot be deformed while keeping its angles fixed. The space of isometry classes of Euclidean conformal simplices is shown to be homeomorphic to hyperbolic 3-space. Thus the moduli space of conformal simplices is a 4-ball. Considered as a subspace of real 4-space it is shown to be non-convex. Analogous to the case of Euclidean conformal simplices the paper also treats the case of conformal simplices in hyperbolic 3-space. Finally the results of the paper are applied to showing rigidity of ball packings with prescribed combinatorics. The latter application is in line with previous work of \textit{Y. Colin de Verdière} [Invent. Math. 104, No. 3, 655-669 (1991; Zbl 0745.52010)].
reviewed by: Volkmar Welker
Mathematics Subject Classification ID: 51M20
Mathematics Subject Classification ID: 52C17
Mathematics Subject Classification ID: 51M05
zbMATH DE Number: 913094
zbMATH Keywords: scalar curvature
zbMATH Keywords: conformal deformation
zbMATH Keywords: ball packing
zbMATH Keywords: conformal simplex
zbMATH Keywords: Euclidean 3-space
zbMATH Keywords: hyperbolic 3-space
zbMATH Keywords: rigidity
MaRDI profile type: Publication
full work available at URL: mrl.1996.v3.n1.a5
OpenAlex ID: W2023551584
DOI: 10.4310/MRL.1996.V3.N1.A5
Recommended article: Triangulations in Mobius Geometry
Recommended article: Q5493802
Recommended article: Duality structures and discrete conformal variations of piecewise constant curvature surfaces
Recommended article: On the global rigidity of sphere packings on \(3\)-dimensional manifolds
Recommended article: 3-dimensional combinatorial Yamabe flow in hyperbolic background geometry
Recommended article: Discrete conformal maps and ideal hyperbolic polyhedra
Recommended article: Globally Rigid Ball-Polyhedra in Euclidean 3-Space
Recommended article: Dihedral angles of convex polyhedra
Recommended article: Rigidity of ball-polyhedra in Euclidean 3-space
Recommended article: Decomposition method related to saturated hyperball packings
description: scientific article; zbMATH DE number 913094