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Combinatorial scalar curvature and rigidity of ball packings. (English) Zbl 0868.51023

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 Y. Colin de Verdière [Invent. Math. 104, No. 3, 655-669 (1991; Zbl 0745.52010)].
Reviewer: V.Welker (Essen)

MSC:

51M20 Polyhedra and polytopes; regular figures, division of spaces
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
51M05 Euclidean geometries (general) and generalizations

Citations:

Zbl 0745.52010
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