Combinatorial scalar curvature and rigidity of ball packings (Q1919325)

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)].