On the global rigidity of sphere packings on $3$-dimensional manifolds

Abstract

In this paper, we prove the global rigidity of sphere packings on 3-dimensional manifolds. This is a 3-dimensional analogue of the rigidity theorem of Andreev-Thurston and was conjectured by Cooper and Rivin in [5]. We also prove a global rigidity result using a combinatorial scalar curvature introduced by Ge and the author in [13]. MSC (2010): 52C25; 52C26

3-dimensional combinatorial Yamabe flow in hyperbolic background geometry

“…which implies cos(β jk (r r r (n) )) > 0. This proves β jk (r r r (n) ) → 0 by (25). Hence the proposition follows.…”

“…The decomposition has an interesting geometric intepretation, see [25,7]. For {i, j, k, l} = {1, 2, 3, 4}, we place three balls B j , B k and B l with radii r j , r k and r l in H 3 , externally tangent to each other, whose centers lie on an embedded totally geodesic hyperbolic plane Π.…”

“…For real and virtual tetrahedra, Xu [25] introduce a natural extension of solid angles. For a tetrahedron τ = {1234}, the extended solid angle α i at the vertex i is defined as…”

“…is locally injective by Cooper and Rivin [3] and globally injective by Xu [25], which generalize Thurston's rigidity about surface circle packings [23].…”

Combinatorial Calabi flow with surgery on surfaces

“…which may be negative and unbounded along the Euclidean combinatorial Calabi flow (1.2) for general initial PL metric. Even through discrete Laplace operators with negative coefficients has many applications in the study of combinatorial curvatures on two and three dimensional manifolds (see [15,28,29,30,33,38,48,49,53] for example), a discrete Laplace operator is generally defined with nonnegative weights [14]. For ∆ E,T , the weight is nonnegative is equivalent to θ ij k + θ ij l ≤ π, which is the Delaunay condition [3].…”