## On the global rigidity of sphere packings on $$3$$-dimensional manifolds.(English)Zbl 1446.52019

The author proves the global rigidity of sphere packings on 3-dimensional manifolds. The main results are as follows:
Theorem 1.3: Suppose $$(M,T)$$ is a closed triangulated 3-manifold.
(1)
A Euclidean sphere packing metric on $$(M,T)$$ is determined by its combinatorial scalar curvature $$K:V\to\mathbb{R}$$ up to scaling.
(2)
A hyperbolic sphere packing metric on $$(M,T)$$ is determined by its combinatorial scalar curvature $$K:V\to\mathbb{R}$$.

Theorem 1.5: Suppose $$(M,T)$$ is a closed triangulated 3-manifold and $$\bar{R}$$ is a given function defined on the vertices of $$(M,T)$$.
(1)
In the case of Euclidean background geometry,
(1a)
if $$\alpha\bar{R}\equiv 0$$, there exists at most one Euclidean sphere packing metric in $$\Omega$$ with combinatorial $$\alpha$$-curvature equal to $$\bar{R}$$ up to scaling.
(1b)
if $$\alpha\bar{R}\leq 0$$ and $$\alpha\bar{R}\not\equiv 0$$, there exists at most one Euclidean sphere packing metric in $$\Omega$$ with combinatorial $$\alpha$$-curvature equal to $$\bar{R}$$.
(2)
In the case of hyperbolic background geometry, if $$\alpha\bar{R}\leq 0$$, there exists at most one hyperbolic sphere packing metric in $$\Omega$$ with combinatorial $$\alpha$$-curvature equal to $$\bar{R}$$.

Theorem 1.3 is a 3-dimensional analogue of the rigidity theorem of Andreev-Thurston and was conjectured by D. Cooper and I. Rivin in [Math. Res. Lett. 3, 51–60 (1996; Zbl 0868.51023)]. The combinatorial scalar curvature used in Theorem 1.5 was introduced by H. Ge and the author in [“A combinatorial Yamabe problem on two and three dimensional manifolds”, Preprint, arXiv:1504.05814].

### MSC:

 52C25 Rigidity and flexibility of structures (aspects of discrete geometry) 52C17 Packing and covering in $$n$$ dimensions (aspects of discrete geometry) 52C26 Circle packings and discrete conformal geometry

Zbl 0868.51023
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