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**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.

Theorem 1.5: Suppose \((M,T)\) is a closed triangulated 3-manifold and \(\bar{R}\) is a given function defined on the vertices of \((M,T)\).

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].

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].

Reviewer: Victor Alexandrov (Novosibirsk)

### 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 |