Rigidity of inversive distance circle packings revisited (Q1639649)

The paper under review deals with the notion of inversive distance circle packing, see [\textit{P. L. Bowers} and \textit{K. Stephenson}, Mem. Am. Math. Soc. 805, 97 p. (2004; Zbl 1080.52511)], [\textit{K. Stephenson}, Introduction to circle packing. The theory of discrete analytic functions. Cambridge: Cambridge University Press (2005; Zbl 1074.52008)]. The main attention is paid to the Bowers-Stephenson conjecture which claims that the inversive distance circle packings are rigid. The main result of the paper affirms the conjecture in the case of inversive distance circle packings with inversive distance in \((-1,+\infty)\) and with Euclidean or hyperbolic background metric. It is shown that an arbitrary circle packing in question is well determined by its combinatorial curvature. The proved statement unifies and extends rigidity results obtained in [\textit{R. Guo}, Trans. Am. Math. Soc. 363, No. 9, 4757--4776 (2011; Zbl 1252.52019)] and [\textit{F. Luo}, Geom. Topol. 15, No. 4, 2299--2319 (2011; Zbl 1242.52027)]. Moreover, the rigidity of inversive distance circle packings involving the notion of combinatorial \(\alpha\)-curvature is discussed too, cf. [\textit{H. Ge} and \textit{X. Xu}, Calc. Var. Partial Differ. Equ. 55, Paper No. 12, 16 p. (2016; Zbl 1336.53078)].