Rigidity of generalized Thurston's sphere packings on 3-dimensional manifolds with boundary

Motivated by Guo–Luo’s generalized circle packings on surfaces with boundary [Guo and Luo, Geom. Topol. 13(3) (2009) 1265–1312], we introduce the generalized Thurston’s sphere packings on 3-dimensional manifolds with boundary. Then we investigate the rigidity of the generalized Thurston’s sphere packings. We prove that the generalized Thurston’s sphere packings are locally determined by the combinatorial scalar curvatures. We further prove the infinitesimal rigidity that the generalized Thurston’s sphere packings cannot be deformed while keeping the combinatorial Ricci curvatures fixed.