On the deformation of hyperbolic ball packings

For a triangulation of a 3-manifold, we prove that if the number of tetrahedra incident to each vertex is at least 23, then there exist ball packings with vanishing discrete scalar curvature, i.e. the solid angle at each vertex is equal to 4{\pi}. In this case, if such a ball packing is real, then the (extended) combinatorial Yamabe flow converges exponentially fast to that ball packing. Moreover, we prove that there is no real or virtual ball packing with vanishing discrete scaler curvature if the number of tetrahedra incident to each vertex is at most 22. This is joint work with Bobo Hua.