  We study the 3-dimensional combinatorial Yamabe flow in hyperbolic background
geometry. 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 real or
virtual ball packings with vanishing (extended) combinatorial scalar curvature,
i.e. the (extended) 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 (extended)
combinatorial scaler curvature if the number of tetrahedra incident to each
vertex is at most 22.
