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We study the 3-dimensional combinatorial Yamabe flow in hyperbolic backgroundgeometry. For a triangulation of a 3-manifold, we prove that if the number oftetrahedra incident to each vertex is at least 23, then there exist real orvirtual ball packings with vanishing (extended) combinatorial scalar curvature,i.e. the (extended) solid angle at each vertex is equal to 4{\pi}. In thiscase, if such a ball packing is real, then the (extended) combinatorial Yamabeflow converges exponentially fast to that ball packing. Moreover, we prove thatthere is no real or virtual ball packing with vanishing (extended)combinatorial scaler curvature if the number of tetrahedra incident to eachvertex is at most 22.

3-dimensional Combinatorial Yamabe Flow in Hyperbolic Background Geometry