Abstract: The combinatorial Yamabe problem connects the geometry and combinatorics of 3-dimentional

ball packing. With the help of combinatorial Yamabe flows, we give some partial results. For Euclidean ball

packings, if the triangulation is regular, then the combinatorial Yamabe flow converges exponentially fast to

a constant curvature packing. For hyperbolic ball packings, if the vertex degree is at least23, then there exist

real or virtual packings with vanishing curvature, i.e. the solid angle at each vertex is equal to 4\pi. In this case,

if such a packing is real, then the (extended) combinatorial Yamabe flow converges exponentially fast to that

packing. Moreover, we show that there is no real or virtual packing with vanishing curvature if the number of

tetrahedrons incident to each vertex is at most 22. These are joint works with Wenshuai Jiang and Liangming Shen, and with Bobo Hua.