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.
