A discrete uniformization theorem for polyhedral surfaces II
  A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying Gauss-Bonnet formula. Furthermore, the hyperbolic polyhedral metric with given curvature can be obtained using a discrete Yamabe flow with surgery. In particular, each hyperbolic polyhedral metric on a closed surface with negative Euler characteristic is discrete conformal to a unique hyperbolic metric. 
Xianfeng Gu, Ren Guo, Feng Luo, Jian Sun, Tianqi Wu
math.GT math.DG math.MG