On the deformation of discrete conformal factors on surfaces

Luo (Commun Contemp Math 6:765–780, 2004 ) conjectured that the discrete Yamabe flow will converge to the constant curvature PL-metric after finite number of surgeries on the triangulation. In this paper, we prove that the flow can always be extended (without surgeries) to a solution for all time. Moreover, we consider the convergence of such solution. We show that the extended solution converges exponentially fast to the constant curvature PL-metric if it exists. In addition, we investigate the geometric meaning of the limit of the extended solution.