On convergence of combinatorial Ricci flow on surfaces with negative weights

Abstract

Chow and Lou in 2003 had shown that the analogue of the Hamilton Ricci flow on surfaces in the combinatorial setting converges to the Thruston’s circle packing metric. The combinatorial setting includes weights defined for edges of a triangulation. Crucial assumption in the paper of Chow and Lou was that the weights are nonnegative. We show that the same results on convergence of Ricci flow can be proved under weaker condition: some weights can be negative and should satisfy certain inequalities. As a consequence we obtain theorem of existence of Thurston’s circle packing metric for a wider range of weights.