Weighted combinatorial Ricci flow and metrics defined by degenerate circle packings

Abstract

Chow and Luo [1] in 2003 had shown that the combinatorial analogue of the Hamilton Ricci flow on surfaces under certain conditions converges to Thruston?s circle packing metric of constant curvature. The combinatorial setting includes weights defined for edges of a triangulation. Crucial assumption in the paper [1] was that the weights are nonnegative. Recently we have shown that same statement on convergence can be proved under weaker condition: some weights can be negative and should satisfy certain inequalities, [3]. Moreover, in [6] notions of degenerate circle packing and corresponding metric were introduced. In [6] theory of combinatorial Ricci flow for such metrics was developed, which includes Chow-Luo theory as a partial case for nondegenerate circle packing and nonnegative weights on edges. On the other hand, in [2] the combinatorial Yamabe flow was introduced and investigated. In [7, 8] we developed weighted modification of Yamabe flow. In this paper we merge ideas from these two theories and introduce weighted combinatorial Ricci flow on metrics defined by degenerate circle packings. We prove that under certain conditions for any initial metric the flow converges to a unique metric of constant curvature.