Chow and Luo [3] showed in 2003 that the combinatorial analogue of the Hamilton Ricci flow on surfaces converges under certain conditions to Thruston’s circle packing metric of constant curvature. The combinatorial setting includes weights defined for edges of a triangulation. A crucial assumption in [3] was that the weights are nonnegative. Recently we have shown that the same statement on convergence can be proved under a weaker condition: some weights can be negative and should satisfy certain inequalities [4].On the other hand, for weights not satisfying conditions of Chow — Luo’s theorem we observed in numerical simulation a degeneration of the metric with certain regular behaviour patterns [5]. In this note we introduce degenerate circle packing metrics, and under weakened conditions on weights we prove that under certain assumptions for any initial metric an analogue of the combinatorial Ricci flow has a unique limit metric with a constant curvature outside of singularities.

中文翻译：

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简并圆堆积度量的组合Ricci流

Chow和Luo [3]在2003年证明，在一定条件下，表面上的Hamilton Ricci流的组合模拟收敛于Thruston的恒定曲率圆堆积度量。组合设置包括为三角剖分的边定义的权重。[3]中的一个关键假设是权重是非负的。最近，我们证明了在较弱的条件下也可以证明相同的收敛性陈述：某些权重可能为负，并且应满足某些不等式[4]。另一方面，对于不满足Chow的权重-罗定理，我们在数值模拟度量具有某些规则行为模式的退化[5]。在本说明中，我们介绍了退化圆包装的指标，