Theodore Popelensky
Leninskie Gory 1, Moscow, 119991 Russia
Moscow State University, Faculty of Mechanics and Mathematics
Publications:
Pepa R. Y., Popelensky T. Y.
Combinatorial Ricci Flow for Degenerate Circle Packing Metrics
2019, vol. 24, no. 3, pp. 298311
Abstract
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. 
Pepa R. Y., Popelensky T. Y.
Equilibrium for a Combinatorial Ricci Flow with Generalized Weights on a Tetrahedron
2017, vol. 22, no. 5, pp. 566578
Abstract
Chow and Lou [2] showed in 2003 that under certain conditions the combinatorial analogue of the Hamilton Ricci flow on surfaces converges to Thruston’s circle packing metric of constant curvature. The combinatorial setting includes weights defined for edges of a triangulation. A crucial assumption in [2] was that the weights are nonnegative.We have recently shown that the same statement on convergence can be proved under weaker conditions: some weights can be negative and should satisfy certain inequalities. In this note we show that there are some restrictions for weakening the conditions. Namely, we show that in some situations the combinatorial Ricci flow has no equilibrium or has several points of equilibrium and, in particular, the convergence theorem is no longer valid.
