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Theodore Popelensky

Leninskie Gory 1, Moscow, 119991 Russia
Moscow State University, Faculty of Mechanics and Mathematics


Pepa R. Y., Popelensky T. Y.
Equilibrium for a Combinatorial Ricci Flow with Generalized Weights on a Tetrahedron
2017, vol. 22, no. 5, pp.  566-578
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Ricci flow has no equilibrium or has several points of equilibrium and, in particular, the convergence theorem is no longer valid.
Keywords: circle packing, combinatorial Ricci flow
Citation: Pepa R. Y., Popelensky T. Y.,  Equilibrium for a Combinatorial Ricci Flow with Generalized Weights on a Tetrahedron, Regular and Chaotic Dynamics, 2017, vol. 22, no. 5, pp. 566-578

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