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 Regul. Chaotic Dyn.: Year: Volume: Issue: Page: Find

 Regul. Chaotic Dyn., 2019, Volume 24, Issue 3, Pages 298–311 (Mi rcd479)

Combinatorial Ricci Flow for Degenerate Circle Packing Metrics

Ruslan Yu. Pepaa, Theodore Yu. Popelenskyb

a Moscow Institute of International Relations, pr. Vernadskogo 76, Moscow, 119454 Russia
b Moscow State University, Faculty of Mechanics and Mathematics, Leninskie Gory 1, Moscow, 119991 Russia

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.

Keywords: combinatorial Ricci flow, degenerate circle packing metric

 Funding Agency Grant Number Russian Science Foundation 16-11-10069 The work was supported by the Russian Science Foundation (grant No. 16-11-10069).

DOI: https://doi.org/10.1134/S1560354719030043

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MSC: 52C26
Accepted:29.04.2019
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Citation: Ruslan Yu. Pepa, Theodore Yu. Popelensky, “Combinatorial Ricci Flow for Degenerate Circle Packing Metrics”, Regul. Chaotic Dyn., 24:3 (2019), 298–311

Citation in format AMSBIB
\Bibitem{PepPop19} \by Ruslan Yu. Pepa, Theodore Yu. Popelensky \paper Combinatorial Ricci Flow for Degenerate Circle Packing Metrics \jour Regul. Chaotic Dyn. \yr 2019 \vol 24 \issue 3 \pages 298--311 \mathnet{http://mi.mathnet.ru/rcd479} \crossref{https://doi.org/10.1134/S1560354719030043} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000470233800004} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85066614483}