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Infinite combinatorial Ricci flow in spherical background geometry
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Since the fundamental work of Chow-Luo \cite{CL03}, Ge \cite{Ge12,Ge17} et al., the combinatorial curvature flow methods became a basic technique in the study of circle pattern theory. In this paper, we investigate the combinatorial Ricci flow with prescribed total geodesic curvatures in spherical background geometry. For infinite cellular decompositions, we establish the existence of a solution to the flow equation for all time. Furthermore, under an additional condition, we prove that the solution converges as time tends to infinity. To the best of our knowledge, this is the first study of an infinite combinatorial curvature flow in spherical background geometry.
Forward citations
Cited by 2 Pith papers
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A prescribed curvature flow on hyperbolic surfaces with infinite topological type
An infinite version of the prescribed curvature flow is shown to converge under side conditions, yielding generalized circle packings and smooth infinite-type hyperbolic surfaces with prescribed total geodesic curvature.
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Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow
This paper proves new convergence results for infinite combinatorial Ricci flow on ideal circle patterns, but the claimed existence of infinite ideal hyperbolic polyhedra is not yet established.
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