Combinatorial p-th Ricci Flows for Inversive Distance Circle Packings

Abstract

In this paper, we consider the combinatorial p-th Ricci flows in the inversive distance circle packing setting. A primary challenge arises from the fact that the solutions to the flow equations may develop three distinct types of boundary singularities, namely “zero boundary”, “infinity boundary” and “triangle inequality invalid boundary” in finite time due to the inversive distance condition I > 1. Adopting the extension techniques, we establish the long time existence and convergence of the solutions to the combinatorial p-th Ricci flows for inversive distance circle packings in Euclidean (resp., hyperbolic) background geometry. Our results partially generalize the work of Ge and Jiang on the deformation of inversive distance circle packings from p = 2 to any p > 1.

Keywords

Combinatorial p-th Ricci flows / inversive distance / combinatorial Ricci potential / 52C26 / 53C15 / 05E45

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