Infinite combinatorial Ricci flow in spherical background geometry

TL;DR

This work studies the infinite-circle-pattern problem in spherical geometry by evolving radii under a combinatorial Ricci flow with prescribed total geodesic curvatures. The authors develop an exhaustive finite-domain approximation scheme, leverage a convex potential for total geodesic curvature, and apply a maximum principle on infinite graphs to obtain global existence for $t\in[0,\infty)$. Under a natural set of positivity and summation conditions (S1)-(S3), the flow not only exists for all time but also converges so that $T(r(t))$ approaches the target $\hat{T}$ as $t\to\infty$. This constitutes the first global existence and convergence result for infinite circle patterns in spherical background geometry, broadening the scope of discrete conformal geometry on noncompact surfaces and extending finite spherical results to the infinite setting.

Abstract

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.

Figures (3)

• Figure 1: A circle pattern on $S$
• Figure 2: Spherical quadrilateral $Q_e$
• Figure 3: The diagonal order

Theorems & Definitions (8)

• Theorem 1.1
• Theorem 1.2: Existence
• Theorem 1.3: Convergence
• Lemma 2.1
• Lemma 2.2
• Lemma 3.1
• Theorem 3.1
• Theorem 3.2