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Combinatorial Calabi flows with ideal circle patterns

Xiaoxiao Zhang

TL;DR

This work extends the framework of combinatorial Ricci flows to combinatorial Calabi flows for ideal circle patterns on surfaces, in both hyperbolic and Euclidean geometries. By formulating the flow as a negative gradient flow of the discrete Calabi energy ${\mathcal C}(r)=\|K\|^2$ with $u_i=\ln\tanh\frac{r_i}{2}$ and a positive definite Jacobian $L=\partial K/\partial u$, the authors prove global existence for all time and exponential convergence to zero-curvature ideal patterns, whenever such patterns exist. The results rely on the injectivity of the curvature map for ideal patterns, a robust edge-angle calculus, and convex potential arguments that drive ${\mathcal C}(t)$ to zero. This provides a canonical procedure for realizing flat cone (Euclidean) or hyperbolic metrics from ideal circle patterns and unifies the Euclidean and hyperbolic theories under the combinatorial Calabi flow. Practically, the findings give a rigorous, time-efficient method to deform initial ideal circle patterns toward canonical geometric structures on surfaces.

Abstract

In this paper, we extend the work of Ge-Hua-Zhou \cite{GHZ} on combinatorial Ricci flows for ideal circle patterns to combinatorial Calabi flows in both hyperbolic and Euclidean background geometry. We prove the solution to the combinatorial Calabi flows with any given initial Euclidean (hyperbolic resp.)ideal circle pattern exists for all time and converges exponentially fast to a flat cone metric (hyperbolic resp.) on a given surface.

Combinatorial Calabi flows with ideal circle patterns

TL;DR

This work extends the framework of combinatorial Ricci flows to combinatorial Calabi flows for ideal circle patterns on surfaces, in both hyperbolic and Euclidean geometries. By formulating the flow as a negative gradient flow of the discrete Calabi energy with and a positive definite Jacobian , the authors prove global existence for all time and exponential convergence to zero-curvature ideal patterns, whenever such patterns exist. The results rely on the injectivity of the curvature map for ideal patterns, a robust edge-angle calculus, and convex potential arguments that drive to zero. This provides a canonical procedure for realizing flat cone (Euclidean) or hyperbolic metrics from ideal circle patterns and unifies the Euclidean and hyperbolic theories under the combinatorial Calabi flow. Practically, the findings give a rigorous, time-efficient method to deform initial ideal circle patterns toward canonical geometric structures on surfaces.

Abstract

In this paper, we extend the work of Ge-Hua-Zhou \cite{GHZ} on combinatorial Ricci flows for ideal circle patterns to combinatorial Calabi flows in both hyperbolic and Euclidean background geometry. We prove the solution to the combinatorial Calabi flows with any given initial Euclidean (hyperbolic resp.)ideal circle pattern exists for all time and converges exponentially fast to a flat cone metric (hyperbolic resp.) on a given surface.
Paper Structure (8 sections, 10 theorems, 49 equations, 4 figures)

This paper contains 8 sections, 10 theorems, 49 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ideal circle patterns
  4. Statement of results
  5. Hyperbolic geometry background
  6. Euclidean geometry background
  7. Proof of Theorem \ref{['exists-for-all-time']}
  8. Proof of Theorem \ref{['Euclidean']}

Key Result

Theorem 3.1

Let $G$ be the 1-skeleton of a cellular decomposition $\mathfrak{D}$ of an oriented closed surface $S$. Assume the edge weight $\Theta: E\rightarrow (0,\pi)$ satisfies $\langle\star\rangle$. Then the solution $r(t)$ to the Calabi flow with any given initial circle pattern $r(0)$ exists for all time

Figures (4)

  • Figure 1: circles meet interior
  • Figure 2: circles meet exterior
  • Figure 3: pattern of two circles
  • Figure 4: two adjacent cells

Theorems & Definitions (13)

  • Definition 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • proof
  • ...and 3 more