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On the Ricci flow on Trees

Shuliang Bai, Bobo Hua, Yong Lin, Shuang Liu

TL;DR

The paper analyzes continuous-time Ricci flow on finite trees using the Lin-Lu-Yau Ollivier curvature, revealing how edge weights and curvatures evolve and establishing long-time limits. By exploiting the combinatorial simplicity of trees and selecting the curvature parameter $oldsymbol{ rac{1}{x}}$, it proves convergence of the flow and characterizes when a zero-curvature limit can occur, showing it is possible only for caterpillar trees. The work provides precise asymptotics for leaf and internal edges, proves monotonicity properties of unnormalized weights, and outlines a conjecture that the normalized flow generally converges to a constant-curvature metric, with a complete zero-curvature case proved for caterpillars. These results illuminate the interplay between topology, discrete curvature, and metric evolution, and pave the way for leveraging tree structure in discrete geometric flows.

Abstract

In this paper, we study the evolution of metrics on finite trees under continuous-time Ricci flows based on the Lin-Lu-Yau version of Ollivier Ricci curvature. We analyze long-time dynamics of edge weights and curvatures, providing precise characterizations of their limiting behaviors. We prove that the Ricci flow converges to metric with zero curvature on edges whose normalized weights converge to positive values only if the tree is a caterpillar tree.

On the Ricci flow on Trees

TL;DR

The paper analyzes continuous-time Ricci flow on finite trees using the Lin-Lu-Yau Ollivier curvature, revealing how edge weights and curvatures evolve and establishing long-time limits. By exploiting the combinatorial simplicity of trees and selecting the curvature parameter , it proves convergence of the flow and characterizes when a zero-curvature limit can occur, showing it is possible only for caterpillar trees. The work provides precise asymptotics for leaf and internal edges, proves monotonicity properties of unnormalized weights, and outlines a conjecture that the normalized flow generally converges to a constant-curvature metric, with a complete zero-curvature case proved for caterpillars. These results illuminate the interplay between topology, discrete curvature, and metric evolution, and pave the way for leveraging tree structure in discrete geometric flows.

Abstract

In this paper, we study the evolution of metrics on finite trees under continuous-time Ricci flows based on the Lin-Lu-Yau version of Ollivier Ricci curvature. We analyze long-time dynamics of edge weights and curvatures, providing precise characterizations of their limiting behaviors. We prove that the Ricci flow converges to metric with zero curvature on edges whose normalized weights converge to positive values only if the tree is a caterpillar tree.

Paper Structure

This paper contains 13 sections, 21 theorems, 134 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Main results.
  3. Preliminaries and Definitions
  4. Ricci curvature on trees.
  5. The parameter function $\gamma(x) = \frac{1}{x}$ and basic facts.
  6. An illustrative example of Ricci Flow on a tree.
  7. Some lemmas
  8. Outline of the proof for the results
  9. Asymptotic behavior of weights under the Ricci flow
  10. Asymptotic behavior of curvature
  11. The limit metric with constant curvature zero
  12. Exploring the Convergence of Curvature and Weights
  13. Appendix

Key Result

Theorem 1

For a tree $T=(V, E)$ with an initial metric, the Ricci flow eq:unnor_continuous converges. Moreover, if $\lim\limits_{t\to\infty} w_e(t)= \infty$, then for sufficiently large time $t\gg 1,$$w_e(t)$ is increasing in $t$.

Figures (9)

  • Figure 1: A simple tree
  • Figure 2: Evolution of unnormalized weight on edge uv for different initial values
  • Figure 3: Caterpillar tree derived from a path by adding pending vertices
  • Figure 4: A non-caterpillar tree: vertex $v_3$ has three internal edges (solid along spine and dashed extra branch), which forces at least one extra leaf $\ell_6$ not counted in the spine formula.
  • Figure 5: An evolution of the unnormalized Ricci flow on the left tree $T_1$ with constant curvature $0$
  • ...and 4 more figures

Theorems & Definitions (48)

  • Theorem 1
  • Definition 1
  • Conjecture 1
  • Theorem 2
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 3
  • Example 1
  • Lemma 1: Barbalat's Lemma
  • ...and 38 more