Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns

TL;DR

This paper studies the combinatorial p-th Calabi flow for ideal circle patterns, establishing convergence criteria in finite cases and long-time existence in infinite cases, advancing the understanding of curvature flows in discrete geometry.

Contribution

It provides new convergence conditions for finite circle patterns and proves long-time existence of the flow in infinite settings, extending the theory of discrete curvature flows.

Findings

Flow converges iff a constant curvature metric exists in finite cases.

Long-time existence of solutions for p ≥ 2 in infinite cases.

Significant progress in discrete curvature flow theory.

Abstract

This paper presents a comprehensive study of the combinatorial $p$-th Calabi flow for both finite and infinite ideal circle patterns. In the finite case, we establish a sharp criterion: the combinatorial $p$-th Calabi flow with $p>1$ converges if and only if a constant curvature metric exists in the underlying geometric background. In the infinite setting, we prove the long-time existence of solutions to the combinatorial $p$-th Calabi flow for $p \geq 2$, representing a significant advance in the theory of curvature flows on infinite structures.