The Calabi flow with prescribed curvature on finite graphs

TL;DR

This paper studies a geometric flow called the Calabi flow on finite graphs with prescribed curvature, establishing conditions for its global existence and convergence, especially for constant curvature cases.

Contribution

It introduces a Calabi flow for a specific curvature on finite graphs and characterizes its convergence and existence conditions, extending geometric analysis to graph settings.

Findings

Calabi flow exists globally and converges if a suitable weight function exists.

For constant curvature weights, convergence is proven under certain topological conditions.

Provides an equivalent characterization linking flow convergence to prescribed curvature realization.

Abstract

In this paper, we investigate the prescribed curvature problem associated with a special Lin-Lu-Yau curvature on finite graphs of girth at least 6. We define the corresponding Calabi flow for this curvature type, and establish an equivalent characterization of the problem, namely, the solution to the Calabi flow exists globally in time and converges if and only if there exists a weight function that realizes the prescribed curvature. In particular, for constant curvature weights, we prove that the solution to the Calabi flow exists globally in time and converges under certain topological conditions.