2-Dimensional Combinatorial Calabi Flow in Hyperbolic Background Geometry

TL;DR

This paper introduces a combinatorial Calabi flow for triangulated hyperbolic surfaces, proving convergence to zero curvature metrics under certain conditions, thus advancing discrete geometric analysis in hyperbolic geometry.

Contribution

It defines a new combinatorial flow in hyperbolic space and establishes convergence criteria, extending discrete geometric flows to hyperbolic background geometry.

Findings

Flow converges to ZCCP-metric with small initial energy.

Flow exists for all time under curvature bounds.

Convergence to ZCCP-metric iff such a metric exists.

Abstract

For triangulated surfaces locally embedded in the standard hyperbolic space, we introduce combinatorial Calabi flow as the negative gradient flow of combinatorial Calabi energy. We prove that the flow produces solutions which converge to ZCCP-metric (zero curvature circle packing metric) if the initial energy is small enough. Assuming the curvature has a uniform upper bound less than $2\pi$, we prove that combinatorial Calabi flow exists for all time. Moreover, it converges to ZCCP-metric if and only if ZCCP-metric exists.