Discrete Quasi-Einstein Metrics and Combinatorial Curvature Flows in 3-Dimension

TL;DR

This paper introduces Discrete Quasi-Einstein metrics on 3D triangulated manifolds, analyzes curvature flows that converge to these metrics, and provides algorithms for computing sphere packings with prescribed curvatures.

Contribution

It defines DQE-metrics, studies their properties via curvature flows, and proves convergence results, offering new computational methods for discrete geometric structures.

Findings

Curvature flows converge to DQE-metrics with small initial energy.

Discrete dual-Laplacians are interpreted as Jacobians of the curvature map.

Algorithm for computing sphere packing metrics with prescribed curvatures.

Abstract

We define Discrete Quasi-Einstein metrics (DQE-metrics) as the critical points of discrete total curvature functional on triangulated 3-manifolds. We study DQE-metrics by introducing some combinatorial curvature flows. We prove that these flows produce solutions which converge to discrete quasi-Einstein metrics when the initial energy is small enough. The proof relies on a careful analysis of discrete dual-Laplacians which we interpret as the Jacobian matrix of the curvature map. As a consequence, combinatorial curvature flow provides an algorithm to compute discrete sphere packing metrics with prescribed curvatures.