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

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.

20 pages, 1 figures

References in corpus (4)

• Geometric triangulations and discrete Laplacians on manifolds
• An extended correction to ``Combinatorial Scalar Curvature and Rigidity of Ball Packings,'' (by D. Cooper and I. Rivin)
• Discrete Laplace-Beltrami Operator Determines Discrete Riemannian Metric
• Combinatorial Calabi flows on surfaces

Cited by in corpus (11)

• 2-Dimensional Combinatorial Calabi Flow in Hyperbolic Background Geometry
• Simplicial Ricci Flow
• $\alpha$-curvatures and $\alpha$-flows on low dimensional triangulated manifolds
• 3-Dimensional Discrete curvature flows and discrete Einstein metric
• Combinatorial Calabi flows on surfaces
• Topological Signals of Singularities in Ricci Flow
• Rigidity of discrete conformal structures on surfaces
• Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, I
• Fractional combinatorial Calabi flow on surfaces
• Equivalence of Simplicial Ricci Flow and Hamilton's Ricci Flow for 3D Neckpinch Geometries
• Combinatorial Calabi flow with surgery on surfaces