Rigidity and deformation of generalized sphere packings on 3-dimensional manifolds with boundary

TL;DR

This paper introduces generalized sphere packings on 3D manifolds with boundary, investigates their rigidity, and develops curvature flows to prescribe scalar curvature, extending concepts from 2D circle packings.

Contribution

It extends the theory of circle packings to 3D manifolds, proving rigidity and introducing curvature flows for prescribed scalar curvature.

Findings

Generalized sphere packing metrics are uniquely determined by scalar curvature.

The paper establishes conditions for the existence and convergence of curvature flows.

New methods for prescribing scalar curvature on 3D manifolds with boundary are developed.

Abstract

Motivated by Guo-Luo's generalized circle packings on surfaces with boundary \cite{GL2}, we introduce the generalized sphere packings on 3-dimensional manifolds with boundary. Then we investigate the rigidity of the generalized sphere packing metrics. We prove that the generalized sphere packing metric is determined by the combinatorial scalar curvature. To find the hyper-ideal polyhedral metrics on 3-dimensional manifolds with prescribed combinatorial scalar curvature, we introduce the combinatorial Ricci flow and combinatorial Calabi flow for the generalized sphere packings on 3-dimensional manifolds with boundary. Then we study the longtime existence and convergence for the solutions of these combinatorial curvature flows.