Rigidity of generalized Thurston's sphere packings on 3-dimensional manifolds with boundary

TL;DR

This paper extends Thurston's sphere packings to 3D manifolds with boundary, proving their local and infinitesimal rigidity based on combinatorial curvatures, thus advancing geometric understanding of such structures.

Contribution

It introduces generalized Thurston's sphere packings on 3D manifolds with boundary and establishes their rigidity properties, a novel extension from surface packings.

Findings

Packings are locally determined by combinatorial scalar curvatures

Infinitesimal rigidity prevents deformation while fixing combinatorial Ricci curvatures

Extends rigidity results from 2D to 3D manifolds with boundary

Abstract

Motivated by Guo-Luo's generalized circle packings on surfaces with boundary \cite{GL2}, we introduce the generalized Thurston's sphere packings on 3-dimensional manifolds with boundary. Then we investigate the rigidity of the generalized Thurston's sphere packings. We prove that the generalized Thurston's sphere packings are locally determined by the combinatorial scalar curvatures. We further prove the infinitesimal rigidity that the generalized Thurston's sphere packings can not be deformed while keeping the combinatorial Ricci curvatures fixed.