# Thurston's sphere packings on 3-dimensional manifolds, I

## TL;DR

This paper establishes local and infinitesimal rigidity results for Thurston's Euclidean sphere packings on 3-manifolds, showing they are uniquely determined by combinatorial scalar curvature up to scaling.

## Contribution

It generalizes previous rigidity results to Thurston's sphere packings on 3-manifolds, proving local determination and infinitesimal rigidity.

## Key findings

- Thurston's Euclidean sphere packing is locally determined by combinatorial scalar curvature.
- Thurston's Euclidean sphere packing cannot be deformed without changing curvature, except by scaling.
- The results extend rigidity theory from 2D circle packings to 3D sphere packings.

## Abstract

Thurston's sphere packing on a 3-dimensional manifold is a generalization of Thusrton's circle packing on a surface, the rigidity of which has been open for many years. In this paper, we prove that Thurston's Euclidean sphere packing is locally determined by combinatorial scalar curvature up to scaling, which generalizes Cooper-Rivin-Glickenstein's local rigidity for tangential sphere packing on 3-dimensional manifolds. We also prove the infinitesimal rigidity that Thurston's Euclidean sphere packing can not be deformed (except by scaling) while keeping the combinatorial Ricci curvature fixed.