On the deformation of ball packings

Abstract

In this paper, we study the geometric aspects of ball packings on (M,T), where T is a triangulation on a 3-manifold M. First, we introduce a combinatorial Yamabe invariant YT, depending on the topology of M and the combinatoric of T. Then we prove that YT is attainable if and only if there is a constant curvature packing, and the combinatorial Yamabe problem can be solved by minimizing the Cooper-Rivin-Glickenstein functional. We also study the combinatorial Yamabe flow introduced by Glickenstein [23–25]. First, we prove a small energy convergence theorem that the flow would converge to a constant curvature metric if the initial energy is close in a quantitative way to the energy of a constant curvature metric. We also show even in case that the flow develops singularities in finite time, there is a natural way to extend the flow through the singularities such that the flow exists for all time. Finally, if the triangulation T is regular (i.e., all the numbers of tetrahedrons surrounding each vertex are equal), the combinatorial Yamabe flow converges exponentially fast to a constant curvature packing.