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In this paper, we study the geometric aspects of ball packings on $(M,mathcal{T})$, where $mathcal{T}$ is a triangulation on a 3-manifold $M$. We introduce a combinatorial Yamabe invariant $Y_{mathcal{T}}$, depending on the topology of $M$ and the combinatoric of $mathcal{T}$. We prove that $Y_{mathcal{T}}$ is attainable if and only if there is a constant curvature packing, and the combinatorial Yamabe problem can be solved by minimizing Cooper-Rivin-Glickenstein functional. We then study the combinatorial Yamabe flow introduced by Glickenstein cite{G0}-cite{G2}. We first prove a small energy convergence theorem which says 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 shall also prove: although the flow may develop singularities in finite time, there is a natural way to extend the solution of the flow so as it exists for all time. Moreover, if the triangulation $mathcal{T}$ is regular (that is, the number of tetrahedrons surrounding each vertex are all equal), then the combinatorial Yamabe flow converges exponentially fast to a constant curvature packing.
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Discrete conformal structure on polyhedral surfaces is a discrete analogue of the conformal structure on smooth surfaces, which includes tangential circle packing, Thurstons circle packing, inversive distance circle packing and vertex scaling as special cases and generalizes them to a very general context. Glickenstein conjectured the rigidity of discrete conformal structures on polyhedral surfaces, which includes Luos conjecture on the rigidity of vertex scaling and Bowers-Stephensons conjecture on the rigidity of inversive distance circle packings on polyhedral surfaces as special cases. We prove Glickensteins conjecture using a variational principle. We further study the deformation of discrete conformal structures on polyhedral surfaces by combinatorial curvature flows. It is proved that the combinatorial Ricci flow for discrete conformal structures, which is a generalization of Chow-Luos combinatorial Ricci flow for circle packings and Luos combinatorial Yamabe flow for vertex scaling, could be extended to exist for all time and the extended combinatorial Ricci flow converges exponentially fast for any initial data if the discrete conformal structure with prescribed combinatorial curvature exists. This confirms another conjecture of Glickenstein on the convergence of the combinatorial Ricci flow and provides an effective algorithm for finding discrete conformal structures with prescribed combinatorial curvatures. The relationship of discrete conformal structures on polyhedral surfaces and 3-dimensional hyperbolic geometry is also discussed. As a result, we obtain some new convexities of the co-volume functions for some generalized 3-dimensional hyperbolic tetrahedra.
A contact graph of a packing of closed balls is a graph with balls as vertices and pairs of tangent balls as edges. We prove that the average degree of the contact graph of a packing of balls (with possibly different radii) in $mathbb{R}^3$ is not greater than $13.92$. We also find new upper bounds for the average degree of contact graphs in $mathbb{R}^4$ and $mathbb{R}^5$.
We show that the geodesic period spectrum of a Riemannian 2-orbifold all of whose geodesics are closed depends, up to a constant, only on its orbifold topology and compute it. In the manifold case we recover the fact proved by Gromoll, Grove and Pries that all prime geodesics have the same length. In the appendix we partly strengthen our result in terms of conjugacy of contact forms and explain how to deduce rigidity on the real projective plane based on a systolic inequality due to Pu. (We do not use a Lusternik-Schnirelmann type theorem on the existence of at least three simple closed geodesics.)
A leafwise Hodge decomposition was proved by Sanguiao for Riemannian foliations of bounded geometry. Its proof is explained again in terms of our study of bounded geometry for Riemannian foliations. It is used to associate smoothing operators to foliated flows, and describe their Schwartz kernels. All of this is extended to a leafwise version of the Novikov differential complex.
We introduce a category of rigid geometries on singular spaces which are leaf spaces of foliations and are considered as leaf manifolds. We single out a special category $mathfrak F_0$ of leaf manifolds containing the orbifold category as a full subcategory. Objects of $mathfrak F_0$ may have non-Hausdorff topology unlike the orbifolds. The topology of some objects of $mathfrak F_0$ does not satisfy the separation axiom $T_0$. It is shown that for every ${mathcal N}in Ob(mathfrak F_0)$ a rigid geometry $zeta$ on $mathcal N$ admits a desingularization. Moreover, for every such $mathcal N$ we prove the existence and the uniqueness of a finite dimensional Lie group structure on the automorphism group $Aut(zeta)$ of the rigid geometry $zeta$ on $mathcal{N}$.
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