In this paper, we prove the global rigidity of sphere packings on 3-dimensional manifolds. This is a 3-dimensional analogue of the rigidity theorem of Andreev-Thurston and was conjectured by Cooper and Rivin. We also prove a global rigidity result using a combinatorial scalar curvature introduced by Ge and the author.

Finding surface mappings with least distortion arises from many applications in various fields. Extremal Teichm ller maps are surface mappings with least conformality distortion. The existence and uniqueness of the extremal Teichm ller map between Riemann surfaces of finite type are theoretically guaranteed [1]. Recently, a simple iterative algorithm for computing the Teichm ller maps between connected Riemann surfaces with given boundary value was proposed in [11]. Numerical results was reported in the paper to show the effectiveness of the algorithm. The method was successfully applied to landmark-matching registration. The purpose of this paper is to prove the iterative algorithm proposed in [11] indeed converges.

Given an open disc D 2 with a Lorentz metric g, we are interested in finding a conformal embedding of (D

We define a new combinatorial class of triangulations of closed 3manifolds, and study them using twisted squares. As an application, we obtain strong restrictions on the topology of a 3manifold from the existence of non-smooth maxima of the volume function on the space of circle-valued angle structures.

Computing uniformization maps for surfaces has been a challenging problem and has many practical applications. In this paper, we provide a theoretically rigorous algorithm to compute such maps via combinatorial Calabi flow for vertex scaling of polyhedral metrics on surfaces, which is an analogue of the combinatorial Yamabe flow introduced by Luo (Commun Contemp Math 6(5):765–780, 2004). To handle the singularies along the combinatorial Calabi flow, we do surgery on the flow by flipping. Using the discrete conformal theory established in Gu et al. (J Differ Geom 109(3):431–466, 2018; J Differ Geom 109(2):223–256, 2018), we prove that for any initial Euclidean or hyperbolic polyhedral metric on a closed surface, the combinatorial Calabi flow with surgery exists for all time and converges exponentially fast after finite number of surgeries. The convergence is independent of the combinatorial structure of the initial triangulation on the surface.