It is proved that the volume of spherical or hyperbolic simplices, when considered as a function of the dihedral angles, can be extended continuously to degenerated simplices. This verifies affirmatively a conjecture of John Minor.

Let G={h_t|t∈ℝ} be a continuous flow on a connected n-manifold M. The flow G is said to be strongly reversible by an involution τ if h_{−t}=τh_tτ for all t∈ℝ, and it is said to be periodic if h_s= identity for some s∈ℝ^∗. A closed subset K of M is called a global section for G if every orbit G(x) intersects K in exactly one point. In this paper, we study how the two properties “strongly reversible” and “has a global section” are related. In particular, we show that if G is periodic and strongly reversible by a reflection, then G has a global section.

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.

Registration, which aims to find an optimal 1-1 correspondence between shapes, is an important process in different research areas. Conformal mappings have been widely used to obtain a diffeomorphism between shapes that minimizes angular distortion. Conformal registrations are beneficial since it preserves the local geometry well. However, when landmark constraints are enforced, conformal mappings generally do not exist. This motivates us to look for a unique landmark matching quasi-conformal registration, which minimizes the conformality distortion. Under suitable condition on the landmark constraints, a unique diffeomporphism, called the Teichmller extremal mapping between two surfaces can be obtained, which minimizes the maximal conformality distortion. In this paper, we propose an efficient iterative algorithm, called the Quasi-conformal (QC) iterations, to compute the Teichmller mapping. The basic idea is to represent the set of diffeomorphisms using Beltrami coefficients (BCs), and look for an optimal BC associated to the desired Teichmller mapping. The associated diffeomorphism can be efficiently reconstructed from the optimal BC using the Linear Beltrami Solver (LBS). Using BCs to represent diffeomorphisms guarantees the diffeomorphic property of the registration. Using our proposed method, the Teichmller mapping can be accurately and efficiently computed within 10 seconds. The obtained registration is guaranteed to be bijective. The proposed algorithm can also be extended to compute Teichmller mapping with soft landmark constraints. We applied the proposed algorithm to real applications, such as brain

We show that the hyperbolic structure on a closed, orientable, hyperbolic 3-manifold can be constructed from a solution to the hyperbolic gluing equations using any triangulation with essential edges. The key ingredients in the proof are Thurston's spinning construction and a volume rigidity result attributed by Dunfield to Thurston, Gromov and Goldman. As an application, we show that this gives a new algorithm to detect hyperbolic structures and small Seifert fibred structures on closed 3-manifolds.