Almost all three dimensional manifolds admit canonical metrics with constant sectional curvature. In this paper we proposed a new algorithm pipeline to compute such canonical metrics for hyperbolic 3-manifolds with high genus boundary surfaces. The computation is based on the discrete curvature flow for 3-manifolds, where the metric is deformed in an angle-preserving fashion until the curvature becomes uniform inside the volume and vanishes on the boundary. We also proposed algorithms to visualize the canonical metric by realizing the volume in the hyperbolic space ³, both in single period and in multiple periods. The proposed methods could not only facilitate the theoretical study of 3-manifold topology and geometry using computers, but also have great potentials in volumetric parameterizations, 3D shape comparison, volumetric biomedical image analysis and etc.

It is a theorem of Casson and Rivin that the complete hyperbolic metric on a cusp end ideal triangulated 3-manifold maximizes volume in the space of all positive angle structures. We show that the conclusion still holds if some of the tetrahedra in the complete metric are flat.

We consider a volume maximization program to construct hyperbolic structures on triangulated 3-manifolds, for which previous progress has lead to consider angle assignments which do not correspond to a hyperbolic metric on each simplex. We show that critical points of the generalized volume are associated to geometric structures modeled on the extended hyperbolic space-the natural extension of hyperbolic space by the de Sitter space-except for the degenerate case where all simplices are Euclidean in a generalized sense.

We define a new combinatorial class of triangulations of closed 3-manifolds, satisfying a weak version of 0-efficiency combined with a weak version of minimality, and study them using twisted squares. As an application, we obtain strong restrictions on the topology of a 3-manifold from the existence of non-smooth maxima of the volume function on the space of circle-valued angle structures.

Surface registration plays a fundamental role in many applications in computer vision and aims at finding a oneto-one correspondence between surfaces. Conformal mapping based surface registration methods conformally map 2D/3D surfaces onto 2D canonical domains and perform the matching on the 2D plane. This registration framework reduces dimensionality, and the result is intrinsic to Riemannian metric and invariant under isometric deformation. However, conformal mapping will be affected by inconsistent boundaries and non-isometric deformations of surfaces. In this work, we quantify the effects of boundary variation and non-isometric deformation to conformal mappings, and give the theoretical upper bounds for the distortions of conformal mappings under these two factors. Besides giving the thorough theoretical proofs of the theorems, we verified them by concrete experiments using 3D human facial scans with dynamic expressions and varying boundaries. Furthermore, we used the distortion estimates for reducing search range in feature matching of surface registration applications. The experimental results are consistent with the theoretical predictions and also demonstrate the performance improvements in feature tracking.