Computing and Visualizing Constant-Curvature Metrics on Hyperbolic 3-Manifolds with Boundaries

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 3manifolds 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 H, 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.