Narrow Results

For any g ≥ 2 we construct a graph Γ_g ⊂ S³ whose exterior M_g = S³\N(Γ_g) supports a complete finite-volume hyperbolic structure with one toric cusp and a connected geodesic boundary of genus g. We compute the canonical decomposition and the isometry group of M_g, showing in particular that any self-homeomorphism of M_g extends to a self-homeomorphism of the pair (S³,Γ_g), and that Γ_g is chiral. Building on a result of Lackenby we also show that any non-meridinal Dehn filling of M_g is hyperbolic, thus getting an infinite family of graphs in S² × S¹ whose exteriors support a hyperbolic structure with geodesic boundary.

Every surface in the Euclidean space ℝ³ admits a canonical Riemannian metric that has constant Gaussian curvature and is conformal to the original metric. Similarly, 3-manifolds can be decomposed into pieces that admit canonical metrics. Such metrics not only have theoretical significance in 3-manifold geometry and topology, but also have potential applications to practical problems in engineering fields such as shape classification.

In this paper we present an algorithm that is based on a discrete curvature flow to compute constant curvature metrics on 3-manifolds that are hyperbolic and have boundaries of a certain type. We also provide an approach to visualize such a metric by embedding the fundamental domain and universal covering in the hyperbolic space ℍ³. Some experimental results are given for both algorithms.

Furthermore, we propose an algorithm to automatically construct truncated tetrahedral meshes for 3-manifolds with boundaries. It can not only generate inputs to the curvature flow algorithm, but could also serve as an automatic tool for geometers and topologists to build simple models for complicated 3-manifolds, and therefore facilitate their research that requires such models.