Discrete curvature and rigidity of Fuchsian manifolds

Abstract

This thesis is devoted to some applications of cone-manifolds and discrete curvature to problems in 3-dimensional hyperbolic geometry. First, we prove a realization and rigidity result for a specific family of hyperbolic cone-3-manifolds. This allows us to give a new variational proof of the existence and uniqueness of a hyperbolic cone-metric on S_g with prescribed curvature in a given discrete conformal class. Here S_g is a closed orientable surface of genus g > 1. This also provides a new proof of the fact that every hyperbolic cusp-metric on S_g can be uniquely realized as a convex surface in a Fuchsian manifold. A Fuchsian manifold is a hyperbolic manifold homeomorphic to S_g ×[0; +∞) with geodesic boundary Sg × {0}. They are known as toy cases for studying geometry of non-compact hyperbolic 3-manifolds and hyperbolic 3-manifolds with boundary. Second, we consider compact Fuchsian manifolds with boundary, i.e., hyperbolic manifolds homeomorphic to S_g × [0; 1] with geodesic boundary S_g × {0}. We use cone-manifolds to prove that a compact Fuchsian manifold with convex boundary is uniquely determined by the induced metric on S_g × {1}. It is distinguishing that except convexity we do not put any other condition on the boundary, so it may be neither smooth nor polyhedral.