Heegaard splittings and hyperbolic geometry

It is well known that every closed 3-manifold has a Heegaard splitting and the combinatorics of the Heegaard splitting identifies the 3-manifold. Yet it has been hard to use Heegaard splittings to obtain information about topology and geometry of the manifold. We develop a new aproach to use hyperbolic geometry and in particular deformation theory of compressible ends of hyperbolic manifolds to study closed 3-manifolds. Using this approach, we have been able to prove that a big class of 3-manifolds which admit a Heegaard splitting with what we call "bounded combinatorics" admit a negatively curved metric with sectional curvatures pinched about -1. This answers some interesting questions about these manifolds and in fact gives a coarse description of the geometry of these manifolds equipped with the negatively curved metrics. The description of these geometries is motivated by work of Minsky in constructing models for hyperbolic manifolds with incompressible boundary. In fact, much of our work is aimed at developing a similar theory in the compressible boundary case.