Mathematical Sciences: Geometry of Hyperbolic 3-Manifolds

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The investigator will attack the problem of the isometric classification of hyperbolic metrics on a given 3-dimensional manifold, as well as study the topology of limit sets of such metrics. These two problems involve understanding the geometry of geometrically infinite ends of hyperbolic 3-manifolds. The approach which he plans to use is based on the technique of pleated surfaces in hyperbolic manifolds. He intends to develop a calculus for pleated surfaces which will enable him to obtain estimates on their geometry and the speed at which they converge to infinity in the 3-manifold. It is a surprising fact that although we live in a three dimensional space, a so-called 3-manifold, and so are blessed with a natural intuition about such geometric objects, in the end this does not carry us as far as we might have expected,for questions which have been settled by algebraic calculations for higher dimensional manifolds still remain baffling in the 3-dimensional case. The most famous of these is the celebrated conjecture of Poincare from around the turn of the century concerning 3- dimensional spheres, where precisely the original 3-dimensional case is the only one still open. The investigator is pursuing a variety of questions about 3-dimensional manifolds with slightly strange notions of distance on them, so-called hyperbolic metrics, but time and time again these questions have been shown to have clear relevance to the case of manifolds with a more familiar notion of distance.