The Topology and Geometry of Hyperbolic 3-Manifolds

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DMS-0203698Richard CanaryThe topology and geometry of hyperbolic 3-manifoldsProfessor Canary will study the space AH(M) of marked hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M. The interior of this space is well-understood due to work of Ahlfors, Bers, Kra, Marden, Maskit, Sullivan and Thurston. Roughly, the components of the interior are enumerated by topological data and each component is parameterized by analytical data. Thurston's Ending Lamination Conjecture predicts that points in the entire space AH(M) can be classified by topological data (coming from the marked homeomorphism type of the hyperbolic 3-manifold) and geometric data (which encodes the asymptotic geometry of the ends.) Yair Minsky has made substantial progress towards the resolution of Thurston's Ending Lamination Conjecture in the case that M has incompressible boundary. Prof. Canary will collaborate with Jeff Brock and Yair Minsky on a program to complete the proof of this conjecture and a number of related conjectures. Recent work, by Anderson, Brock, Bromberg, Canary, Holt, McCullough and McMullen, has revealed surprising new phenomena concerning the global topology of AH(M). Components of the interior may bump (i.e. have intersecting closures) and individual components may bump themselves. Prof. Canary proposes to further study the global topology of AH(M). In particular, the techniques used in the attack on the Ending Lamination Conjecture will be used to study the continuity of the ending invariants.The study of 3-dimensional manifolds employs tools from geometry, topology, dynamics and complex analysis. It is the interplay of these tools which gives the subject its rich flavor. A 2-dimensionalmanifold or surface is a space which looks locally like 2-dimensionalEuclidean space; examples are given by the surfaces of familiar 3-dimensional objects such as doughnuts. Similarly, a 3-manifold is a space that looks locally like 3-dimensional Euclidean space. A Riemannian metric is a way of measuring distances and angles in a manifold. For example, the universe we live in is a 3-manifold with a Riemannian metric. Surfaces were classified in the 19th century, and it was shown that all surfaces admit a beautiful Riemannian metric of one of three types: Euclidean, spherical or hyperbolic. The study of all possible such metrics on a fixed surface was begun in the 19th century and has played a key rolein topology, complex analysis, Riemannian geometry and algebraic geometry. Thurston revolutionized the study of 3-manifolds by conjecturing, and proving in many cases, that every 3-manifoldmay be canonically decomposed into pieces which admit metrics ofone of 8 geometric types. Hyperbolic metrics are the most common and least understood class of such metrics. Prof. Canary will study the space of all hyperbolic metrics on a fixed 3-manifold.