Structure of hyperbolic 3-manifolds

Yale University, New Haven CT

Investigators

Abstract

The field of hyperbolic 3-manifolds and Kleinian groups has seen considerable progress in the last three years, with the resolution of most of the main motivational conjectures, such as Tameness, the Ending Lamination Conjecture, and the Density conjecture. These advances confirm much about our expected picture of hyperbolic 3-manifolds and their deformation spaces, and place the field in a moment of transition and opportunity. the techniques introduced in the proofs have much potential for further applications. Minsky will focus on deepening our understanding of the structure and deformation theory of hyperbolic 3-manifold, applying in particular the tools that have come out of his contribution to the solution of the Ending Lamination Conjecture. The models and estimates provided by these tools should provide an approach to a number of open questions, notably that of local connectivity of limit sets, geometric description of closed manifolds from the Heegaard decompositions, and uniformity theorems for deformation spaces (some of this work will be in collaboration with Brock, Bromberg and Canary). Another area of applications (jointly with Brock and Masur) involves the structure of geodesics in the Teichmuller space endowed with its Weil-Petersson metric. These have up till now resisted analysis but appear to be quite intimately connected to the geometry of 3-manifolds. The interactions between geometry, topology and dynamics have been a beautiful and powerful feature of mathematics and physics for more than a hundred years. Dynamics is the study of time-evolution of mathematical or physical systems, whereas geometry and topology involve "static" objects such as surfaces or higher-dimensional analogues, often the background for a dynamical process. Henri Poincare already knew that the standard round sphere, the setting of classical analysis and geometry, functioned also as a "horizon at infinity" for an exotic non-Euclidean geometry that we now call Hyperbolic space. Dynamical properties of transformations of the sphere translate to geometric properties of rigid motions of this space, and give rise to families of symmetric tilings whose structure we can study by geometric and topological methods. The complexity of these systems can constrain them so much that a combinatorial (or topological) description suffices to determine them uniquely, and this is what we call rigidity. This phenomenon occurs in many guises throughout geometry and dynamics, and is relevant to issues such as classification of systems, mapping out regions of stability and instability, deformation and bifurcation of families of systems, and probabilistic properties such as ergodicity, all of which have significance in both pure and applied mathematics. The particular aspects studied in this project are typical in some ways and special in others. They focus on the intricate relationships between geometry in two and three dimensions, and also on the ways in which topology, particularly of systems of curves within surfaces, determines geometry. There is also a strong emphasis on studying families of geometric structures on surfaces and three-dimensional manifolds, which are closely analogous to other families of dynamical systems.