Hyperbolic Geometry

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9803445 Bonahon The leading theme of this project is the study of rigidity problems for hyperbolic structures on 3-manifolds. In particular, it aims at proving that the geometry of a 3-dimensional hyperbolic convex hull is completely determined by partial data on the geometry of its boundary. The project also involves several connected problems on hyperbolic and euclidean cone manifolds, and on a generalization of curves on a surface called geodesic laminations. These problems are of interest by themselves, but they are also a laboratory for developing and testing new technical tools for application to a wider range of problems in low-dimensional geometry. It is anticipated that a solution to these problems will deepen our understanding of hyperbolic geometry in low dimensions and have applications to other branches of mathematics as well. The project will study several geometric problems on 3-manifolds, namely on spaces of dimension 3. Twenty years ago, revolutionary developments in the field of low-dimensional topology showed that hyperbolic (non-euclidean) geometry was a very powerful tool to analyze the topology of 3-manifolds. This breakthrough also led to the establishment of unexpected connections between low-dimensional topology and other branches of mathematics, such as differential geometry, complex analysis and dynamical systems. ***