Geometría Hiperbólica: Variedades de Dimensión 3, Convexidad e Integrales de Curvatura

Project Details

Description

This project deals with there aspects of hyperbolic geometry. Hyperbolic geometry plays a central role in three.dimensional topology, by Thurston's works. One of the main conjectures in the field deals with equipping three-manifolds with hyperbolic metrics. A special case is the so called Orbifold Theorem, stated in 1981 but remaining to be proved. We want to prove the Orbifold Theorem in full generality, A special case of it has already been proved by members of the group jointly with M. Boileau from Toulouse (France). The works (GR85) and (GR99) of two members of the group have improved the understanding of the asymptotic behavior of convex, h-convex and ^convex subsets of hyperbolic plane. The ratio area/longitude can take any value between 0 and 1 when the convex sets grow till filing the plane. The behavior of h-convex (with respect to horocycles) sets in Hn is also known (BM99). In this case the limit value is 1/(n-1). We want to analyze the asymptotic behavior of the ratio volume/(area of the boundary) for complete sumply connected manifolds with negative curvature. The question of bounding the total absolute curvature for immersions of compact manifolds in Euclidean and elliptic spaces has been solved by Chern and Lashof (CL58) and by Langevin and Rosenberg (LR96). It remains to understand the hyperbolic case. For compact immersed manifolds in hyperbolic spaces, we want to give lower bounds for the total absolute curvature, so that the bounds depend on the topology of the manifold.
StatusFinished
Effective start/end date19/12/0019/12/03

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