2π theorem

Article summary

In mathematics, the theorem of Gromov and Thurston states a sufficient condition for Dehn filling on a cusped hyperbolic 3-manifold to result in a negatively curved 3-manifold. Let be a cusped hyperbolic 3-manifold. Disjoint horoball neighborhoods of each cusp can be selected. The boundaries of these neighborhoods are quotients of horospheres and thus have Euclidean metrics. A slope, i.e. unoriented isotopy class of simple closed curves on these boundaries, thus has a well-defined length by taking the minimal Euclidean length over all curves in the isotopy class. The theorem states: a Dehn filling of with each filling slope greater than results in a 3-manifold with a complete metric of negative sectional curvature. In fact, this metric can be selected to be identical to the original hyperbolic metric outside the horoball neighborhoods. The basic idea of the proof is to explicitly construct a negatively curved metric inside each horoball neighborhood that matches the metric near the horospherical boundary. This construction, using cylindrical coordinates, works when the filling slope is greater than . See for complete details. According to the geometrization conjecture, these negatively curved 3-manifolds must actually admit a complete hyperbolic metric.…

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