Positively oriented ideal triangulations on hyperbolic three-manifolds
In this paper the (smooth but not necessarily complete) hyperbolic structures with positively oriented ideal triangulations on non-compact 3-manifolds are studied. The hyperbolic structure on a manifold \(M\) is obtained from a triangulation of \(M\) by hyperbolic ideal tetrahedra in such a way that their hyperbolic metrics consistently define a smooth hyperbolic metric on \(M\). This requires that certain consistency equations be satisfied, and a solution leads to a hyperbolic structure of \(M\) when the orientations of the tetrahedra are compatible, or equivalently positive orientations (i.e., parametrized by complex numbers with positive imaginary part). The author proves that if \(M\) admits a complete hyperbolic structure of finite volume, then the gluing variety defined by the gluing consistency equations is a smooth complex manifold with dimension equal to the number of boundary components (ends) of \(M\). This theorem was first formulated by W. Thurston in the early 1980s, but in this paper all details of the proof are provided. Moreover, the author shows that the complex lengths of any collection of non-trivial boundary curves, one from each boundary component, give a local holomorphic parameterization of the gluing variety. As an application, some estimates for the size of the hyperbolic Dehn surgery space of once-punctured torus bundles are given.
scientific article; zbMATH DE number 2117934