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Pseudo-developing maps for ideal triangulations. II: Positively oriented ideal triangulations of cone-manifolds. (English) Zbl 1371.57015

In this paper, the authors study hyperbolic cone-manifold structures on ideal triangulated \(3\)-manifolds, with positively oriented \(3\)-simplices. This work generalizes the work in [Y.-E. Choi, Topology 43, No. 6, 1345–1371 (2004; Zbl 1071.57012)]. Choi’s work studies \(3\)-manifolds with tori boundary, while the work in this paper studies 3-manifolds with arbitrary boundary.
An ideal triangulated \(3\)-manifold \(M\) is obtained from finitely many \(3\)-simplices, by pasting faces of \(3\)-simplices pairwise, then deleting vertices (which may not be manifold points). If each \(3\)-simplex is endowed with the metric of an ideal hyperbolic \(3\)-simplex, then they induce a hyperbolic cone-manifold structure on \(M\), where the cone-angles near \(1\)-simplices may not be \(2\pi\).
In this paper, the authors prove the following main results. For a topological ideal triangulated \(3\)-manifold \(M\) (with arbitrary boundary), if the set of hyperbolic cone-manifold structures on \(M\) with some prescribed cone-angles is not empty, then this set is a smooth complex manifold whose dimension equals the sum of the genera of the vertex links of \(M\) (boundary components of \(M\)). Moreover, for each vertex link of \(M\) with genus \(g\), if we take \(g\) disjoint homologically independent normal simple closed curves on it, then the complex lengths of these curves give a local holomorphic parametrization of the complex manifold.
The authors also compute the set of hyperbolic cone-manifold structures with prescribed cone-angles on two explicit manifolds.

MSC:

57M50 General geometric structures on low-dimensional manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)

Citations:

Zbl 1071.57012

Software:

Regina
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References:

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