The convex core of quasifuchsian manifolds with particles
A quasifuchsian manifold is a complete hyperbolic manifold \(M\) diffeomorphic to \(S \times \mathbb{R}\), where \(S\) is a closed oriented surface of genus at least \(2\), which contains a nonempty compact convex subset. Such a manifold has a boundary at infinity formed by two copies of \(S\), which come with conformal structures induced by the hyperbolic metric on \(M\). The celebrated double uniformization theorem of \textit{L. Bers} [Bull. Am. Math. Soc. 66, 94--97 (1960; Zbl 0090.05101)] asserts these two conformal structures uniquely determine the quasifuchsian metric on \(M\), thus defining a homeomorphism between the product of two copies of the Teichmüller space of \(S\) and the space of quasifuchsian metrics. The smallest nonempty convex subset of \(M\) is called the convex core \(C(M)\). Its boundary is again the union of two copies of \(S\), each being a pleated surface in \(M\) with an induced hyperbolic metric and a measured bending lamination. The well known results of \textit{F. Bonahon} and \textit{J.-P. Otal} [Ann. Math. (2) 160, No. 3, 1013--1055 (2005; Zbl 1083.57023)] establish the relation between the quasifuchsian metric on \(M\) and the induced boundary metrics and laminations.NEWLINENEWLINEThe main goal of the present paper is an extension of these results to quasifuchsian manifolds with ``particles'', i.e. cone singularities of fixed angle less than \(\pi\) along curves joining the two boundary components. Along the way the authors also obtain some new proofs of known theorems. Among the key ingredients of the proofs are a compactness lemma in Section 3 and a local rigidity result for hyperbolic cone-manifolds of Hodgson and Kerckhoff.NEWLINENEWLINEThe paper is clearly written with many details explained very carefully. Besides proving new interesting results, it may also serve as a good reference for an introduction to the subject.
scientific article; zbMATH DE number 6356610
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