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Variations of the boundary geometry of $$3$$-dimensional hyperbolic convex cores. (English) Zbl 0937.53020
In this paper, $$M$$ is a connected hyperbolic 3-manifold whose fundamental group is finitely generated and is not a finite extension of an abelian group. The convex core $$C_M$$ of $$M$$ is the smallest non-empty closed convex subset of $$M$$. Its boundary $$\partial C_M$$ is a surface of finite topological type which is almost everywhere totally geodesic, and which is bent along a family of geodesics called its pleating locus. The geometry of $$\partial C_M$$ has been studied extensively by W. Thurston. The author investigates here, how the geometry of $$\partial C_M$$ varies as one deforms the metric of $$M$$. To state the results, let $${\mathcal {QD}}(M)$$ be the space of quasi-isometric deformations of the metric of $$M$$ and let $${\mathcal T}(\partial C_M)$$ be the Teichmüller space of $$\partial C_M$$. There are two natural maps, $$\mu : {\mathcal {QD}}(M) \to {\mathcal T}(\partial C_M)$$ (defined by taking the induced path metric on $$\partial C_M$$, which is hyperbolic) and $$\beta : {\mathcal {QD}}(M) \to {\mathcal {ML}}(\partial C_M)$$ (defined by taking the measured lamination which determines the bending of $$\partial C_M$$). The author proves the following:
Theorem 1. The map $$\mu$$ is continuously differentiable.
Theorem 2. The map $$\beta: {\mathcal {QD}}(M) \to {\mathcal {ML}}(\partial C_M)$$ admits a tangent map everywhere (note that one cannot talk about the differentiability of $$\beta$$ since the target space is not a smooth manifold, but a PL manifold).
As a by-product of the proofs of these theorems, the author obtains a result on the space $${\mathcal P}(S)$$ of complex projective structures on a surface $$S$$. This result concerns the homeomorphism between $${\mathcal T}(S)\times {\mathcal ML}(S)$$ and $${\mathcal T}(S)$$, which has been defined by Thurston. The author proves
Theorem 3. The Thurston homeomorphism and its inverse admit a tangent map everywhere.

##### MSC:
 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 57N10 Topology of general $$3$$-manifolds (MSC2010) 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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