Variations of the boundary geometry of \(3\)-dimensional hyperbolic convex cores.

*(English)*Zbl 0937.53020In 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.

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.

Reviewer: A.Papadopoulos (Strasbourg)

##### 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) |