Variations of the boundary geometry of 3--dimensional hyperbolic convex cores

Abstract

A fundamental object in a hyperbolic 3-manifold M is its convex core C(M), defined as the smallest closed non-empty convex subset of M. We investigate the way the geometry of the boundary S of C(M) varies as we vary the hyperbolic metric of M. Thurston observed that the intrinsic metric of S is hyperbolic, and that its bending is described by a measured geodesic lamination. We show that, as the hyperbolic metric of the 3--manifold M varies differentiably, the hyperbolic metric of the surface S varies in a C^1, but usually not C^2, manner. Differentiability properties for the bending measured lamination are conceptually less simple, because the space ML(S) of measured geodesic laminations on S has no natural differentiable structure. However, ML(S) is a piecewise linear manifold, and we also show that the bending measured geodesic lamination varies differentiably in a piecewise linear sense. The two results are proved simultaneously, mixing the differentiable and piecewise linear contexts. In particular, the 'corners' of the piecewise linear structure of ML(S) account for the fact that the metric of S is not a C^2 function of the metric of M. A by-product of the proof is a similar differentiablity property for Thurston's parametrization of complex projective structures on a surface by hyperbolic metrics and measured laminations.