A Schläfli-type Formula for Convex Cores of Hyperbolic 3–manifolds

Let M be a (connected) hyperbolic 3–manifold, namely a complete Riemannian manifold of dimension 3 and of constant sectional curvature −1, with finitely generated fundamental group. A fundamental subset of M is its convex core CM , which is the smallest non-empty convex subset of M . The condition that the volume of CM is finite is open in the space of hyperbolic metrics on M , provided we restrict attention to cusp-respecting deformations. In this paper, we give a formula which, for a cusp-preserving variation of the hyperbolic metric of M , expresses the variation of the volume of the convex core CM in terms of the variation of the bending measure of its boundary. This formula is analogous to the Schläfli formula for the volume of an n–dimensional hyperbolic polyhedron P ; see [Sc1][Kne][AVS] and §1. If the metric of P varies, the Schläfli formula expresses the variation of the volume of P in terms of the variation of the dihedral angles of P along the (n− 2)–faces of its boundary and of the (n− 2)–volumes of these faces. The analogy stems from the fact that the boundary ∂CM of CM is almost polyhedral, in the sense that it is totally geodesic almost everywhere. However, the pleating locus , where ∂CM is not totally geodesic, is not a finite collection of edges any more. Typically, it will consist of uncountably many infinite geodesics. In addition, the topology of this pleating locus can drastically change as we vary the metric of M . So the situation is much more complex. The path metric induced on the surface ∂CM by the metric of M is hyperbolic with finite area. On this hyperbolic surface, the pleating locus λ forms a compact geodesic lamination, namely is compact and is the union of disjoint simple geodesics. The surface ∂CM is bent along λ, and the amount of this bending can be measured, not by dihedral angles any more, but by a transverse measure for λ. Endowing λ with this transverse measure, we get a measured lamination b, called the bending measured lamination of M ; see [Thu][EpM]. Let M be a hyperbolic 3–manifold which is geometrically finite, namely such that the convex core CM has finite volume and such that the fundamental group π1 (M) is finitely generated. Consider a deformation of M , namely a differentiable 1–parameter family of hyperbolic manifolds Mt, t ∈ [0, ε[, such that M0 =M ; when M has cusps, we also require that the cusps of each Mt precisely correspond to the cusps of M . Then, Mt is also geometrically finite for t small enough [Mar]. We showed in [Bo4] that, if bt is the bending measured lamination ofMt, then the family bt, t ∈ [0, ε[, admits a tangent vector ḃ0 at t = 0, in the piecewise linear manifold ML (∂CM ) of all measured geodesic laminations on ∂CM . In addition, in [Bo1][Bo2], we showed that such a tangent vector can be geometrically interpreted as a geodesic lamination endowed with a certain type of transverse distribution, called a transverse Hölder distribution. On a hyperbolic surface, a geodesic lamination with transverse distribution a admits a certain length [Bo2][Bo3]. This length is designed so that it varies continuously with a and coincides with the usual length when a consists of a simple closed geodesic endowed with the Dirac transverse distribution. In particular, on the hyperbolic surface ∂CM0 , we can consider the length l0 ( ḃ0 ) of the tangent vector ḃ0.