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In this paper we consider hyperbolic manifolds with incompressible convex core boundary. We show that total bending along a geodesic arc on the boundary of the convex core is bounded above by a function of its length. Integrating this function over the unit tangent bundle of the boundary of the convex core we obtain a new universal upper bound on the total bending of the convex core boundary. Furthermore, we produce a new universal upper bound on the lipschitz constant for the map from the convex core boundary to the hyperbolic structure at infinity. These results improve on earlier bounds of Bridgeman and Canary. Let N = H/Γ be an orientable hyperbolic manifold with domain of discontinuity Ω(Γ) and limit set LΓ. In this paper we restrict ourselves to the case when all the components of Ω(Γ) are simply connected. This is a natural restriction to make and includes the set of quasi-fuchsian groups. Let CH(LΓ) be the convex hull of Γ and βΓ be the bending lamination on ∂CH(LΓ). Let C(N) = CH(LΓ)/Γ be the convex core, and βN be the bending lamination on ∂C(N). Then we observe that ∂C(N) is incompressible if and only if the components of Ω(Γ) are all simply connected. If α is a geodesic arc in CH(LΓ), the average bending B(α) is defined to be the bending per unit length, or specifically

Bounds on the average bending of the convex hull boundary of a Kleinian group