Constant curvature surfaces and volumes of convex co-compact hyperbolic manifolds

We investigate the properties of various notions of volume for convex co-compact hyperbolic 3-manifolds, and their relations with the geometry of the Teichmüller space. We prove a first-order variation formula for the dual volume of the convex core, as a function over the space of quasi-isometric deformations of a convex co-compact hyperbolic 3-manifold. For quasi-Fuchsian manifolds, we show that the dual volume of the convex core is bounded from above by a linear function of the Weil-Petersson distance between the pair of hyperbolic structures on the boundary of the convex core. We prove that, as we vary the convex co-compact structure on a fixed hyperbolic 3-manifold with incompressible boundary, the infimum of the dual volume of the convex core coincides with the infimum of the Riemannian volume of the convex core. We study various properties of the foliation by constant Gaussian curvature surfaces (k-surfaces) of convex co-compact hyperbolic 3-manifolds. We present a description of the renormalized volume of a quasi-Fuchsian manifold in terms of its foliation by k-surfaces. We show the existence of a Hamiltonian flow over the cotangent space of Teichmüller space, whose flow lines corresponds to the immersion data of the k-surfaces sitting inside a fixed hyperbolic end, and we determine a generalization of McMullen’s Kleinian reciprocity, again by means of the constant Gaussian curvature surfaces foliation.