Notes on the Schwarzian tensor and measured foliations at infinity of quasifuchsian manifolds

Abstract

The boundary at infinity of a quasifuchsian hyperbolic manifold is equiped with a holomorphic quadratic differential. Its horizontal measured foliation $f$ can be interpreted as the natural analog of the measured bending lamination on the boundary of the convex core. This analogy leads to a number of questions. We provide a variation formula for the renormalized volume in terms of the extremal length \ext(f) of $f$, and an upper bound on \ext(f). We then describe two extensions of the holomorphic quadratic differential at infinity, both valid in higher dimensions. One is in terms of Poincar\'e-Einstein metrics, the other (specifically for conformally flat structures) of the second fundamental form of a hypersurface in a "constant curvature" space with a degenerate metric, interpreted as the space of horospheres in hyperbolic space. This clarifies a relation between linear Weingarten surfaces in hyperbolic manifolds and Monge-Amp\`ere equations.Comment: Notes aiming at clarifying the relations between different points of view and introducing one new notion, no real result. Not intended to be submitted at this poin