  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.
