  In a seminal paper, Epstein introduced the theory of what are now called
Epstein surfaces, which construct surfaces in $\mathbb{H}^3$ associated to a
conformal metric on a domain in $\hat{\mathbb{C}}$. More recently, these
surfaces have been used by Krasnov-Schlenker to define the W-volume and
renormalized volume associated with a convex co-compact hyperbolic 3-manifold.
In this paper we consider Epstein surfaces, W-volume and renormalized volume in
two main parts. In the first, we develop an alternate construction of Epstein
surfaces using the Osgood-Stowe differential, a generalization of the
Schwarzian derivative. Krasnov-Schlenker showed that the metric and shape
operator of a surface in hyperbolic space is naturally dual to a conformal
metric and shape operator on a projective structure via the hyperbolic Gauss
map. We show that this projective shape operator can be derived from the
Osgood-Stowe differential. This approach allows us to give a comprehensive and
self-contained development of Epstein surfaces, W-volume, and renormalized
volume. In the second part, we use the theory developed in the first to prove a
number of new results including a generalization of Epstein's univalence
criterion, a variational formula for W-volume in terms of the Osgood-Stowe
differential, and a use of W-volume to relate the length of the bending
lamination of the convex core to the norm of the Schwarzian derivative of the
associated univalent map on the conformal boundary.
