Epstein Surfaces, $W$-Volume, and the Osgood-Stowe Differential

Abstract

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.