Epstein surfaces and applications to Teichmüller theory

Given any Jordan domain U in the Riemann sphere equipped with the hyperbolic metric, one can construct the Epstein surface in hyperbolic 3 space corresponding to it. One important property of the Epstein surface is that its principal curvatures are related to the Schwarzian derivative of the uniformization map of U. In this talk, we will exploit this property to relate the bending lamination of a pleated surface with the Schwarzian derivative. Furthermore, if U is a domain for a Weil-Peterson curve, we will show that the hyperbolic metric and the quasi-hyperbolic metric are arbitrarily close and the two corresponding Epstein surfaces are arbitrarily close.