In this talk I will discuss about the geometry of constant Gaussian curvature (CGC) surfaces inside hyperbolic ends, and how they relate to the structures of their pleated boundary and their boundary at infinity. In particular, I will describe how the “classical” Thurston’s and Schwarzian parametrizations of the space of hyperbolic ends can be recovered as the limit of two families of parametrizations, introduced by Labourie in terms of the data of immersions of the CGC-surfaces. In addition, I will mention a series of consequences of this phenomenon in relation to the notions of dual and W-volume, such as a new characterization of the renormalized volume in terms of the CGC-foliation, and a generalization of McMullen's Kleinian reciprocity theorem.