The study of hyperbolic geometry in dimensions two and three has long been intertwined with holomorphic quadratic differentials on Riemann surfaces. We will focus on three settings in which this connection appears: the Bers embedding of Teichmüller space, Wolf’s parametrization via harmonic maps, and the description of almost Fuchsian manifolds through the second fundamental forms of their minimal surfaces. In this talk, I will present work in progress with Nathaniel Sagman showing that these three topics, traditionally regarded as unrelated, can instead be understood as different restrictions of a more general "holomorphic" framework.
