  In [15] Labourie develops a theory of immersed surfaces of prescribed
extrinsic curvature which has since found widespread applications in hyperbolic
geometry, general relativity, Teichm\"uller theory, and so on. In this chapter,
we present a quaternionic reformulation of these ideas. This yields simpler
proofs of the main results whilst pointing towards the higher-dimensional
generalisation studied by the author in [25].
