Quaternions, Monge–Ampère Structures and k-Surfaces

In Labourie (Geom Funct Anal 7: 496–534, 1997) Labourie developed a theory of immersed surfaces of prescribed extrinsic curvature which has since found widespread applications in hyperbolic geometry, general relativity, Teichmüller 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 Smith (Math Ann 335(1): 57–95, 2013).