Quaternions, Monge-Ampère structures and \(k\)-surfaces (Q6610767)

Let \((X, h)\) be a complete, oriented, \(3\)-dimensional Riemannian manifold, let \(TX\) be its tangent bundle, and \(SX\) be its unit sphere subbundle. An (oriented) immersed surface in \(X\) is a pair \((S, e)\), where \(S\) is an oriented surface, and \(e : S \rightarrow X\) is a smooth immersion. Let \(\nu_{e} : S \rightarrow SX\) denote its unit normal vector field compatible with the orientation. The immersed surface \((S,e)\) is said to be infinitesimally strictly convex (ISC) whenever its second fundamental form is positive definite, and it is quasicomplete whenever it is complete with respect to the Riemannian metric, given by the sum of its first and third fundamental form. The Gauss lift of \((S, e)\) is the pair \((S,\nu_{e})\).\N\N\textit{F. Labourie} [Geom. Funct. Anal. 7, No. 3, 496--534 (1997; Zbl 0885.32013)] showed that the Gauss lift of any ISC immersed surface of prescribed extrinsic curvature is a pseudo-holomorphic curve for some suitable almost complex structure. The book chapter under review presents a quaternionic reformulation of Labourie's ideas, which leads to a simpler proof of his results.\N\NFor the entire collection see [Zbl 1537.51001].