A twistorial interpretation of the Weierstrass representation formulae
The theory of minimal surfaces represents an important chapter in the study of global analysis and remains a testing ground for our understanding of the non-linear partial differential equations of geometry. Perhaps its greatest charm lies in its mercurial avoidance of isolation. Today we see profound applications to such diverse fields as 3-manifold topology and non-abelian gauge theory, to name two ; see [En] for a recent survey and extensive bibliography. (Very recently there have been exciting new applications of the theory of periodic minimal surfaces in 30 to crystallography, see [T&A&H&H].) Consequently, the principal aim of this thesis, which is to establish the groundwork for the investigation of new interactions between minimal surface theory in V, algebraic geometry and soliton theory, see §4.F, is very much in the traditional spirit of the subject.