Pluriharmonic maps, twisted loops and twistors

摘要

A minimal surface in euclidean space has two very special properties: (A) It allows a twisted circle of isometric deformations preserving the tangent plane (Associated family), and (B) it is just the real part of a holomorphic map (Weierstrass representation). In fact, these two properties hold more generally for a pluriharmonic map f of a simply connected complex manifold into euclidean space. If instead the target space is a Rie-mannian symmetric space P, Property (A) essentially remains true, however by lack of global parallel displacements a parallel isomorphism between the tangent spaces of the associated family is needed. Consequently Property (B) gets more complicated: f arises by projecting a "superhorizontal" holomorphic map f into a certain infinite dimensional flag manifold (adjoint orbit) fibering over P. the "universal twistor space". The map f takes values in a finite dimensional sub-twistor space iff the associated family is trivial.