Pluriharmonic maps into Kähler symmetric spaces and Sym's formula
It is well know that the Gauss map of a constant mean curvature surface \( f: M \to \mathbb R^{3}\) is a harmonic map \(h :M \to S^2\). Conversely, the surface \(f(M)\) can be reconstructed from its Gauss map using a famous formula by Sym and Bobenko. The aim of the authors is to generalize this Sym-Bobenko construction to higher dimension and codimension. More precisely, the authors change \(S^2\) by a Kähler symmetric space \(P = G/K\) considered as an adjoint orbit \( P = \mathrm {Ad}_Gx \subset \mathfrak g\). They establish a one-to-one correspondence between pluriharmonic maps \(h : M \to P\) of a complex manifold \(M\) and pairs of maps \(f_+, f_- : M \to \mathfrak g\) which are ``quasi-holomorphic'' along the common normal vector \(h = \frac12 (f_+ - f_-)\). All regular points of the Riemannian metrics on \(M\) induced by \(f_{\pm}\) are Kähler. Moreover, both immersions are ``pluri-constant mean curvature'', that is, their restriction to a complex one-dimensional submanifold behave like a constant mean curvature surface.
scientific article; zbMATH DE number 5671548
scientific article