Pluriharmonic maps into Kähler symmetric spaces and Sym's formula
type: Item
instance of: scholarly article
title: Pluriharmonic maps into Kähler symmetric spaces and Sym's formula
zbMATH Open document ID: 1189.31009
author: J.-H. Eschenburg
author: Peter Quast
published in: Mathematische Zeitschrift
publication date: 2010-02-15T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 31C10
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 53C43
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 5671548
zbMATH Keywords: pluriharmonic maps
zbMATH Keywords: Kähler symmetric spaces
zbMATH Keywords: constant mean curvature surfaces
zbMATH Keywords: Gauss map
zbMATH Keywords: Sym-Bobenko formula
reviewed by: Dmitri Alekseevsky
MaRDI profile type: Publication
full work available at URL: s00209-008-0472-9
OpenAlex ID: W4253827913
Wikidata QID: Q125022990
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DOI: 10.1007/S00209-008-0472-9
description: scientific article; zbMATH DE number 5671548