Associated families of the pluriharmonic maps and isotropy
type: Item
instance of: scholarly article
title: Associated families of the pluriharmonic maps and isotropy
zbMATH Open document ID: 0902.53037
published in: Manuscripta Mathematica
publication date: 1998-12-07T00:00:00Z
full work available at URL: https://eudml.org/doc/156377
review text: Minimal surface immersions in \({\mathbb R}^3\) always admit a 1-parameter family of isometric deformations rotating the differential, known as an associated family. There is a well-known connection between minimal surfaces and harmonic maps. The authors consider pluriharmonic maps from Kähler manifolds into Riemannian manifolds, showing that such maps are characterized by the existence of an associated family. The special case where the associated family of maps is constant is of particular interest. The authors also look at a generalization of immersions of surfaces in \({\mathbb R}^3\) with constant mean curvature. These are immersions of Kähler manifolds into symmetric spaces that are \((1,1)\)-parallel. Let \(M\) be a Kähler manifold and \(S\) any Riemannian manifold. A smooth map \(f:M \rightarrow S\) is pluriharmonic if the \((1,1)\)-part of its Hessian vanishes. Thus, the restriction of \(f\) to any complex curve in \(M\) is a harmonic map. Define \(R_\theta : TM \rightarrow TM\) by \(R_\theta(X) = \cos(\theta)X + \sin(\theta)JX\) for \(0 \leq \theta \leq 2\pi\). For any smooth map \(f:M \rightarrow S\), an associated family for \(f\) is a smooth family of maps \(f_\theta:M \rightarrow S\) such that \(\Phi_\theta \circ df_\theta = df \circ R_\theta\) for all \(\theta\) and for some (family of) parallel bundle isomorphisms \(\Phi_\theta: f^*_\theta TS \rightarrow f^* TS\) preserving the full curvature tensor of \(S\). The authors' main theorem asserts that a smooth map \(f\) is pluriharmonic if and only if there exists an associated family for \(f\). The special case of isotropic pluriharmonic maps consists of those pluriharmonic maps \(f\) such that \(f_\theta = f\) for all \(\theta\). This case is related to the work of \textit{F. E. Burstall} and \textit{J. H. Rawnsley} [`Twistor theory for Riemannian symmetric spaces' (Lect. Notes Math. 1424, Springer-Verlag, Berlin etc.) (1990; Zbl 0699.53059)]. Many interesting isotropic examples are associated to Kähle…
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 31C10
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 1143108
zbMATH Keywords: pluriharmonic maps
zbMATH Keywords: associated families
zbMATH Keywords: isotropic pluriharmonic maps
zbMATH Keywords: minimal surfaces
zbMATH Keywords: immersions of Kähler manifolds into symmetric spaces
zbMATH Keywords: \((1,1)\)-parallel immersions
zbMATH Keywords: weak associated families
author: J.-H. Eschenburg
author: R. Tribuzy
reviewed by: Edward G. Dunne
MaRDI profile type: Publication
OpenAlex ID: W4242806881
DOI: 10.1007/S002290050030
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description: scientific article; zbMATH DE number 1143108