Associated families of the pluriharmonic maps and isotropy
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 Kähler manifolds fibered over a Riemannian symmetric space with complex fibers. For \((1,1)\)-parallel immersions, the authors show that an adapted isometric immersion is \((1,1)\)-parallel if and only if there is a weak associated family.
immersions of Kähler manifolds into symmetric spaces
scientific article; zbMATH DE number 1143108
scientific article