The associated family. (English) Zbl 1168.53032
Author’s abstract: Minimal surfaces in Euclidean 3-space, i.e. conformal harmonic maps, enjoy two important properties: They allow a circle of isometric deformations rotating the principal curvature directions, the so called associated family, and they are obtained as the real part of holomorphic functions into \(\mathbb C^3\). These properties are shared by arbitrary (pluri-)harmonic maps into Euclidean \(n\)-space. Replacing \(\mathbb R^n\) by an arbitrary symmetric space leads to similar results, but the role of \(\mathbb C^n\) is played by an infinite dimensional complex homogeneous space acted on by a twisted loop group. We give a survey of the development of this theory from our view point, and we discuss applications to the construction of (pluri-)harmonic maps into symmetric spaces and their rank restrictions.
Reviewer: Constantin Udrişte (Bucureşti)
MSC:
53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
22E67 | Loop groups and related constructions, group-theoretic treatment |
53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |