Integrability of generalized pluriharmonic maps

TL;DR

This paper constructs examples of pluriharmonic maps from almost complex domains to symmetric spaces that are not integrable, demonstrating the sharpness of previous results on associated families.

Contribution

It provides explicit examples of non-integrable pluriharmonic maps from various complex structures to symmetric spaces, clarifying the limits of existing integrability results.

Findings

Examples of non-integrable pluriharmonic maps are constructed.

Shows the sharpness of previous integrability theorems.

Highlights differences between nearly Kähler and complex manifold sources.

Abstract

In this paper we provide examples of maps from almost complex domains into pseudo-Riemannian symmetric targets, which are pluriharmonic and not integrable, i.e. do not admit an associated family. More precisely, for one class of examples the source has a non-integrable complex structure, like for instance a nearly Kaehler structure and the target is a Riemannian symmetric space and for the other class the source is a complex manifold and the target is a pseudo-Riemannian symmetric space. These examples show, that a former result on the existence of associated families is sharp.