Complete lifts of harmonic maps and morphisms between Euclidean spaces

TL;DR

This paper introduces the concept of complete lifts of maps between Euclidean spaces, exploring their properties related to harmonicity and conformality, and uses this to characterize holomorphic maps and construct new harmonic morphisms.

Contribution

It defines complete lifts for maps between Euclidean spaces and applies this to characterize holomorphic maps and generate new examples of harmonic morphisms.

Findings

Complete lifts preserve harmonicity and conformality properties.

Characterization of holomorphic maps via complete lifts.

Construction of new harmonic morphisms not derived from Kähler structures.

Abstract

We introduce the complete lifts of maps between (real and complex) Euclidean spaces and study their properties concerning holomorphicity, harmonicity and horizontal weakly conformality. As applications, we are able to use this concept to characterize holomorphic maps $\phi:{\Bbb C}^{m}\supset U\longrightarrow {\Bbb C}^{n}$ (Proposition 2.3) and to construct many new examples of harmonic morphisms (Theorem 3.3). Finally we show that the complete lift of the quaternion product followed by the complex product is a simple and explicit example of a harmonic morphism which does not arise (see Definition 4.8 in \cite{BaiWoo95}) from any K{\"a}hler structure.