[Paper Review] Complete lifts of harmonic maps and morphisms between Euclidean spaces

Ye‐Lin Ou|ArXiv.org|Nov 7, 1995

Geometric Analysis and Curvature Flows11 references3 citations

TL;DR

This paper introduces the concept of complete lifts of maps between Euclidean spaces to study holomorphicity, harmonicity, and horizontal weakly conformality. It characterizes holomorphic maps via complete lifts and constructs explicit harmonic morphisms, including one not derivable from Kähler structures, offering new examples in harmonic map theory.

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 $ϕ:{\Bbb C}^{m}\supset U\longrightarrow {\Bbb C}^{n}$ (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{ä}hler structure.

Motivation & Objective

• To define and study the properties of complete lifts of maps between real and complex Euclidean spaces.
• To investigate how complete lifts relate to holomorphicity, harmonicity, and horizontal weak conformality.
• To use complete lifts as a tool to characterize holomorphic maps between complex spaces.
• To construct new explicit examples of harmonic morphisms using the complete lift construction.
• To provide a harmonic morphism that does not arise from any Kähler structure, challenging existing classification frameworks.

Proposed method

• The complete lift of a map between Euclidean spaces is defined as a higher-dimensional extension preserving geometric and analytic properties.
• The construction involves lifting maps via tensor products and algebraic operations, particularly the quaternion and complex products.
• The method applies differential geometric techniques to analyze harmonicity and weakly conformality of the lifted maps.
• Theoretical analysis is used to verify that the complete lift preserves harmonicity and horizontal weak conformality under specific conditions.
• The paper uses complex and quaternionic algebra to build explicit examples of harmonic morphisms from the complete lift of the quaternion product followed by the complex product.
• Theoretical characterization theorems (e.g., Proposition 2.3 and Theorem 3.3) are derived to establish conditions under which maps are holomorphic or harmonic morphisms.

Experimental results

Research questions

• RQ1 How can the complete lift of a map between Euclidean spaces be defined and what geometric properties does it preserve?
• RQ2 Under what conditions does the complete lift of a map preserve harmonicity and horizontal weak conformality?
• RQ3 Can the complete lift construction be used to characterize holomorphic maps between complex Euclidean spaces?
• RQ4 What new examples of harmonic morphisms can be generated using this lift construction?
• RQ5 Is there a harmonic morphism constructed via complete lifts that does not arise from a Kähler structure?

Key findings

• The complete lift construction successfully characterizes holomorphic maps between complex Euclidean spaces, as shown in Proposition 2.3.
• The paper constructs explicit new examples of harmonic morphisms using the complete lift of the quaternion product followed by the complex product.
• The resulting harmonic morphism is shown to not arise from any Kähler structure, as defined in Bai and Wong (1995), providing a novel example.
• The complete lift preserves harmonicity and horizontal weak conformality under the specified algebraic constructions.
• The method provides a systematic way to generate harmonic morphisms beyond those derived from Kähler geometry.
• Theoretical results confirm that the complete lift is a powerful tool for constructing and classifying harmonic morphisms in Euclidean spaces.