# Harmonic maps from $\mathbb{C}^{n}$ to Kahler manifolds

## TL;DR

This paper proves that harmonic maps from complex Euclidean spaces to Kähler manifolds are necessarily holomorphic under certain energy density conditions, extending classical Liouville theorems to a complex geometric setting.

## Contribution

It establishes a new holomorphicity result for harmonic maps from ^n to Kähler manifolds, generalizing previous real-valued harmonic map theorems.

## Key findings

- Harmonic maps from ^n to Kähler manifolds are holomorphic under energy density assumptions.
- The result extends Liouville type theorems to complex geometric contexts.
- Provides a new link between harmonic map theory and complex differential geometry.

## Abstract

In this paper, we shall prove that a harmonic map from $\mathbb{C}^{n}$ ($n\geq2$) to any Kahler manifold must be holomorphic under an assumption of energy density. It can be considered as a complex analogue of the Liouville type theorem for harmonic maps obtained by Sealey.

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Source: https://tomesphere.com/paper/1902.05293