# Higher-power harmonic maps and sections

## TL;DR

This paper develops a variational theory for higher-power energy of maps between Riemannian manifolds and classifies certain vector fields on 3D unimodular Lie groups, advancing geometric analysis.

## Contribution

It introduces a comprehensive variational framework for higher-power energies and provides a complete classification of invariant vector fields on specific Lie groups.

## Key findings

- Complete classification of left-invariant vector fields on 3D unimodular Lie groups.
- Development of a variational theory for higher-power energy in Riemannian geometry.
- Application to sections of Riemannian vector bundles and sphere subbundles.

## Abstract

The variational theory of higher-power energy is developed for mappings between Riemannian manifolds, and more generally sections of submersions of Riemannian manifolds, and applied to sections of Riemannian vector bundles and their sphere subbundles. A complete classification is then given for left-invariant vector fields on 3-dimensional unimodular Lie groups equipped with an arbitrary left-invariant Riemannian metric.