k-harmonic maps into a Riemannian manifold with constant sectional curvature

TL;DR

This paper explores the properties of k-harmonic maps into Riemannian manifolds with constant curvature, establishing non-existence results for 3-harmonic maps and proposing conjectures for k-harmonic submanifolds.

Contribution

It introduces the relationship between biharmonic and k-harmonic maps, proves non-existence of 3-harmonic maps, and defines and studies k-harmonic submanifolds and curves in Euclidean spaces.

Findings

Non-existence of 3-harmonic maps under certain conditions

Definition and analysis of k-harmonic submanifolds in Euclidean spaces

A conjecture on the characterization of k-harmonic submanifolds

Abstract

J. Eells and L. Lemaire introduced k-harmonic maps, and Wang Shaobo showed the first variational formula. When, k=2, it is called biharmonic maps (2-harmonic maps). There have been extensive studies in the area. In this paper, we consider the relationship between biharmonic maps and k-harmonic maps, and show non-existence theorem of 3-harmonic maps. We also give the definition of k-harmonic submanifolds of Euclidean spaces, and study k-harmonic curve in Euclidean spaces. Futhermore, we give a conjecture for k-harmonic submanifolds of Euclidean spaces.