k-Harmonic immersion and submersion into a sphere

TL;DR

This paper investigates k-harmonic immersions and submersions into spheres, establishing relationships between the radius and k, generalizing existing results, and constructing non-harmonic examples via Hopf maps.

Contribution

It extends the theory of k-harmonic maps to immersions and submersions into spheres, providing new relationships and generalizations of prior results.

Findings

Derived relationship between radius and k for k-harmonic immersions

Generalized Oniciuc's results to k-harmonic submersions

Constructed non-harmonic k-harmonic maps using Hopf maps

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 study k-harmonic immersion into a sphere, and get the rerationship between radious and "k" of k-harmonic. And we also consider k-harmonic submersion, and genelarize Oniciuc's results. Futhermore we construct non harmonic k-harmonic by Hopf map.