Geometry of k-harmonic maps and the second variational formula of the k-energy

TL;DR

This paper develops the second variational formula for k-harmonic maps, explores their properties in non-positive curvature spaces, and analyzes specific harmonic curves in spheres, revealing new non-trivial solutions.

Contribution

It provides the second variation formula for k-harmonic maps and investigates existence, non-existence, and explicit solutions in specific geometric contexts.

Findings

Derived the second variation formula for k-harmonic maps.

Proved non-existence of proper k-harmonic maps into non-positive curvature manifolds for k >= 2.

Described differential equations and found non-trivial solutions for 3- and 4-harmonic curves in spheres.

Abstract

J.Eells and L. Lemaire introduced k-harmonic maps, and T. Ichiyama, J. Inoguchi and H.Urakawa showed the first variation formula. In this paper, we give the second variation formula of k-harmonic maps, and show non-existence theorem of proper k-harmonic maps into a Riemannian manifold of non-positive curvature (k >= 2). We also study k-harmonic maps into the product Riemannian manifold, and describe the ordinary differential equations of 3-harmonic curves and 4-harmonic curves into a sphere, and show their non-trivial solutions.