The second variational formula of the $k$-energy and $k$-harmonic curves

Abstract

In [4], J. Eells and L. Lemaire introduced $k$-energy and $k$-harmonic maps. In 1989, S.B. Wang [17] showed the first variation formula of the $k$-energy. In this paper, we give the second variation formula of $k$-energy and a notion of weakly stable and unstable. We also study $k$-harmonic maps into product Riemannian manifolds and $k$-harmonic curves into Riemannian manifolds with constant sectional curvature. Moreover, we give some non-trivial solutions of $3$-harmonic curves.