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

Abstract

J. Eells and L. Lemaire introduced k-harmonic maps, and Shaobo Wang 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 we show the non-existence theorem of 3-harmonic maps. We also give the definition of k-harmonic submanifolds of Euclidean spaces and study k-harmonic curves in Euclidean spaces. Furthermore, we give a conjecture for k-harmonic submanifolds of Euclidean spaces.

Biharmonic Curves in Finsler Spaces

“…Since in the case when κ 1 = 0 (i.e., c is a geodesic), the curve is trivially biharmonic, we immediately get: A similar result is known to hold in Riemannian geometry (see [2], [6], [14], [18]). …”

“…In Riemannian spaces, biharmonic curves and, more generally, biharmonic maps, were examined quite in detail (see [6], [8], [12], [13], [14], [17], [18], [19]). As a first remark, geodesics of ∇ are always biharmonic (actually, they are minimum points for the bienergy), but the converse is generally not true.…”

New examples of r-harmonic immersions into the sphere

“…Here we recall some basic facts whose proofs can be found in [4] and [9]. Let ϕ : (M, g M ) → (N, g N ) be a smooth map between two Riemannian manifolds M and N of dimension m and n respectively.…”

Properr-harmonic submanifolds into ellipsoids and rotation hypersurfaces

“…Therefore, we say that an immersed submanifold of N is a proper r-harmonic submanifold if the immersion is an r-harmonic map which is not harmonic (or, equivalently, not minimal). As a general fact, when the ambient has nonpositive sectional curvature there are several results which assert that, under suitable conditions, an r-harmonic submanifold is minimal (see [5], [15], [18] and [25], for instance), but the Chen conjecture that a biharmonic submanifold of R n must be minimal is still open (see [6] for recent results in this direction). More generally, the Maeta conjecture (see [15]) that any r-harmonic submanifold of the Euclidean space is minimal is open.…”

“…As a general fact, when the ambient has nonpositive sectional curvature there are several results which assert that, under suitable conditions, an r-harmonic submanifold is minimal (see [5], [15], [18] and [25], for instance), but the Chen conjecture that a biharmonic submanifold of R n must be minimal is still open (see [6] for recent results in this direction). More generally, the Maeta conjecture (see [15]) that any r-harmonic submanifold of the Euclidean space is minimal is open. By contrast, in our recent paper [23] we produced several new proper r-harmonic submanifolds of the Euclidean unit sphere S m (r ≥ 4, extending the previous results of [16] for r = 3).…”