New examples of r-harmonic immersions into the sphere

Abstract

Polyharmonic, or r-harmonic, maps are a natural generalization of harmonic maps whose study was proposed by Eells-Lemaire in 1983. The main aim of this paper is to construct new examples of proper r-harmonic immersions into spheres. In particular, we shall prove that the canonical inclusion i : S n−1 (R) ֒→ S n is a proper r-harmonic submanifold of S n if and only if the radius R is equal to 1/ √ r. We shall also prove the existence of proper r-harmonic generalized Clifford's tori into the sphere.

Properr-harmonic submanifolds into ellipsoids and rotation hypersurfaces

“…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). The aim of this paper is to continue the work started in [23] and describe some extensions to cases where the ambient manifold does not have constant sectional curvature.…”

“…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). The aim of this paper is to continue the work started in [23] and describe some extensions to cases where the ambient manifold does not have constant sectional curvature. In particular, we shall construct new examples of r-harmonic submanifolds into Euclidean ellipsoids.…”

“…According to Proposition 3.3 ε r (α * ) is, up to a constant, the r-energy of ϕ α * . Now, a direct computation as in Proposition 3.1 of [23] (see this reference for details) shows that, in the spirit of [26],…”

“…Remark 4.12. The case r = 2 of Corollary 4.10 was proved in [21] by different methods, while the spherical case a = b, r ≥ 3 was first obtained in [16,23].…”

“…Therefore, for all r, m ≥ 2 and b > 0, we have the existence of two proper r-harmonic parallel hyperspheres which lie in a symmetric position with respect to the equator plane x m+1 = 0. The case r = 2 of Theorem 1.1 was proved in [21] by different methods, while the spherical case a = b, r ≥ 3 was first obtained in [16,23].…”