Non-existence of quadratic harmonic maps of $S^{4}$ into $S^{5}$ or $S^{6}$

Abstract

In this paper, we settle the last two open cases of non-existence of full quadratic harmonic maps from S 4 to S 5 or S 6 . Assume that there exist full quadratic harmonic maps from S 4 to S n for some integer n. As a consequence of our theorem we obtain that the sufficient and necessary condition of the existence of such maps is that n satisfy 4 ≤ n ≤ 13 and n = 5, 6.

Biharmonic homogeneous polynomial maps between spheres

“…Inspired by these examples whose components, as functions in R n+1 , are homogeneous polynomials of degree 2, in this paper we study the biharmonicity of the maps φ : S m → S n whose components are non-harmonic homogeneous polynomials of degree k. We note that the homogeneous polynomial maps between Euclidean spheres are very important in the harmonic map theory because, when they are harmonic as functions between Euclidean spaces, they provide all harmonic maps with constant energy density between the corresponding Euclidean spheres of codimension 1 (see, for example, [1] and [6]). Especially when k = 2, many interesting results have been obtained in [11], [15], [21], [24], [25], [27] etc.…”