Non-existence of quadratic harmonic maps of s into s or s

Let S denote the unit sphere in the Euclidean space R. A quadratic harmonic map f : S → S is the restriction of a map F : R → R whose components are harmonic polynomials of homogeneous degree 2. Such a map f is called full if the image of f spans R. A spherical harmonic on S of order p is an eigenfunction of the spherical Laplacian with eigenvalue λp = p(p +m − 1). It is well known that a spherical harmonic of order p is the restriction to S of a harmonic polynomial of homogeneous degree p in R. So a quadratic harmonic map f : S → S is also called a λ2-eigenmap, and generally one can investigate a λp-eigenmap. A λp-eigenmap is a harmonic map with constant energy density λp/2. Up to isometries on the domain and the range, how many equivalence classes are there for the given m,n and p? The range dimension n obviously depends on the given m and p. What are the possible values of n for the given m and p? These problems are far from being solved even for p = 2 [2, 3, 5, 10, 11]. Besides the classical examples such as the Hopf constructions and the Veronese maps, there are several available effective ways of constructing new eigenmaps out of the old ones [4, 7, 8, 10, 11]. Calabi proved that any full λ2-eigenmap f : S 2 → S is rigid; that is, any such map is equivalent to the Veronese map S → S. In 1987, G. Toth [6] gave a complete classification of full λ2-eigenmaps from S 3 to S. In 2003, after giving a rigidity result for a λ2-eigenmap from S 4 to itself, Huixia He, Hui Ma, and Feng Xu [7] completely solved the existence problem of λ2-eigenmaps from S 2n−k to S for k = 1, · · · , 5. Gauchman, Toth, Lam, Tang, Ueno and Yiu have done much work on quadratic harmonic maps between spheres; see [5, 6, 7, 8, 9, 10, 11, 12] for more details. The existence problem of λp-eigenmaps between the Euclidean spheres constantly generates great interest from researchers since many challenging problems are still open. In 1994, Gauchman and Toth [4] showed that full λ2-eigenmaps f : S 4 → S