Constructions of harmonic maps between Hadamard manifolds

CHAPTER 2
Equivariant harmonic maps
In this chapter, we shall construct equivariant harmonic maps between noncompact, complete
Riemannian manifolds by making use of the results established in the previous chapter.
We begin with fixing our notation and a brief review of relevant theorems for eigenmaps.
2.1 Eigenmaps
Let (S m ,g S m) be the m-dimensional standard unit sphere. A map ϕ :(S m ,g S m) → (S n ,g S n)
is called an eigenmap if it is a harmonic map with constant energy density. It is a well-known
result that from Takahashi’s theorem, we have the following
Lemma 2.1.1. Let ϕ :(S m ,g S m) → (S n ,g S n) be an eigenmap. Then all components of the
map Φ=i ◦ ϕ : S m → R n+1 are harmonic homogeneous polynomials on R m+1 of the same
polynomial degree, where i : S n → R n+1 is the inclusion map.
Thus, if we can find a family of harmonic homogeneous polynomials of the same degree,
say {Φ i } n+1
i=1 , satisfying ∑n+1
Φ 2 i (x) = 1 for x ∈ S m ,
i=1
then we obtain an eigenmap ϕ : S m → S n . An eigenmap ϕ : S m → S n is said to be of degree
k if all components of ϕ are harmonic homogeneous polynomial of degree k.
There is no general theory of constructing an eigenmap ϕ : S m → S n for a given degree.
It should be remarked that almost all known results have been for the case of degree two.
We shall give an algorithm of constructing a new eigenmap of degree two from the old ones
of the same degree.
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