Constructions of harmonic maps between Hadamard manifolds

2.1. EIGENMAPS 47
Then the restriction ϕ |S m+n+1 of ϕ to S m+n+1 gives rise to an eigenmap of degree two from
S m+n+1 into S p+q+r+1 . Moreover if f,g and F are full, then so is ϕ.
Most significant case is that of m = p =1. In this case, f : S 1 → S 1 is given by the Hopf
map, that is, f(x 1 ,x 2 )=(|x 1 | 2 −|x 2 | 2 , 2x 1 x 2 ), and Proposition 2.1.2 ensures the existence
of a full orthogonal multiplication F : R 2 × R n+1 → R r for even r. Thus we obtain the
following
Proposition 2.1.3. Let g : S n → S q be a full eigenmap of degree two. Then we have a new
full eigenmap ϕ : S n+2 → S q+r+2 of degree two, where r is even and n +1≤ r ≤ 2n +2.
By making use of this proposition, we can prove the following
Theorem 2.1.4. (1) (i)
7 ≤ n ≤ 19.
A full eigenmap ϕ : S 5 → S n of degree two exists for n =4or
(ii) A full eigenmap ϕ : S 6 → S n of degree two exists for 11 ≤ n ≤ 26.
(2) Let k ≥ 3.
(i) A full eigenmap ϕ : S 2k+1 → S n of degree two exists for k 2 +3k −10 ≤ n ≤ 2k 2 +5k +1.
(ii) A full eigenmap ϕ : S 2k+2 → S n of degree two exists for k 2 +5k − 7 ≤ n ≤ 2k 2 +7k +4.
Note. The space of harmonic polynomials of degree two on R m+1 has the dimension
m(m +3)/2. Thus, in Theorem 2.1.4, the upper bound of the dimension of target manifold
is best possible to assure the existence of a full eigenmap of degree two.
Note. Gauchman and Toth ([18]) proved that a full eigenmap of degree two ϕ : S 4 → S n
exists for n =4, 7or9≤ n ≤ 13.
In order to prove Theorem 2.1.4, we use the following result.
Lemma 2.1.5 (Gauchman and Toth [18]). From a full eigenmap g : S m → S n of degree
two, one can construct a full eigenmap ˜g : S m+1 → S m+n+2 of the same degree.