Constructions of harmonic maps between Hadamard manifolds

56 CHAPTER 2. EQUIVARIANT HARMONIC MAPS
there exists a global, strictly monotone increasing, and unbounded solution r = r(t) to the
equation (2.2.4). Therefore, from Theorem 2.1.4, we obtain the following
Theorem 2.3.4. (1) (i) A full equivariant harmonic map u : RH 6 → RH n exists for n =5
or 8 ≤ n ≤ 20.
(ii) A full equivariant harmonic map u : RH 7 → RH n exists for 12 ≤ n ≤ 27.
(2) Let k ≥ 4.
(i) A full equivariant harmonic map u : RH 2k → RH n exists for k 2 +k−12 ≤ n ≤ 2k 2 +k−2.
(ii) A full equivariant harmonic map u : RH 2k+1 → RH n exists for k 2 +3k − 11 ≤ n ≤
2k 2 +3k − 1.
Note. From the eigenmap S 1 ∋ z ↦→ z k ∈ S 1 (k ∈ Z), we obtain an equivariant harmonic
map u : RH 2 → RH 2 . However, u is holomorphic or anti-holomorphic. Indeed, using the
Poincaré disc model D 2 , it is given by D 2 ∋ z ↦→ z k ∈ D 2 .
Case 2. Let f(t) = sinh t and h(r) =r. Then M and N are isometric to the real hyperbolic
space and the real Euclidean space, respectively. Since these f and h satisfy the conditions
in Theorem 2.3.3, there exists a global solution r = r(t) to the equation (2.2.4). Moreover,
it holds that ∫ ∞
dr
h(r) = ∞.
Thus, from Theorem 2.1.4, we obtain the following
Theorem 2.3.5. There exists a family of equivariant harmonic maps from RH m+1
R n(m)+1 , each of which has a bounded image.
into
Recently, Nagasawa and Tachikawa studied the case where
∫
√
∞
µ 2
f 1 (t) +
ν2
dt = ∞,
2 f 2 (t)
2
and proved, for example, the following