Constructions of harmonic maps between Hadamard manifolds

2.3. CONSTRUCTIONS OF EQUIVARIANT HARMONIC MAPS 55
then M and N are smooth Riemannian manifolds diffeomorphic to Euclidean space.
Let ϕ : S m → S n be an eigenmap with 2e(ϕ) =µ 2 . Then a product map u(t, x) =
(r(t),ϕ(x)) is a harmonic map if and only if r = r(t) is a solution to the following ordinary
differential equation:
f(t)
(2.2.4) ¨r(t)+m ˙
f(t)ṙ(t) − h(r(t))h′ (r(t))
µ2 =0.
f(t) 2
In order for u to be continuous, we require that r = r(t) satisfies
lim r(t) =0.
t→0
Then Theorem A and Theorem B in Chapter 1 imply the following result.
Theorem 2.3.3. Let f = f(t) and h = h(r) be smooth functions on [0, ∞) and R, respectively.
Assume that f and h satisfy the following conditions:
⎧
f(0) = 0, f(0) ˙ = 1, f(t) > 0 and f(t) ˙ ≥ 0 for t>0,
⎪⎨
f(t)
0 ≤ m ˙
∫ ∞
f(t) − 1 for t ≥ 0, dt
f(t) < ∞,
⎪⎩ h(0) = 0, h ′ (0) = 1, h(r) > 0 for r>0 and (h 2 ) ′′ (r) ≥ 0 for r ∈ R.
Then, if there exists an eigenmap ϕ : S m
harmonic map
→ S n , then we can construct an equivariant
u : ([0, ∞) × S m ,dt 2 + f(t) 2 g S m) ∋ (t, x) ↦→ (r(t),ϕ(x)) ∈ ([0, ∞) × S n ,dr 2 + h(r) 2 g S n).
maps.
As applications of this theorem, we now illustrate few examples of equivariant harmonic
Case 1.
Let f(t) = sinh t and h(r) = sinh r. Then M and N are isometric to the real
hyperbolic spaces of dimension m + 1 and n +1, respectively. Since these f and h satisfy
the conditions in Theorem 2.3.3 and
∫ ∞
dr
h(r) < ∞,