Constructions of harmonic maps between Hadamard manifolds

On the other hand, the (m + 1)-dimensional real hyperbolic space RH m+1 is represented
as the warped product manifold
([0, ∞) × S m ,dt 2 + (sinh 2 t)g S m).
Thus, for an eigenmap ϕ : S m → S n , we may consider an equivariant map
u : ([0, ∞) × S m ,dt 2 + (sinh 2 t)g S m) ∋ (t, θ)
↦→ (r(t),ϕ(θ)) ∈ ([0, ∞) × S n ,dr 2 + (sinh 2 r)g S n).
Then the harmonicity of u is also reduced to the following ordinary differential equation in
r = r(t) defined on [0, ∞):
¨r(t)+m cosh t sinh r(t) cosh r(t)
ṙ(t) − µ2
sinh t sinh 2 t
where e(ϕ) =µ 2 /2 denotes the energy density function of ϕ.
In Chapter 1, we shall investigate the following ordinary differential equation defined on
[0, ∞), which stems from the study of such harmonic maps:
{
f
¨r(t)+ p ˙ 1 (t)
f 1 (t) + q f ˙ } {
2 (t)
ṙ(t) − µ 2 h 1(r(t))h ′ 1(r(t))
+ ν 2 h }
2(r(t))h ′ 2(r(t))
=0,
f 2 (t)
f 1 (t) 2
f 2 (t) 2
where p, q, µ and ν are some constants.
We first prove the short time existence of solutions following the method due to Baird
([3]). However, it seems to the present author that his proof needs some refinements, in
particular, on the life span of solutions. In fact, under appropriate conditions for the given
functions f i (t) and h i (r), we can elaborate his method and obtain a complete proof of the
short time existence. Moreover, if we pose a condition on the growth rate of the function
f i (t) ast goes to infinity, then such a short time solution can be extended to a global
solution, which ensures the existence of an equivariant harmonic map as well as a harmonic
map defined by the join. Applying these results, we can prove that there exists a family
of harmonic maps from the real hyperbolic space RH m onto itself for m ≥ 3, which is
parameterized by Z.
=0,
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