Constructions of harmonic maps between Hadamard manifolds

1.1. LOCAL EXISTENCE OF SOLUTIONS TO (1.1.1) WITH (1.1.2) 17
Corollary 1.1.3. Let r = r(t) and ρ = ρ(t) be solutions to the equation (1.1.3) with the
boundary condition (1.1.2). Ifr(t 0 )=ρ(t 0 ) holds for some t 0 ∈ (0,T), then we have
r(t) =ρ(t) on [0,T),
where [0,T) is the common life span of r(t) and ρ(t). In particular, if r(t 0 )=0, then r ≡ 0.
To prove the existence part of Theorem 1.1.1, we employ a method due to Baird ([3,
Chapter 6]).
We replace the variable t in the equation (1.1.3) with τ = log t to remove the singularity
at t =0. Then the equation (1.1.3) becomes
(1.1.4)
where
d 2 r dr
(τ)+P (τ) (τ) − Q(τ,r(τ))=0,
dτ
2
dτ
P (τ) =e τ F (e τ ) − 1,
Note that P and Q satisfy the following conditions:
⎧
P ∈ C ∞ (R), P(τ) ≥ 0 on R,
Q(τ,s)=e 2τ G(e τ ,s).
Q ∈ C ∞ (R × R), Q(τ,s) > 0 for (τ,s) ∈ R × (0, ∞)
⎪⎨
lim Q(τ,s) > 0 for any s>0, lim Q(τ,s)=0,
τ→−∞ τ→−∞,s→0
Q(τ,0) = 0
for any τ ∈ R, and,
We also set τ 0 = log t 0 .
⎪⎩
s 1 ≤ s 2 =⇒ Q(τ,s 1 ) ≤ Q(τ,s 2 ) for any τ ∈ R.
Under these conditions, we prove the following
Theorem 1.1.4. For any τ 0 ∈ R and r 0 > 0, there exists a solution r = r(τ) to the equation
(1.1.4) on (−∞,τ 0 ] satisfying the following conditions:
(1) r(τ) > 0,
(2) lim r(τ) =0, lim
τ→−∞
dr
dτ (τ) > 0 on (−∞,τ 0], and
τ→−∞
dr
(τ) =0and
dτ lim
τ→−∞
d 2 r
(τ) =0.
dτ
2