Constructions of harmonic maps between Hadamard manifolds

1.2. GLOBAL PROPERTIES OF SOLUTIONS 37
Lemma 1.2.7. The function φ(r 0 ,t 0 ) is well-defined for r 0 > 0 and t 0 > 0, and
lim φ(r 0,t 0 )=0.
r 0 →∞
Proof. It follows from (H-3) that (h 2 i ) ′ (s) ≥ (h 2 i ) ′ (s − r 0 ) for s>r 0 . Integrating both
sides with respect to s from r 0 to s, we get
Therefore, it holds that for K>0
∫ ∞
r 0 +K
h i (s) 2 − h i (r 0 ) 2 ≥ h i (s − r 0 ) 2 .
[γ 1 (t 0 ) 2 {h 1 (s) 2 − h 1 (r 0 ) 2 } + γ 2 (t 0 ) 2 {h 2 (s) 2 − h 2 (r 0 ) 2 } + β(r 0 ,t 0 ) 2 ] −1/2 ds
≤ C(t 0 )
≤ C(t 0 )
∫ ∞
r 0 +K
∫ ∞
K
[h 1 (s − r 0 ) 2 + h 2 (s − r 0 ) 2 ] −1/2 ds
ds
h(s) .
Note that it follows from the condition (1.2.5) that for any ε>0
for sufficiently large K.
∫ ∞
K
ds
h(s) r 0
where ρ i ∈ (r 0 ,s). It follows from (H-3) that
Therefore it holds that
∫ r0 +K
r 0
h i (s) 2 − h i (r 0 ) 2 =2h i (ρ i )h ′ i (ρ i)(s − r 0 ),
h i (s) 2 − h i (r 0 ) 2 ≥ 2h i (r 0 )h ′ i(r 0 )(s − r 0 ) > 0.
[γ 1 (t 0 ) 2 {h 1 (s) 2 − h 1 (r 0 ) 2 } + γ 2 (t 0 ) 2 {h 2 (s) 2 − h 2 (r 0 ) 2 } + β(r 0 ,t 0 ) 2 ] −1/2 ds
≤ [2(γ 1 (t 0 ) 2 h 1 (r 0 )h ′ 1(r 0 )+γ 2 (t 0 ) 2 h 2 (r 0 )h ′ 2(r 0 ))] −1/2 ∫ r0 +K
≤ C(t 0 ) √ K min { [h 1 (r 0 )h ′ 1 (r 0)] −1/2 , [h 2 (r 0 )h ′ 2 (r 0)] −1/2} .
r 0
ds
√ s − r0