Constructions of harmonic maps between Hadamard manifolds

1.1. LOCAL EXISTENCE OF SOLUTIONS TO (1.1.1) WITH (1.1.2) 21
Since A(r 0 ) is an empty set, r(τ) > 0 for τ ∈ (τ 1 ,τ 0 ]. Choose ¯s 0 so that
{
¯s 0 > max s 0 ,C 1 (τ 0 − τ)+ r }
0
,
τ 0 − τ 1
where C 1 := max τ∈[τ1 ,τ 0 ] Q(τ,r 0 ). We shall show that ¯s 0 ∈A(r 0 ).
Let ρ = ρ(τ) be a solution to the equation (1.1.4) satisfying
dρ
ρ(τ 0 )=r(τ 0 ) and
dτ (τ 0)=¯s 0 > dr
dτ (τ 0),
and (τ 2 ,τ 0 ] the life span of ρ. Then from Lemma 1.1.6 it holds that
as long as both r and ρ exist.
ρ(τ)
dτ dτ (τ)
When τ 1 ≤ τ 2 , we have ρ(τ) →−∞as τ → τ 2 +0. Hence there exists τ 3 ∈ (τ 2 ,τ 0 ] such
that
dρ dr
ρ(τ 3 ) = 0 and (τ) >
dτ dτ (τ) > 0 on (τ 3,τ 0 ].
Therefore ¯s 0 ∈A(r 0 ), contradicting the fact A(r 0 )=∅.
On the other hand, when τ 1 >τ 2 ,ρsatisfies (dρ/dτ)(τ) > (dr/dτ)(τ) > 0on(τ 1 ,τ 0 ]. Let
Then C 2 ≥ 0, and it holds that
C 2 :=
min P (τ).
τ∈[τ 1 ,τ 0 ]
d 2 ρ
(τ) =−P (τ)dρ
dτ
2
dτ (τ)+Q(τ,ρ(τ))
≤−P (τ) dρ
dτ (τ)+Q(τ,ρ(τ 0))
dρ
≤−C 2
dτ (τ)+C 1
on [τ 1 ,τ 0 ]. Integrating this inequality from τ ∈ [τ 1 ,τ 0 ]toτ 0 , we have
− dρ (τ) ≤−dρ
dτ dτ (τ 0) − C 2 {ρ(τ 0 ) − ρ(τ)} + C 1 (τ 0 − τ)
≤−¯s 0 + C 1 (τ 0 − τ)
< − r 0
τ 0 − τ 1
.