Constructions of harmonic maps between Hadamard manifolds

1.1. LOCAL EXISTENCE OF SOLUTIONS TO (1.1.1) WITH (1.1.2) 25
Thus
d 2 r
dτ (τ 2)=−P (τ 2 2 ) dr
dτ (τ 2)+Q(τ 2 ,r(τ 2 ))
= Q(τ 2 ,r(τ 2 )) > 0.
Therefore, r(τ 2 ) is locally a minimum value. On the other hand, we can show that
dr
dτ (τ) < 0
on (τ 1 ,τ 2 ). Indeed, suppose that there exists τ 3 ∈ (τ 1 ,τ 2 ) such that
dr
dτ (τ 3) = 0
and
dr
dτ (τ) < 0 on (τ 3,τ 2 ).
Then we have
d 2 r
dτ (τ 3) ≤ 0.
2
However, from the equation (1.1.4), it holds that
d 2 r
dτ 2 (τ 3)=Q(τ 3 ,r(τ 3 )) > 0,
which is a contradiction. Hence, r(τ 2 ) is positive and is the minimal value of r on [τ 1 ,τ 0 ],
which contradicts the assumption r(τ 1 )=0. Thus r>0on(τ ∗ ,τ 0 ].
Second, we show that
dr
dτ (τ) > 0 on (τ ∗,τ 0 ].
Assume that there exists τ 1 ∈ (τ ∗ ,τ 0 ) such that
dr
dτ (τ 1)=0.
Then, by the same argument used for proving r>0, we have
dr
dτ (τ) < 0 on (τ ∗,τ 1 ) and
dr
dτ (τ) > 0 on (τ 1,τ 0 ],
which implies that r(τ 1 ) is the minimal value of r on (τ ∗ ,τ 0 ]. Take τ 2 ∈ (τ ∗ ,τ 1 ) and τ 3 ∈ (τ 1 ,τ 0 )
so that r(τ 2 )=r(τ 3 ) and I =[τ 2 ,τ 3 ]. It then follows from the continuous dependence of