Constructions of harmonic maps between Hadamard manifolds

1.1. LOCAL EXISTENCE OF SOLUTIONS TO (1.1.1) WITH (1.1.2) 19
On the other hand, since r 1 and r 2 are solutions to the equation (1.1.4) and the function
s ↦→ Q(·,s) is monotone, it holds that
d 2 w
dτ (τ 2)=−P (τ 2 2 ) dw
dτ (τ 2)+Q(τ 2 ,r 1 (τ 2 )) − Q(τ 2 ,r 2 (τ 2 ))
= Q(τ 2 ,r 1 (τ 2 )) − Q(τ 2 ,r 2 (τ 2 ))
≤ 0,
which yields a contradiction. Hence w(τ) < 0, that is, r 1 (τ) ¯τ) toτ 0 , we obtain
˜P (τ 0 )(r 1 (τ 0 ) − r 2 (τ 0 )) d
dτ (r 1(τ 0 ) − r 2 (τ 0 ))
≥ ˜P (τ)(r 1 (τ) − r 2 (τ)) d
dτ (r 1(τ) − r 2 (τ)) +
> ˜P (τ)(r 1 (τ) − r 2 (τ)) d
dτ (r 1(τ) − r 2 (τ)),
from which it follows that
∫ τ0
˜P (τ)(r 1 (τ) − r 2 (τ)) d
dτ (r 1(τ) − r 2 (τ))
< ˜P (τ 0 )(r 1 (τ 0 ) − r 2 (τ 0 )) d
dτ (r 1(τ 0 ) − r 2 (τ 0 ))
≤ 0.
τ
}
{ } 2 d
˜P (τ)
dτ (r 1(τ) − r 2 (τ)) dτ
Since r 1 (τ) dr 2
dτ (τ)