Constructions of harmonic maps between Hadamard manifolds

1.1. LOCAL EXISTENCE OF SOLUTIONS TO (1.1.1) WITH (1.1.2) 23
Since the solutions depend continuously on initial values, it follows that for the above δ>0,
there exists an ¯ε >0 such that if | ¯s 0 − s 0 | < ¯ε, then
∣ sup |ρ(τ) − r(τ)| + sup
dρ dr ∣∣∣
∣ (τ) −
τ∈I
τ∈I dτ dτ (τ)
dτ dτ (τ) − δ>η>0 on [τ 1 − ε, τ 0 ],
ρ(τ 1 − ε) 0. Suppose that inf A(r 0 )=0. For s 0 ∈A(r 0 ), let r = r(τ)
be a solution to the equation (1.1.4) with the initial condition (1.1.5). Then
d 2 r
dτ (τ 0)=−P (τ 2 0 )s 0 + Q(τ 0 ,r 0 ).
Since Q(τ 0 ,r 0 ) > 0, for sufficiently small δ>0, one can take s 0 > 0 so that
d 2 r
dτ (τ 0) > 2δ.
2
Hence there exists an ε = ε(s 0 ) > 0 such that
d 2 r dr
(τ) >δ,
dτ
2
dτ (τ) > 0 and r(τ) > 0 on (τ 0 − 2ε, τ 0 +2ε).
For ¯s 0 ∈A(r 0 ) such that ¯s 0 r(τ) and
Thus we obtain
dρ dr
(τ) <
dτ dτ (τ).
d 2 ρ
(τ) =−P (τ)dρ
dτ
2
dτ (τ)+Q(τ,ρ(τ))
≥−P (τ) dr
dτ (τ)+Q(τ,r(τ)) = d2 r
(τ) >δ
dτ
2