Constructions of harmonic maps between Hadamard manifolds

1.2. GLOBAL PROPERTIES OF SOLUTIONS 31
(1) Assume that r(t 0 ) > 0 and ṙ(t 0 ) > 0 for some t 0 ∈ (0,T). Then it holds that
ṙ(t) > 0
on (0,T). Hence the solution constructed in the previous section is strictly monotone increasing
as long as it exists.
(2) If T 0on(0,t 0 ]. We shall show that ṙ(t) > 0on(t 0 ,T). If
this is not the case, we have a point t 1 ∈ (t 0 ,T) such that
r(t 1 ) > 0, ṙ(t 1 ) = 0 and ¨r(t 1 ) ≤ 0.
On the other hand, the equation (1.1.3) asserts that
¨r(t 1 )=G(t 1 ,r(t 1 )) > 0,
which is a contradiction. Thus ṙ(t) > 0on[t 0 ,T).
(2) Let ˜F (t) = exp ∫ t F (s)ds. Then, from the equation (1.1.3), we have
d
{ }
˜F (t)ṙ(t) =
dt
˜F (t)G(t, r(t)).
Integrating both sides from t 0 to t(> t 0 ), we obtain
˜F (t) |ṙ(t)| ≤ ˜F (t 0 ) |ṙ(t 0 )| +
∫ t
˜F (t)|G(t, r(t))|dt.
Since G ∈ C ∞ ([0, ∞) × R), if r(t) is bounded when t tends to T, then so is ṙ(t). Thus the
assertion holds.
From now on, we shall consider the original ordinary differential equation (1.1.1).
t 0