Constructions of harmonic maps between Hadamard manifolds

3.2. HARMONIC MAPS BETWEEN DAMEK-RICCI SPACES 67
Proof. The assumption on f implies that the fourth term in the equation (2) of Lemma
3.2.3 never vanishes. Thus u 0 0 > 0, which implies
u α 0 =0, u α 00 = 0 and
∑n ′
α,β=1
Γ γ−n′
αβ
u α j u β j =0
from the equation (4) of Lemma 3.2.3 and (3.2.4). Finally, (3.2.3) implies u β k0 =0.
When S and S ′ are real hyperbolic planes, this proposition means that a proper harmonic
map must be conformal at the ideal boundary.
By Proposition 3.2.4 we get the following
Corollary 3.2.5. We have
⎧
⎪ ⎨
⎪ ⎩
u α 0 = O(y) (1 ≤ α ≤ n′ ),
u α 0 = o(y) (n ′ +1≤ α ≤ n ′ + m ′ ).
We shall now complete the proof of Theorem 3.2.1.
If we write
u(y, n) =(ȳ(u), ¯n(u)), w(y, n) =(ȳ(w), ¯n(w)), f(n) =¯n(f),
then it holds that
(3.2.5)
d(u, w) ≤ d((ȳ(u), ¯n(u)), (ȳ(u), ¯n(f)))
+d((ȳ(u), ¯n(f)), (ȳ(w), ¯n(f))) + d((ȳ(w), ¯n(f)), (ȳ(w), ¯n(w))).
From the explicit expression for the metrics and ¯n(f) =¯n(u(0,n)), we see that the first term
on the right hand side of (3.2.5) is
∫ y
∣ ∫ ∂¯n ∣∣∣ y
∣
0 ∂t (u(t, n)) ∑n dt ≤
[t ′
n
∑
′ +m ′
−1 |u α 0 | + t −2
0
α=1
α=n ′ +1
]
|u α 0 | dt = o(1).
The last estimate in the above follows from Corollary 3.2.5. Similarly, the third term is O(y).
On the other hand, the second term on the right hand side of (3.2.5) is
∣ ȳ(u)
∣∣∣
∣log ȳ(w) ∣ = log u0 (y, n)
w 0 (y, n) ∣ .