Constructions of harmonic maps between Hadamard manifolds

3.2. HARMONIC MAPS BETWEEN DAMEK-RICCI SPACES 69
Then there exists an h ∈ C 2,ε ( ¯S, ¯S ′ ) such that h = f at the ideal boundary and ‖τ(h)‖ =
O(y ε ), where ‖τ(h)‖ is the norm of the tension field in the Riemannian metric.
Proof.
Let φ>0 be a unique solution of
(n +2m)φ 4 −
n∑ ∑n ′
(f β j )2 φ 2 − 2
j=1 β=1
n+m
∑
n
∑
′ +m ′
j=n+1 β=n ′ +1
(f β j )2 =0.
Since f ∈ C 3,ε , we have φ ∈ C 2,ε (∂S). Set h(y, n) :=(yφ(n),f(n)). Then h ∈ C 2,ε (∂S,∂S ′ ).
Moreover, it is easy to verify that h satisfies
⎧
h 0 0 = φ,
⎪⎨ h α 0 =0 (1≤ α ≤ n ′ + m ′ ),
h β 00 =0 (n ′ +1≤ β ≤ n ′ + m ′ ),
⎪⎩ h γ j0 =0 (1≤ j ≤ n, n′ +1≤ γ ≤ n ′ + m ′ ),
and h = f at the ideal boundary. Lemma 3.2.6 and the explicit expression for the metric
then imply ‖τ(h)‖ = O(y ε ).
Next we construct a comparison function.
Lemma 3.2.8. For sufficiently large r 0 and some constant s, define
⎧
⎪⎨ e −sr 0
(r ≤ r 0 ),
ψ(r) :=
⎪⎩ e −sr (r ≥ r 0 ).
If 0