Constructions of harmonic maps between Hadamard manifolds

3.2. HARMONIC MAPS BETWEEN DAMEK-RICCI SPACES 65
3.2.2 Uniqueness theorem
Let S and S ′ be Damek-Ricci spaces and assume that they are represented as in Section
3.2.1. We denote by ¯S (resp. ¯S ′ ) the Eberlein-O’Neill compactification of S (resp. S ′ ). Then
{y =0}×N (resp. {ȳ =0}×N ′ ) represents the ideal boundary, ∂S (resp. ∂S ′ ), except a
point in ∂S (resp. ∂S ′ ) (cf. [6]).
Let u ∈ C 2 (S, S ′ ) be a proper harmonic map. First, we investigate a necessary condition
for the existence of a C 2 -extension u : ¯S → ¯S ′ of u, and then prove the following uniqueness
theorem.
Theorem 3.2.1 (Uniqueness theorem).
Let u and w be proper harmonic maps between
Damek-Ricci spaces S and S ′ . Suppose u, w ∈ C 2 ( ¯S, ¯S ′ ) and f := u |∂S = w |∂S . If
then u = w on ¯S.
n+m
∑
n
∑
′ +m ′
j=n+1 γ=n ′ +1
(f γ j )2 > 0 on ∂S,
In order to prove this theorem, the following lemma plays an important role.
Lemma 3.2.2 ([12], [26]). Suppose that ω ∈ C 1 Λ 1 S ∩ C 0 Λ 1 ¯S is a 1-form defined on a
n+m
∑
neighborhood of p ∈ ∂S. Let ω = ω i e ∗ i , where {e ∗ i } is the dual coframe of {e i }. Then
there exists a sequence of points {q k }⊂S such that q k → p and
(
)
∑
g jj ω jj (q k )=0.
lim
k→∞
i=0
y −1 n+m
j=0
Using this lemma together with τ(u) =0, we obtain the following necessary condition.
Lemma 3.2.3. Let u ∈ C 2 ( ¯S, ¯S ′ ) be a proper harmonic map. Then at the ideal boundary
we have the following:
(1)
n∑
n
∑
′ +m ′
j=0 β=n ′ +1
(u β j )2 =0,