Constructions of harmonic maps between Hadamard manifolds

76 CHAPTER 4. NON-EXISTENCE OF PROPER HARMONIC MAPS
From the formula ∂/∂¯z j = ¯X j + z j ¯N, we then have
(∗1) = (1 −|z| 2 )Lu α
+2a(u)(z) −1
=(1−|z| 2 )Lu α
n∑
β=1 j=1
+(1−|z| 2 2
)
a(u)(z)
+
2
a(u)(z)
m∑
j=1 β=1
m∑
{(X j u α )u β ( ¯X j u β + z j ¯Nu β )
+(X j u β )u β ( ¯X j u α + z j ¯Nu α )+(X j u β )u α ( ¯X j u β + z j ¯Nu β )}
n∑
{(Nu α )(u β ¯Nu β )+(¯Nu α )(u β Nu β ) − u α (Nu β )( ¯Nu β )}
β=1
n∑
{(X j u α )(u β ¯Xj u β )+(¯X j u α )(u β X j u β ) − u α (X j u β )( ¯X j u β )}.
In order to investigate the asymptotic behavior of proper harmonic maps, Li and Ni
proved the following
Lemma 4.1.5 ([24]). Assume f ∈ C 2 (B m ) ∩ C 1 (B m ). Then we have
1 −|z| 2
lim
(
z→∂B m ε(z)
∫B(z,ε(z))
¯fLf)(ζ)dζ =0,
2m
where ε(z) =(1−|z|)/2 and B(z,ε(z)) is the open ball in (C m ,g C m) with the radius ε(z)
centered at z, and dζ is the volume element of (C m ,g C m).
proof.
As a corollary of this lemma, we have the following result, which is in essential use in our
Corollary 4.1.6. Assume u ∈ C 2 (B m , C n ) ∩ C 1 (B m , C n ). Then for any point p ∈ ∂B m
there exists a sequence {z j } ∞ j=1 ⊂ B m satisfying the following properties:
(1) z j → p (j →∞).
(2) lim
j→∞
(1 −|z j | 2 )〈Lu, u〉(z j )=0.
We first prove the following result.