Constructions of harmonic maps between Hadamard manifolds

74 CHAPTER 4. NON-EXISTENCE OF PROPER HARMONIC MAPS
ural smooth structure inherited from C m (resp. R n ), and regard it as a submanifold with
boundary in C m (resp. R n ). With these understood, we have the following
Lemma 4.1.1. (1) Let g i¯j , Γ i
j k and ∆ B m
be the components, the Christoffel symbols and
the Laplace-Beltrami operator of Bergman metric g, respectively. Then we have
⎧
g i¯j =(1−|z| 2 ) −2 {(1 −|z| 2 )δ ij +¯z i z j },
⎪⎨
Γ i
j k =(1−|z|2 ) −1 (¯z j δ ik +¯z k δ ij ),
⎪⎩
∆ B m =(1−|z| 2 )
m∑
(δ ij − z i¯z j ∂ 2
)
∂z i ∂¯z . j
i,j=1
(2) Let g αβ ′ , α ˜Γ β γ and ∆ D n be the components, the Christoffel symbols and the Laplace-
Beltrami operator of Poincaré metric g ′ , respectively. Then we have
⎧
g αβ ′ = 1
(1 −|x| 2 ) δ αβ,
2
⎪⎨
˜Γ α
β γ = 2
(1 −|x| 2 ) (xβ δ αγ + x γ δ αβ − x α δ βγ ),
⎪⎩ ∆ D n =(1−|x| 2 ) 2
n∑
α=1
∂ 2
∂x α ∂x α +2(n − 2)(1 −|x|2 )
n∑
α=1
x α
∂
∂x . α
Following [24], we define a family of global vector fields N and X j , 1 ≤ j ≤ m, on C m
in the following manner:
⎧
N = ⎪⎨
⎪⎩ X j =
m∑
i=1
m∑
i=1
z i
∂
∂z , i
(δ ij − z i¯z j ) ∂
∂z i =
∂
∂z −〈 ∂
j ∂z ,N〉N,
j
where 〈·, ·〉 denotes the standard Hermitian inner product on C m . Then we get
Lemma 4.1.2 ([24]). (1) N + ¯N is the outer normal vector field on ∂B m .
(2) {X j + ¯X j , √ −1(X j − ¯X j ), √ −1(N − ¯N)} m j=1 is a basis of the tangent space T p∂B m at
each p ∈ ∂B m .