Constructions of harmonic maps between Hadamard manifolds

CHAPTER 4
Non-existence of proper harmonic maps from complex
hyperbolic spaces into real hyperbolic spaces
In this chapter, we investigate proper harmonic maps from the m-dimensional complex
hyperbolic space B m to the n-dimensional real hyperbolic space D n . Our goal is to prove
the following non-existence result.
Theorem 4.1.0. Let m, n ≥ 2. Then there is no proper harmonic map u ∈ C 2 (B m , D n )
which has C 1 -regularity up to the ideal boundary.
The C 1 -regularity assumption up to the ideal boundary, supposed for proper harmonic
maps in Theorem 4.1.0, plays a crucial role in our proof. Indeed, in the last section, we show
that there exists a counter example to this theorem if we relax its regularity condition up to
the ideal boundary.
4.1 Proof of Theorem
Let (B m ,g) and (D n ,g ′ ) be the ball models of the m-dimensional complex hyperbolic space
and the n-dimensional real hyperbolic space, respectively. Namely,
(
m
)
B m = {z ∈ C m 1 ∑ {
||z| < 1}, g=
(1 −|z| 2 )δ
(1 −|z| 2 ) 2 ij +¯z i z j} dz i d¯z j ,
(
D n = {x ∈ R n ||x| < 1}, g ′ =
i,j=1
)
4
n∑
(dx i ) 2 ,
(1 −|x| 2 ) 2
where z =(z 1 ,... ,z m ) ∈ B m and x = (x 1 ,... ,x n ) ∈ D n are the canonical Euclidean
coordinates. We endow the Eberlein-O’Neill compactification B m (resp. D n ) with the nat-
73
i=1