Constructions of harmonic maps between Hadamard manifolds

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vanishing sectional curvatures for certain 2-planes (see Theorem 3.1.2). We shall prove that
Donnelly’s existence and uniqueness results can be extended to the case of Damek-Ricci
spaces. Thus the geometric conditions such as symmetry and strict negativity of sectional
curvatures are proved to be not essential, and it is worthwhile studying the problem for
Hadamard manifolds in further generality.
In the last section, we shall prove a non-existence result for proper harmonic maps from
complex hyperbolic spaces to real hyperbolic spaces. To be precise, let B m and D n be
the ball models of the m-dimensional complex hyperbolic space and the n-dimensional real
hyperbolic space, respectively. Then we prove that there exists no proper harmonic map
u ∈ C 2 (B m , D n ) which has C 1 -regularity up to the ideal boundary, where m, n ≥ 2. Recently,
Li and Ni [24] obtained the same result in the case of n =2. Furthermore, we show that
there exists a counter example to this result if we relax the regularity condition up to the
ideal boundary.
It should be remarked that Donnelly [12] proved that, for a suitable boundary map, there
exists a solution to the Dirichlet problem for harmonic maps from complex hyperbolic spaces
to real hyperbolic spaces, which has sufficiently high regularity up to the ideal boundary.
Hence our theorem appears to contradict his result. However, this is caused by the different
choice of the models of complex hyperbolic spaces. For the upper half space model, Donnelly
used the one defined by
M 1 =
(R + × R 2m−1 ,h 1 = dy2
y + 1 2 y g 2 1 + 1 )
y g 4 2 ,
where y ∈ R + and g 1 + g 2 is a Riemannian metric on R 2m−1 . On the other hand, our model
is given by
M 2 =
(R + × R 2m−1 ,h 2 = dη2
4η + 1 2 η g 1 + 1 )
η g 2 2 ,
where η ∈ R + . If one define a map f : M 2 → M 1 by
f(η, x i )=( √ η, x i ) for (η, x i ) ∈ R + × R 2m−1 ,