Constructions of harmonic maps between Hadamard manifolds

68 CHAPTER 3. DAMEK-RICCI SPACES
Since u 0 0 ((0,n)) and w0 0 (w(0,n)) are uniquely determined by f and positive, we have
∣ ȳ(u)
∣∣∣
∣log ȳ(w) ∣ = log u0 0(0,n)y + o(y)
w0(0,n)y 0 + o(y) ∣ = o(1).
Therefore d(u, w) = 0 at the ideal boundary. Hence the maximal principle ([33]) implies
Theorem 3.2.1.
3.2.3 Existence theorem
To show the existence of a solution of the Dirichlet problem at infinity, we use the heat flow
method due to Li and Tam [25]. As we see in the previous section, Damek-Ricci spaces are
homogeneous spaces of nonpositive sectional curvature and have positive bottom spectrum
of the Laplace-Beltrami operator. Therefore, we can apply the general existence theory [25,
Theorem 5.2] if there exists a suitable initial map. Indeed, h in Proposition 3.2.7 can be
taken as our initial map.
The decay order of the tension field of the initial map can be estimated by the following
Lemma 3.2.6. Let h ∈ C 2,ε (∂S,∂S ′ )(0