Constructions of harmonic maps between Hadamard manifolds
Constructions of harmonic maps between Hadamard manifolds
Constructions of harmonic maps between Hadamard manifolds
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
3.2. HARMONIC MAPS BETWEEN DAMEK-RICCI SPACES 71<br />
By virtue <strong>of</strong> [25, Lemma 5.1], we have<br />
‖v t (x, t)‖ ≤c 1 e −c 2t<br />
for some constants c 1 ,c 2 > 0. Then for any T>0<br />
d S ′(u(x),h(x)) ≤<br />
∫ ∞<br />
‖v t (x, t)‖dt ≤<br />
≤ ce −εr(x) T + c 1 e −c2T .<br />
Choosing T = εr, we get<br />
∫ T<br />
‖v t (x, t)‖dt +<br />
∫ ∞<br />
0<br />
0<br />
T<br />
‖v t (x, t)‖dt<br />
ce −εr(x) T and c 1 e −c 2T → 0<br />
as r →∞.<br />
Therefore, d S ′(u(x),h(x)) → 0asx → ∂S and u = h = f at the ideal boundary.<br />
Note. Our result remains true, with necessary modifications, when the target manifold is<br />
replaced by the real hyperbolic space.
Change language