Constructions of harmonic maps between Hadamard manifolds

2.3. CONSTRUCTIONS OF EQUIVARIANT HARMONIC MAPS 57
Theorem 2.3.6 ([28]). Let M =(R + × S m ,dt 2 + f(t) 2 dθ 2 ) and N =(R + × S n ,dr 2 +
h(r) 2 dϕ 2 ). Assume that, in addition to the conditions (F-1) through (F-3) and (H-1) through
(H-3), f and h satisfy
∫ ∞
∫
dt
∞
f(t) dt = ∞ and dr
dr < ∞.
h(r)
Then there exists no equivariant harmonic map from M to N except constant maps.
As a corollary, there is no equivariant harmonic map from R m to RH n , which was
independently proved by Tachikawa([35]). See also [2] and [36].
In consequence, it is observed that the existence or non-existence of equivariant harmonic
maps is closely related to the growth orders of warping function, f and h, when t and r tend
to ∞, respectively. We shall investigate a related non-existence result in the last chapter.