Constructions of harmonic maps between Hadamard manifolds

52 CHAPTER 2. EQUIVARIANT HARMONIC MAPS
Lemma 2.2.2.
⎧
⎪⎨
τ(u) 1
(
)
f˙
1 (t)
=¨r(t)+ m 1
f 1 (t) + m f˙
2 (t)
2 ṙ(t)
f 2 (t)
−2e(ϕ) h′ 1 (r(t))h 1(r(t))
f 1 (t) 2
⎪⎩ τ(u) α = 1
f 2 (t) 2 τ(ψ)α−n 1−1
− 2e(ψ) h′ 2 (r(t))h 2(r(t))
f 2 (t) 2 ,
τ(u) α = 1
f 1 (t) 2 τ(ϕ)α−1 (2 ≤ α ≤ n 1 +1),
(n 1 +2≤ α ≤ n 1 + n 2 +1),
where e(ϕ) and τ(ϕ) denote the energy density function and the tension field of ϕ :(M 1 ,g 1 ) →
( ˜M 1 , ˜g 1 ), and e(ψ) and τ(ψ) denote the energy density function and the tension field of
ψ :(M 2 ,g 2 ) → ( ˜M 2 , ˜g 2 ), respectively.
Proposition 2.2.3. Let u :(I × M 1 × M 2 ,dt 2 + f 1 (t) 2 g 1 + f 2 (t) 2 g 2 ) → (Ĩ × ˜M 1 × ˜M 2 ,dr 2 +
h 1 (r) 2˜g 1 + h 2 (r) 2˜g 2 ) be a map as in (2.2.1). Then u is a harmonic map if and only if the
following two conditions hold:
(1) r = r(t) is a solution to the following ordinary differential equation
(2.2.2)
(
)
f˙
1 (t)
¨r(t)+ m 1
f 1 (t) + m f˙
2 (t)
2 ṙ(t)
f 2 (t)
−2e(ϕ) h′ 1 (r(t))h 1(r(t))
f 1 (t) 2
− 2e(ψ) h′ 2 (r(t))h 2(r(t))
f 2 (t) 2 =0.
(2) ϕ :(M 1 ,g 1 ) → ( ˜M 1 , ˜g 1 ) and ψ :(M 2 ,g 2 ) → ( ˜M 2 , ˜g 2 ) are harmonic maps with constant
energy density, that is, ϕ and ψ are eigenmaps.
Now, making use of Proposition 2.2.3, we shall construct equivariant harmonic maps
between noncompact space forms.