Constructions of harmonic maps between Hadamard manifolds

4
Introduction
Let (M,g) and (M ′ ,g ′ ) be complete Riemannian manifolds, and u :(M,g) → (M ′ ,g ′ )
a C 2 -map from M to M ′ . For a relatively compact domain D ⊂ M, we define the energy
E D (u) ofu over D by
E D (u) := 1 ∫
2
D
|d x u| 2 dv g ,
where |d x u| is the Hilbert-Schmidt norm of the differential d x u : T x M → T u(x) M ′ of u at
x ∈ M, and dv g denotes the volume measure of (M,g). We call u a harmonic map if it is a
critical point of E D , considered as a functional defined on the space C 2 (M,M ′ )ofC 2 -maps
from M to M ′ , for all variations with compact support for any D. In other words, u is a
harmonic map if and only if it satisfies the Euler-Lagrange equation for the energy functional
E D for any D, that is,
τ(u)(x) :=
n∑
( ˜∇ ei du(e i ) − du(∇ ei e i ))(x) =0, x ∈ M,
i=1
where {e i } n i=1
is an orthonormal frame field of M, ∇ is the Levi-Civita connection on the
tangent bundle TM of M, and ˜∇ is the induced connection on the pull-back bundle u −1 TM ′
of the tangent bundle TM ′ of M ′ by u. Geometrically, τ(u) defines a section, called the
tension field of u, of u −1 TM ′ . It should be remarked that any C 2 harmonic map u is smooth,
since τ(u) = 0 is in fact a system of semilinear elliptic partial differential equations of second
order.
In the case where (M,g) and (M ′ ,g ′ ) are compact and without boundaries, a remarkable
existence result of harmonic maps was established in 1964 by Eells-Sampson [17].
They
proved that if the target manifold (M ′ ,g ′ ) has nonpositive sectional curvature everywhere,
then there exists an energy minimizing harmonic map in each homotopy class of smooth
maps from (M,g) to(M ′ ,g ′ ).
On the other hand, when (M,g) and (M ′ ,g ′ ) are noncompact, complete manifolds, not
much has been known for the existence of harmonic maps between them. However, in the