A map 
,
between two compact Riemannian
manifolds, is a harmonic map if it is a critical point for the energy functional
 |
The norm of the differential 
is given by the metric on 
and 
and 
is the measure on 
. Typically, the class of allowable maps lie in a fixed homotopy class of maps.
The Euler-Lagrange differential equation for the energy functional is a non-linear elliptic
partial differential equation. For example, when 
is the circle, then the Euler-Lagrange equation is the same
as the geodesic equation. Hence, 
is a closed geodesic iff 
is harmonic. The map from the circle to the equator of the
standard 2-sphere is a harmonic map, and so are the maps that take the circle and
map it around the equator 
times, for any integer 
.
Note that these all lie in the same homotopy class.
A higher-dimensional example is a meromorphic
function on a compact Riemann surface, which
is a harmonic map to the Riemann sphere.
A harmonic map may not always exist in a homotopy class, and if it does it may not be unique. When 
is negatively curved, a harmonic representative exists for
each homotopy class, and is also unique. For surfaces,
the harmonic maps have been classified, and are precisely the holomorphic maps and
the anti-holomorphic maps. Thus by Hodge's theorem
for surfaces, there are no non-trivial harmonic maps from the sphere
to the torus.
A harmonic map between Riemannian manifolds can be viewed as a generalization of a geodesic when the domain dimension is one, or of a harmonic function when the range is a Euclidean space.
