A harmonic map is a map 
,
between two compact Riemannian
manifolds, 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 unique unless it is
constant or its image is a
closed geodesic. For Riemann
surfaces, holomorphic functions and antiholomorphic
maps are harmonic, but not every harmonic map is of these
types. For example, 
on the flat torus 
is harmonic and is neither holomorphic nor antiholomorphic.
There are no nonconstant 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.
