A (smooth) map φ:M
between Riemannian manifolds M
is called harmonic
if it is a critical point
of the Dirichlet energy
- <math>E(varphi) = int_M |dvarphi|^2, doperatorname.</math>
This functional E
will be defined precisely below—one way of understanding it is to imagine that M
is made of rubber
made of marble
(their shapes given by their respective metrics
), and that the map φ:M
prescribes how one "applies" the rubber onto the marble: E
(φ) then represents the total amount of elastic potential energy
resulting from tension in the rubber. In these terms, φ is a harmonic map if the rubber, when "released" but still constrained to stay everywhere in contact with the marble, already finds itself in a position of equilibrium and therefore does not "snap" into a different shape.
Harmonic maps are the 'least expanding' maps in orthogonal directions.
Existence of harmonic maps from a complete Riemannian manifold to a complete Riemannian manifold of non-positive sectional curvature was proved by .
Given Riemannian manifolds (M,g)
and φ as above, the energy density of φ at a point x
is defined as
- <math>e(varphi) = frac12|dvarphi|^2</math>
where the <math>|dvarphi|^2</math> is the squared norm of the differential
of φ, with respect to the induced metric on the bundle... Read More