 Harmonic Map

# Harmonic map

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Description:
A (smooth) map φ:MN between Riemannian manifolds M and N is called harmonic if it is a critical point of the Dirichlet energy functional

[itex]E(varphi) = int_M |dvarphi|^2, doperatorname.[/itex]

This functional E will be defined precisely below&mdash;one way of understanding it is to imagine that M is made of rubber and N made of marble (their shapes given by their respective metrics), and that the map φ:MN 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 .

## Mathematical definition

Given Riemannian manifolds (M,g), (N,h) and φ as above, the energy density of φ at a point x in M is defined as

[itex]e(varphi) = frac12|dvarphi|^2[/itex]

where the [itex]|dvarphi|^2[/itex] is the squared norm of the differential of &phi;, with respect to the induced metric on the bundle...

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