Harmonic map

转自：http://en.wikipedia.org/wiki/Harmonic_map

This article is about harmonic maps between Riemannian manifolds. For harmonic functions, seeharmonic function.

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

This functional E will be defined precisely below—one way of understanding it is to imagine thatM is made of rubber and N made of marble (their shapes given by their respective metrics), and that the map φ:M→N 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 byEells & Sampson (1964).

Contents

• 1Mathematical definition
• 2Examples
• 3Problems and applications
• 4Harmonic maps between metric spaces
• 5References
• 6External links