Harmonic Map

All Updates

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

This functional*E* will be defined precisely below—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:*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 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

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

- <math>E(varphi) = int_M |dvarphi|^2, doperatorname.</math>

This functional

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 .

- <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

No messages found

about this page

for companies, colleges, celebrities or anything you like.Get updates on MyPage.

Create a new Page