Harmonic map

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

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 in 1964 by J. Eells and J.H. Sampson. [J. Eells and J.H. Sampson, Harmonic mappings of Riemannian manifolds, "Amer. J. Math." 86 (1964), 109–160] [J. Eells and L. Lemaire, A report on harmonic maps, "Bull. London Math. Soc." 10 (1978), 1–68] [J. Eells and L. Lemaire, Another report on harmonic maps, "Bull. London Math. Soc." 20 (1988), 385–524]

Mathematical definition

Given Riemannian manifolds "(M,g)", "(N,h)" and φ as above, the energy of φ at a point "x" in "M" is defined as "e"(φ)("x")=$extstylefrac12$trace_"g"φ^*h.

Using the Einstein summation convention, in local coordinates, the right hand side of this equality reads .

If "M" is compact, define the total energy of the map φ as "E"(φ)=$extstyleint$_"M""e"(φ)dv_"g" (where dv_"g" denotes the measure on "M" induced by its metric).

Then φ is called a harmonic map if it is a critical point of the energy functional "E". This definition is extended to the case where "M" is not compact by requiring the restriction of φ to every compact domain to be harmonic.

Equivalently, the map φ is harmonic if it satisfies the Euler-Lagrange equations associated to the functional "E". These equations read trace_"g"∇dφ^*h=0, where ∇ is the connection on the vector bundle T^*"M"⊗φ^-1(T"N") induced by the Levi-Civita connections on "M" and "N".

Examples

* Idenitity and constant maps are harmonic.
* Assume that the source manifold "M" is the real line R (or the circle "S"¹), i.e. that φ is a curve (or a closed curve) on "N". Then φ is a harmonic map if and only if it is a geodesic. (In this case, the rubber-and-marble analogy described above reduces to the usual elastic band analogy for geodesics.)
* Assume that the target manifold "N" is Euclidean space R^"n" (with its standard metric). Then φ is a harmonic map if and only if it is a harmonic function in the usual sense (i.e. a solution of the Laplace equation). This follows from the Dirichlet principle.
* Every minimal immersion is a harmonic map.
* Every totally geodesic map is harmonic (in this case, ∇dφ^*h itself vanishes, not just its trace).
* Every holomorphic map between Kähler manifolds is harmonic.

Problems and applications

* If, after applying the rubber "M" onto the marble "N" via some map φ, one "releases" it, it will try to "snap" into a position of least tension. This "physical" observation leads to the following mathematical problem: given a homotopy class of maps from "M" to "N", does it contain a representative that is a harmonic map?
* Existence results on harmonic maps between manifolds has consequences for their curvature.
* Once existence is known, how can a harmonic map be constructed explicitly? (One fruitful method uses twistor theory.)
* In theoretical physics, harmonic maps are also known as sigma models.

References

External links

* [http://mathworld.wolfram.com/HarmonicMap.html MathWorld: Harmonic map]
* [http://people.bath.ac.uk/masfeb/harmonic.html Harmonic maps bibliography]